NYLogic

Working in the theory of algebraically closed valued fields, Hrushovski and Loeser used the space $\widehat{V}$ of generically stable types concentrating on $V$ to study the topology of Berkovich analytification $V^{an}$ of $V$. In this talk we will present an analogous construction which provides a model-theoretic counterpart $\widetilde{V}$ of the Huber's analytification of $V$. We show that, the same as for $\widehat{V}$, the space $\widetilde{V}$ is strict pro-definable. Furthermore, we will discuss canonical liftings of the deformation retraction developed by Hrushovski and Loeser. This is a joint project with Pablo Cubides Kovacsics.

Issues of set theoretic compactness frequently arise when considering questions from homological algebra about derived functors. In particular, the non-vanishing of such derived functors is often witnessed by a concrete combinatorial instance of set theoretic incompactness, so that homological questions can be translated into questions about combinatorial set theory. In this talk, we will discuss some recent results about the derived functors of the inverse limit functor. We will focus on a specific inverse system of abelian groups, $\mathbf{A}$, that arose in Mardešić and Prasolov's work on the additivity of strong homology and has since arisen independently in a number of contexts. Our main result states that, relative to the consistency of a weakly compact cardinal, it is consistent that the $n$-th derived limits $\lim^n \mathbf{A}$ vanish simultaneously for all $n \geq 1$. We will sketch a proof of this theorem and then discuss the extent to which certain generalizations of the result can hold. The arguments will be purely set theoretic, and no knowledge of homological algebra will be assumed. This is joint work with Jeffrey Bergfalk.

I will continue to speak about Jim Schmerl's recent paper on the pentagon lattice $\mathbf{N}_5$. In this talk, I will outline the main result that no model of PA has a 'mixed' elementary extension such that the resulting interstructure lattice is isomorphic to the pentagon.

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). We extend it to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces, which implies, e.g., several Hausdorff-Kuratowski-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

Our talk concerns axiomatic theories of truth predicates. They are theories obtained by adding to Peano Arithmetic (${\rm PA}$) a fresh predicate $T(x)$ with the intended reading '$x$ is (a code of) a true sentence in the language of arithmetic' together with some axioms governing newly added predicate.

The canonical example of such a theory is ${\rm CT}^-$ (Compositional Truth). Its axioms state that the truth predicate is compositional. For instance, a conjunction is true iff both conjuncts are. If we add to ${\rm CT}^-$ full induction in the extended language, we call the resulting theory ${\rm CT}$.

It is easy to check that ${\rm CT}$ is not conservative over ${\rm PA}$, i.e., it proves new arithmetical sentences. On the other hand, by a nontrivial theorem of Kotlarski, Krajewski, and Lachlan, ${\rm CT}^-$ extends ${\rm PA}$ conservatively.

In our talk, we will discuss results on the strength of theories between ${\rm CT}^-$ and ${\rm CT}$. It turns out that the natural axioms concerning purely truth theoretic properties of the newly added predicate (as opposed to axiom schemes which are consequences of induction in more general context) are typically either conservative or exactly equal to ${\rm CT}_0$, the theory of compositional truth with $\Delta_0$-induction. Thus ${\rm CT}_0$ turns out to be a surprisingly robust theory and, arguably, the minimal 'natural' non-conservative theory of truth.

The goal of Gödel’s argument that the theory (T) of Peano Arithmetic is not complete, was to show that the Gödel sentences, $G$ , and it’s negation, are not provable in T, unless T is inconsistent. In this paper we examine the first half of this argument, namely, that from a hypothetical derivation, $P_{G}$, of $G$, a derivation, $P_{f}$, can be constructed that ends in a contradiction. We make the observation that the Gödel argument depends on the metatheory concept of representability that, in turn, depends on the metatheory concept of soundness. Our analysis leads to two main observations, the first well know, and the second, a challenge to the standard undecidability argument.

1 – The existence of $P_G$ implies that T is unsound. This conclusion does not require the further construction, from $P_G$, of the derivation $P_f$.

2 - We argue that effectuation of the construction of $P_f$ is obstructed, because that effectuation requires acceptance of a contradiction in the metatheory regarding the soundness of T.

This is joint work with Catherine Hennix.

In our paper, we construct a Hardy field that embeds, via a map representing asymptotic expansion, into the field of transseries as described by Aschenbrenner, van den Dries and van der Hoeven in the recent seminal book. This Hardy field extends that of the o-minimal structure generated by all restricted analytic functions and the exponential function, and it contains Ilyashenko's almost regular germs. I will describe how this Hardy field arises quite naturally in the study of Hilbert's 16th problem and give an outline of its construction. (Joint work with Zeinab Galal and Tobias Kaiser.)

We prove that $\omega_1$ cannot be embedded into any preordering $\Sigma$-definable with parameters in the hereditarily finite superstructure over the ordered field of real numbers, HF(R). As corollaries, we obtain characterizations of $\Sigma$-presentable ordinals and Gödel constructive sets of kind $L_\alpha$. It also follows that there are no $\Sigma$-presentations for structures of $T$-, $m$-, $1$-, and $tt$-degrees over HF(R).

I will continue to speak about the construction in Jim Schmerl's paper on the pentagon lattice.

In this talk we revisit the problem of describing the 'finer' structure of geometrically trivial strongly minimal sets in $DCF_0$. In particular, I will explain how recent work joint with Guy Casale and James Freitag on automorphic functions, has lead to intriguing questions around the $\omega$-categoricity conjecture. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular $j$-function. I will explain how their counter-example fits into the larger context of arithmetic automorphic functions and has allowed us to 'propose' refinements to the original conjecture.

In this talk, we will review Morley's 1963 article 'Omitting Classes of Elements' (in modern parlance, omitting types). In this paper, Morley investigates the question of how large a structure's cardinality must be before it is forced to realize a type (for example, an ordered field with cardinality greater than the continuum must contain non-Archimedean elements). Given a fixed language, there is a cardinal $\kappa$ such that for every theory $T$ and any type $\Sigma$, if $T$ has a model of size $\kappa$ omitting $\Sigma$ then it has such a model in every cardinality. Morley's main result is that given a countable language $L$, if for every $\alpha < \omega_1$ there is a model of $T$ of size $\geq \beth_\alpha$ which omits $\Sigma$, then there is such a model in every cardinality (and if $L$ has at most $\aleph_\beta$ symbols, we replace '$\omega_1$' in this statement by '$\omega_{\gamma + 1}$' where $2^{\aleph_\beta} = \aleph_\gamma$). The proof uses the Erdős–Rado partition theorem and indiscernible sequences.

It's been known since work of Harrington in the early 1970s that computable differential fields have computable differential closures. Recently Calvert, Frolov, Harizanov, Knight, McCoy, Soskova, and Vatev showed that the countable saturated differentially closed field is computable. Their proof involves first creating an effective listing of all types and then using a result of Morley's on existence of computable saturated models. I will give a significant simplification of the enumeration result and, for completeness, sketch Morley's priority construction of a saturated model. Pillay has also given an alternative enumeration argument though ours seems more robust and generalizes to quantifier free types in ACFA.

A Turing degree is low for isomorphism if whenever it can compute an isomorphism between two countably presented structures, there is already a computable isomorphism between them and thus there is no need to use the degree as an oracle at all. I will discuss the types of degrees that are low for isomorphism and the extent to which this class of degrees has the same properties as other lowness classes.

This work is joint with Reed Solomon.

The class of random graphs famously satisfies a zero-one law: every first-order sentence in the language of graphs is such that the proportion of finite graphs on n vertices which satisfy this sentence goes either to zero or to one as n goes to infinity. The 'almost-sure' theory of the class of finite graphs matches the generic theory of its Fraisse limit - the Rado graph. Interestingly, the almost-sure theory of the class of finite triangle-free graphs does not match the theory of the generic triangle free graph. In this talk, we will discuss another class of graphs which are Fraisse limits defined by forbidden configurations, and we examine two such graphs in particular. We show that for one of the them, its generic theory does match the corresponding almost-sure theory, and that for the other, the generic theory does not match the corresponding almost-sure theory.

We present two different elementary algebraic constructions that are as complicated as possible and whose complexity vastly exceeds those typically found in the elementary algebra literature. The first is the prime radical of a noncommutative ring, while the second is the radical of a module. These constructions contrast similar constructions in more familiar contexts that we will also mention along the way. We will spend most of our time describing how to construct radicals that are as complicated as possible from a computability point of view.

When $A \subseteq \mathcal{M} \vDash PA^{\ast}$, we say that $A$ is a basis for $\mathcal{M}$ if for all $X \subseteq A$, the submodel $M_{X}$ generated by $X$ is the unique $\mathcal{N} \prec \mathcal{M}$ such that $X = \mathcal{N} \cap A$, and that such $A$ is solid if for all finite $X,Y \subseteq A$, whenever $f: \mathcal{M}_{X} \to \mathcal{M}_{Y}$ is an isomorphism, then $f \upharpoonright_{X}$ is one-to-one onto $Y$. We will discuss what role a solid basis can play in controlling the amount of indiscernibility and the automorphisms of a model. Also, using a set of types called an 'AH-set', we will construct a solid basis.

PA is the first order fragment of Peano's axiomatization of the natural numbers. The natural numbers, N, is called the standard model of PA. But by compactness theorem in first order logic, there are also models of PA different from N, which are called non-standard models of arithmetic. Like in N, every element of a non-standard model M has a binary expansion, which can be regarded as the characteristic function of a subset of N. The standard system of M is the collection of all such subsets of N. It is known that standard systems of non-standard models are always Scott sets and every Scott set of cardinality less than or equal to the first uncountable cardinal is the standard system of some non-standard model. However, the general Scott set problem, whether every Scott set is the standard system of some non-standard model, remains one of the major open problems in the model theory of arithmetic. This talk will present some history of Scott set problem, as well as two constructions of non-standard models with uncountable standard systems.

Slides

An important advancement in iterated forcing was Jensen’s proof that CH does not imply $\diamondsuit$ by iteratively specializing Aronszajn trees with countable levels without adding reals thus producing a model of CH plus 'all Aronszajn trees are special'. This proof was improved by Shelah who developed a general method around the notion of dee-complete forcing. This class (under certain circumstances) can be iterated with countable support and does not add reals. However, neither Jensen's nor Shelah's posets will specialize trees of uncountable width and it remains unclear when one can iteratively specialize wider trees. Indeed a very intriguing example, due to Todorčević, shows that there is always a wide Aronszajn tree which cannot be specialized without adding reals. By contrast the ccc forcing for specializing Aronszajn trees makes no distinction between trees of different widths (but may add many reals). In this talk we will provide a general criteria a wide trees Aronszajn tree can have that implies the existence of a dee-complete poset specializing it. Time permitting we will discuss applications of this forcing to forcing axioms compatible with CH and some open questions related to set theory of the reals.

Slides

Recall that given a complete theory $T$ and a type $p(x)$ the Hanf number for $p(x)$ is the least cardinal $\kappa$ so that any model of $T$ of size $\kappa$ realizes $p(x)$ (if such a $\kappa$ exists and $\infty$ otherwise). The Hanf number for $T$, denoted $H(T)$, is the supremum of the successors of the Hanf numbers for all possible types $p(x)$ whose Hanf numbers are $\lt\infty$. We have seen so far in the seminar that for any complete, consistent $T$ in a countable language $H(T) \leq \beth_{\omega_1}$ (a result due to Morley). In this talk I will present the following theorems: (1) The Hanf number for true arithmetic is $\beth_{\omega}$ (Abrahamson-Harrington-Knight) but (2) the Hanf number for False Arithmetic is $\beth_{\omega_1}$ (Abrahamson-Harrington)

Starting from a model of ZFC with two supercompact cardinals, I will discuss how to force and construct a model in which the first two strongly compact cardinals $\kappa_1$ and $\kappa_2$ are also the first two measurable cardinals. In this model, $\kappa_1$'s strong compactness is indestructible under arbitrary $\kappa_1$-directed closed forcing, and $\kappa_2$'s strong compactness is indestructible under ${\rm Add}(\kappa_2, \lambda)$ for any ordinal $\lambda$. This answers a generalized version of a question of Sargsyan.

Slides

The subject of this two-part talk is a 1975 Barwise-Schlipf landmark paper, whose main theorem asserts that a nonstandard model M of PA is recursively saturated iff M has an expansion to a model of the subsystem $\Delta^{1}_{1}-CA_0$ of second order arithmetic. The impression one gets from reading the Barwise-Schlipf paper is that the left-to-right direction of the theorem is deep since it relies on sophisticated techniques from admissible set theory, and that the other direction is fairly routine.

As it turns out, the exact opposite is the case: the left-to-right direction of the Barwise-Schlipf theorem lends itself to a proof from first principles (as observed independently by Jonathan Stavi and Sol Feferman not long after the appearance of the Barwise-Schmerl paper); and moreover, as recently shown in my joint work with Jim Schmerl, there is a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, an error that can be circumvented by a rather nontrivial argument. As I will explain, certain results from the joint work of Matt Kaufmann and Jim Schmerl in the mid-1980s on 'lofty' models of arithmetic come in handy for the analysis of the error, and for circumventing it.

In part I, after going over some history, and preliminaries, I will discuss (1) the gap in the Barwise-Schlipf paper, and (2) the aforementioned Feferman-Stavi proof. In part II, I will focus on how the gap can be circumvented with a proof strategy very different from that Barwise and Schlipf.​

Slides

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In May 2019 Trevor Wilson proved that the Weak Vopenka Principle (WVP), which asserts that the opposite of the category of Ordinals cannot be fully embedded into the category of Graphs, is equivalent to the class of ordinals being Woodin. In particular this implies that WVP is not equivalent to Vopenka’s Principle, thus solving an important long-standing open question in category theory. I will report on a joint ensuing work with Trevor Wilson in which we analyse the strength of WVP for definable classes of full subcategories of Graphs, obtaining exact level-by-level characterisations in terms of a natural hierarchy of strong cardinals.

Slides

The subject of this two-part talk is a 1975 Barwise-Schlipf landmark paper, whose main theorem asserts that a nonstandard model M of PA is recursively saturated iff M has an expansion to a model of the subsystem $\Delta^{1}_{1}-CA_0$ of second order arithmetic. The impression one gets from reading the Barwise-Schlipf paper is that the left-to-right direction of the theorem is deep since it relies on sophisticated techniques from admissible set theory, and that the other direction is fairly routine.

As it turns out, the exact opposite is the case: the left-to-right direction of the Barwise-Schlipf theorem lends itself to a proof from first principles (as observed independently by Jonathan Stavi and Sol Feferman not long after the appearance of the Barwise-Schmerl paper); and moreover, as recently shown in my joint work with Jim Schmerl, there is a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, an error that can be circumvented by a rather nontrivial argument. As I will explain, certain results from the joint work of Matt Kaufmann and Jim Schmerl in the mid-1980s on 'lofty' models of arithmetic come in handy for the analysis of the error, and for circumventing it.

In part I, after going over some history, and preliminaries, I will discuss (1) the gap in the Barwise-Schlipf paper, and (2) the aforementioned Feferman-Stavi proof. In part II, I will focus on how the gap can be circumvented with a proof strategy very different from that Barwise and Schlipf.​

Slides

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Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel's analysis of the constructible universe $L$. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.

We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals $\kappa$. Finally, we will comment on obstacles, questions, and conjectures for lifting these results higher up in the large cardinal hierarchy.

Slides

There is a well-known close logical connection between PA and finite set theory. Is there a set theory that corresponds in an analogous way to bounded arithmetic $I\Delta_0$? I propose a candidate for such a theory, called $I\Delta_0S$, and consider the questions: what set-theoretic axioms can it prove? And given a model M of $I\Delta_0$ is there a model of $I\Delta_0S$ whose ordinals are isomorphic to M? The answer is yes if M is a model of Exp; to obtain the answer we use a new way of coding sets by numbers.

Slides

A model $M$ of set theory is said to be 'condensable' if there is an 'ordinal' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is both isomorphic to $M$, and an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M$. Clearly if $M$ is condensable, then $M$ is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus 'there is an inaccessible cardinal', there is a condensable model $M$ of ZFC which has the property that every definable element of $M$ is in the well-founded part of $M$ (in particular, $M$ is $\omega$-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model $M$ of ZF of any cardinality:
(a) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension.
(b) There is a cofinal subset of 'ordinals' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M.$
Moreover, if $M$ is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:
(c) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension + $\Sigma^1_1$-Choice.

Slides

A model $M$ of set theory is said to be 'condensable' if there is an 'ordinal' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is both isomorphic to $M$, and an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M$. Clearly if $M$ is condensable, then $M$ is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus 'there is an inaccessible cardinal', there is a condensable model $M$ of ZFC which has the property that every definable element of $M$ is in the well-founded part of $M$ (in particular, $M$ is $\omega$-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model $M$ of ZF of any cardinality:
(a) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension.
(b) There is a cofinal subset of 'ordinals' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M.$
Moreover, if $M$ is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:
(c) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension + $\Sigma^1_1$-Choice.

Slides

Truth theories investigate the notion of truth with axiomatic methods. To a fixed base theory (typically Peano Arithmetic ${\rm PA}$) we add a unary predicate $T(x)$ with the intended interpretation '$x$ is a (code of a) true sentence.' Then we analyse how adding various possible sets of axioms for that predicate affects its behaviour.

One of the aspects we are trying to understand is which truth-theoretic principles make the added truth predicate 'strong' in that the resulting theory is not conservative over the base theory. Ali Enayat proposed to call this 'demarcating line' between conservative and non-conservative truth theories 'the Tarski boundary.'

Research on Tarski boundary revealed that natural truth theoretic principles extending compositional axioms tend to be either conservative over ${\rm PA}$ or exactly equivalent to the principle of global reflection over ${\rm PA}$. It says that sentences provable in ${\rm PA}$ are true in the sense of the predicate $T$. This in turn is equivalent to $\Delta_0$ induction for the compositional truth predicate which turns out to be a surprisingly robust theory.

In our talk, we will try to sketch proofs representative of research on Tarski boundary. We will present the proof by Enayat and Visser showing that the compositional truth predicate is conservative over ${\rm PA}$. We will also try to discuss how this proof forms a robust basis for further conservativeness results.

On the non-conservative side of Tarski boundary, the picture seems less organised, since more arguments are based on ad hoc constructions. However, we will try to show some themes which occur rather repeatedly in these proofs: iterated truth predicates and the interplay between properties of good truth-theoretic behaviour and induction. To this end, we will present the argument that disjunctive correctness together with the internal induction principle for a compositional truth predicate yields the same consequences as $\Delta_0$-induction for the compositional truth predicate (as proved by Ali Enayat) and that it shares arithmetical consequences with global reflection. The presented results are currently known to be suboptimal.

This talk is intended as a continuation of 'Tarski boundary' presentation. However, we will try to avoid excessive assumptions on familiarity with the previous part.

An inner model is a ground if V is a set forcing extension of it. The intersection of the grounds is the mantle, an inner model of ZFC which enjoys many nice properties. Fuchs, Hamkins, and Reitz showed that the mantle is highly malleable. Namely, they showed that every model of set theory is the mantle of a bigger, better universe of sets. This then raises the possibility of iterating the definition of the mantle—the mantle, the mantle of the mantle, and so on, taking intersections at limit stages—to obtain even deeper inner models. Let's call the inner models in this sequence the inner mantles.

In this talk I will present some results about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz. Specifically, I will present the following results, analogues of classic results about the sequence of iterated HODs.

1. (Joint with Reitz) Consider a model of set theory and consider an ordinal eta in that model. Then this model has a class forcing extension whose eta-th inner mantle is the model we started out with, where the sequence of inner mantles does not stabilize before eta.

2. It is consistent that the omega-th inner mantle is an inner model of ZF + ¬AC.

3. It is consistent that the omega-th inner mantle is not a definable class, and indeed fails to satisfy Collection.

In the 1973 paper 'Countable models of set theory', H. Friedman's investigation of embeddings between countable models of subsystems of ZF yields the following two striking results:

1. Every countable nonstandard model of PA is isomorphic to a proper initial segment of itself.
2. Every countable nonstandard model of a sufficiently strong subsystem of ZF is isomorphic to a proper initial segment that is a union of ranks of the original model.

Note that, in contrast to PA, in the context of set theory there are three alternative notions of 'initial segment': transitive subclass, transitive subclass that is closed under subsets and rank-initial segment. Paul Gorbow, in his Ph.D. thesis, systematically studies versions of H. Friedman's self-embedding that yield isomorphisms between a countable nonstandard model of set theory and a rank-initial segment of itself. In these two talks I will discuss recent joint work with Ali Enayat that investigates models of set theory that are isomorphic to a transitive subclass of itself. We call the maps witnessing these isomorphisms 'initial self-embeddings'. I will outline a proof of a refinement of H. Friedman's Theorem that guarantees the existence of initial self-embeddings for certain subsystems of ZF without the powerset axiom. I will then discuss several examples including a nonstandard model of ZFC minus the powerset axiom that admits no initial self-embedding, and models that separate the three different notions of self-embedding for models of set theory. Finally, I will discuss two interesting applications of our version of H. Freidman's Theorem. The first of these is a refinement of a result due to Quinsey that guarantees the existence of partially elementary proper transitive subclasses of non-standard models of ZF minus the powerset axiom. The second result shows that every countable model of ZF with a nonstandard natural number is isomorphic to a transitive subclass of the hereditarily countable sets of its own constructible universe.​

Recursive saturation, introduced by J. Barwise and J. Schlipf is a robust notion which has proved to be important for the study of nonstandard models (in particular, it is ubiquitous in the model theory of axiomatic theories of truth, e.g. in the topic of satisfaction classes, where one can show that if $M \models ZFC$ is a countable $\omega$-nonstandard model, then $M$ admits a satisfaction class iff $M$ is recursively saturated). V. Gitman and J. Hamkins showed in A Natural Model of the Multiverse Axioms that the collection of countable, recursively saturated models of set theory satisfy the so-called Hamkins's Multiverse Axioms. The property that forces all the models in the Multiverse to be recursively saturated is the so-called Well-Foundedness Mirage axiom which asserts that every universe is $\omega$-nonstandard from the perspective of some larger universe, or to be more precise, that: if a model $M$ is in the multiverse then there is a model $N$ in the multiverse such that $M$ is a set in $N$ and $N \models 'M \text{ is }\omega-\text{nonstandard.'}$. Inspection of the proof led to a question if the recursive saturation could be avoided in the Multiverse by weakening the Well-Foundedness Mirage axiom. Our main results answer this in the positive. We give two different versions of the Well-Foundedness Mirage axiom - what we call Weak Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse such that $M \in N$ and $N \models 'M \text{ is nonstandard.'}$.) and Covering Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse with $K \in N$ such that $K$ is an end-extension of $M$ and $N \models 'K \text{ is } \omega-\text{nonstandard.'}$). I will present constructions of two different Multiverses satisfying these two weakened axioms. This is joint work with V. Gitman. T. Meadows and K. Williams.

Recursive saturation, introduced by J. Barwise and J. Schlipf is a robust notion which has proved to be important for the study of nonstandard models (in particular, it is ubiquitous in the model theory of axiomatic theories of truth, e.g. in the topic of satisfaction classes, where one can show that if $M \models ZFC$ is a countable $\omega$-nonstandard model, then $M$ admits a satisfaction class iff $M$ is recursively saturated). V. Gitman and J. Hamkins showed in A Natural Model of the Multiverse Axioms that the collection of countable, recursively saturated models of set theory satisfy the so-called Hamkins's Multiverse Axioms. The property that forces all the models in the Multiverse to be recursively saturated is the so-called Well-Foundedness Mirage axiom which asserts that every universe is $\omega$-nonstandard from the perspective of some larger universe, or to be more precise, that: if a model $M$ is in the multiverse then there is a model $N$ in the multiverse such that $M$ is a set in $N$ and $N \models 'M \text{ is }\omega-\text{nonstandard.'}$. Inspection of the proof led to a question if the recursive saturation could be avoided in the Multiverse by weakening the Well-Foundedness Mirage axiom. Our main results answer this in the positive. We give two different versions of the Well-Foundedness Mirage axiom - what we call Weak Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse such that $M \in N$ and $N \models 'M \text{ is nonstandard.'}$.) and Covering Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse with $K \in N$ such that $K$ is an end-extension of $M$ and $N \models 'K \text{ is } \omega-\text{nonstandard.'}$). I will present constructions of two different Multiverses satisfying these two weakened axioms. This is joint work with V. Gitman. T. Meadows and K. Williams.

In the context of arithmetic, a reflection principle for a theory Th is a formal way of expressing that all theorems of Th are true. In the presence of a truth predicate for the language of Th this principle can be expressed as a single sentence (called the Global Reflection principle over Th) but most often is met in the form of a scheme consisting of all sentences of the form

$\forall x \bigl( \text{Prov}_{\text{Th}}(\phi(\dot{x}))\rightarrow \phi(x)\bigr).$ Obviously such a scheme is not provable in a consistent theory Th. Nevertheless, such soundness assertions are said to provide a natural and justified way of extending ones initial theory.

This perspective is nowadays very fruitfully exploited in the context of formal theories of truth. One of the most basic observations is that strong axioms for the notions of truth follow from formally weak types of axiomatizations modulo reflection principles. In such a way compositional axioms are consequences of the uniform disquotational scheme for for the truth predicate, which is

$\forall x\, \, T(\phi(\dot{x}))\equiv \phi(x).$

The above observation is also used in the recent approach to ordinal analysis of theories of predicative strength by Lev Beklemishev and Fedor Pakhomov. The assignment of ordinal notations to theories proceeds via partial reflection principles (for formulae of a fixed $\Sigma_n$ complexity) over (iterated) disquotational scheme. It becomes important to relate theories of this form to fragments of standard theories of truth, in particular the ones based on induction for restricted classes of formulae such as $\text{CT}_0$ (the theory of compositional truth with $\Delta_0$-induction for the extended language. The theory was discussed at length in Bartek Wcisło's talk). Beklemishev and Pakhomov leave the following open question:

Is $\Sigma_1$-reflection principle over the uniform disquotational scheme provable in $\text{CT}_0$?

The main goal of our talk is to present the proof of the affirmative answer to this question. The result significantly improves the known fact on the provability of Global Reflection over PA in $\text{CT}_0$. During the talk, we explain the theoretical context described above including the information on how the result fits into Beklemishev-Pakhomov project. In the meantime we give a different proof of their characterisation of $\Delta_0$-reflection over the disquotational scheme.

Despite the proof-theoretical flavour of these results, our proofs rests on essentially model-theoretical techniques. The important ingredient is the Arithmetized Completeness Theorem.

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The notion of a guessing model introduced by Viale and Weiss. The principle ${\rm GM}(\omega_2,\omega_1)$ asserts that there are stationary many guessing models of size $\aleph_1$ in $H_\theta$, for all large enough regular $\theta$. It follows from ${\rm PFA}$ and implies many of its structural consequences, however it does not settle the value of the continuum. In search of higher of forcing axioms it is therefore natural to look for extensions and higher versions of this principle. We formulate and prove the consistency of one such statement that we call ${\rm SGM}^+(\omega_3,\omega_1)$.
It has a number of important structural consequences:

- the tree property at $\aleph_2$ and $\aleph_3$
- the failure of various weak square principles
- the Singular Cardinal Hypothesis
- Mitchell’s Principle: the approachability ideal agrees with the non stationary ideal on the set of $\text{cof}(\omega_1)$ ordinals in $\omega_2$
- Souslin’s Hypothesis
- The negation of the weak Kurepa Hypothesis
- Abraham’s Principles: every forcing which adds a subset of $\omega_2$ either adds a real or collapses some cardinals, etc.

The results are joint with my PhD students Rahman Mohammadpour.

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Truth theories investigate the notion of truth using axiomatic methods. To a fixed base theory (typically Peano Arithmetic ${\rm PA}$) we add a unary predicate $T(x)$ with the intended interpretation '$x$ is a (code of a) true sentence.' Then we analyse how adding various possible sets of axioms for that predicate affects its behaviour.

One of the aspects which we are trying to understand is which truth-theoretic principles make the added truth predicate 'strong' in that the resulting theory is not conservative over the base theory. Ali Enayat proposed to call this demarcating line between conservative and non-conservative truth theories 'the Tarski boundary.'

Research on Tarski boundary revealed that natural truth theoretic principles extending compositional axioms tend to be either conservative over ${\rm PA}$ or exactly equivalent to the principle of global reflection over $
A$. It says that sentences provable in ${\rm PA}$ are true in the sense of the predicate $T$. This in turn is equivalent to $\Delta_0$ induction for the compositional truth predicate which turns out to be a surprisingly robust theory.

The equivalences between nonconservative truth theories are typically proved by relatively direct ad hoc arguments. However, certain patterns seem common to these proofs. The first one is construction of various arithmetical partial truth predicates which provably in a given theory have better properties than the original truth predicate. The second one is deriving induction for these truth predicates from internal induction, a principle which says that for any arithmetical formula, the set of those elements for which that formula is satisfied under the truth predicate satisfies the usual induction axioms.

As an example of this phenomenon, we will present two proofs. First, we will show that global reflection principle is equivalent to local induction. Global reflection expresses that any sentence provable in ${\rm PA}$ is true. Local induction says that any predicate obtained by restricting truth predicate to sentences of a fixed syntactic complexity $c$ satisfies full induction. This is an observation due to Mateusz Łełyk and the author of this presentation.

The second example is a result by Ali Enayat who showed that ${\rm CT}_0$, a theory compositional truth with $\Delta_0$ induction, is arithmetically equivalent to the theory of compositional truth together with internal induction and disjunctive correctness.

This talk is intended as a continuation of 'Tarski boundary II' presentation at the same seminar. However, we will try to avoid excessive assumptions on familiarity with the previous part.

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Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$ in his 1930 quasi-categoricity result asserting that the models of $\text{ZFC}_2$ are precisely those isomorphic to a rank-initial segment $V_\kappa$ of the cumulative set-theoretic universe $V$ cut off at an inaccessible cardinal $\kappa.$ I shall discuss the extent to which Zermelo's quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the $V_\kappa$ universes are categorically characterized by their sentences or theories. For example, if $\kappa$ is the smallest inaccessible cardinal, then up to isomorphism $V_\kappa$ is the unique model of $\text{ZFC}_2$ plus the sentence 'there are no inaccessible cardinals.' This cardinal $\kappa$ is therefore an instance of what we call a first-order sententially categorical cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of $\text{ZFC}_2$ by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford).

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In the 1973 paper 'Countable models of set theory', H. Friedman's investigation of embeddings between countable models of subsystems of ZF yields the following two striking results:

1. Every countable nonstandard model of PA is isomorphic to a proper initial segment of itself.
2. Every countable nonstandard model of a sufficiently strong subsystem of ZF is isomorphic to a proper initial segment that is a union of ranks of the original model.

Note that, in contrast to PA, in the context of set theory there are three alternative notions of 'initial segment': transitive subclass, transitive subclass that is closed under subsets and rank-initial segment. Paul Gorbow, in his Ph.D. thesis, systematically studies versions of H. Friedman's self-embedding that yield isomorphisms between a countable nonstandard model of set theory and a rank-initial segment of itself. In these two talks I will discuss recent joint work with Ali Enayat that investigates models of set theory that are isomorphic to a transitive subclass of itself. We call the maps witnessing these isomorphisms 'initial self-embeddings'. I will outline a proof of a refinement of H. Friedman's Theorem that guarantees the existence of initial self-embeddings for certain subsystems of ZF without the powerset axiom. I will then discuss several examples including a nonstandard model of ZFC minus the powerset axiom that admits no initial self-embedding, and models that separate the three different notions of self-embedding for models of set theory. Finally, I will discuss two interesting applications of our version of H. Freidman's Theorem. The first of these is a refinement of a result due to Quinsey that guarantees the existence of partially elementary proper transitive subclasses of non-standard models of ZF minus the powerset axiom. The second result shows that every countable model of ZF with a nonstandard natural number is isomorphic to a transitive subclass of the hereditarily countable sets of its own constructible universe.​

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We will discuss some recent ZFC results concerning the generalized Baire spaces, and more specifically the generalized bounding number, relatives of the generalized almost disjointness number, as well as generalized reaping and domination.

A Kaufmann model is an $\omega_1$-like, recursively saturated, rather classless model of PA. Such models were constructed by Kaufmann under the $\diamondsuit$ assumption and then shown to exist in ZFC by Shelah using an absoluteness argument involving the logic $L_{\omega_1, \omega}(Q)$ where $Q$ is the quantifier 'there exists uncountably many…'. It remains an intriguing, if vague, open problem whether one can construct a Kaufmann model in ZFC 'by hand' i.e. without appealing to some form of absoluteness or other very non-constructive methods. In this talk I consider the related problem of axiomatizing Kaufmann models in $L_{\omega_1, \omega}(Q)$ and show that this is independent of ZFC. Along the way we'll see that it is also independent of ZFC whether there is an $\omega_1$-preserving forcing notion adding a truth predicate to a Kaufmann model.

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Many prominent large cardinal notions up to measurability can be characterized by the existence of certain ultrafilters for small models of set theory. Most prominently, this includes weakly compact, ineffable, Ramsey and completely ineffable cardinals, but there are many more, and our characterization schemes also give rise to many new natural large cardinal concepts. Moreover, these characterizations allow for the uniform definition of ideals associated to these large cardinals, which agree with the ideals from the set-theoretic literature (for example, the weakly compact, the ineffable, the Ramsey or the completely ineffable ideal) whenever such had been previously established. For many large cardinal notions, we can show that their ordering with respect to direct implication, but also with respect to consistency strength corresponds in a very canonical way to certain relations between their corresponding large cardinal ideals. This is all material from a fairly extensive joint paper with Philipp Luecke, and I will try to provide an overview as well as present some particular results from this paper.

In the context of arithmetic, a reflection principle for a theory Th is a formal way of expressing that all theorems of Th are true. In the presence of a truth predicate for the language of Th this principle can be expressed as a single sentence (called the Global Reflection principle over Th) but most often is met in the form of a scheme consisting of all sentences of the form

$\forall x \bigl( \text{Prov}_{\text{Th}}(\phi(\dot{x}))\rightarrow \phi(x)\bigr).$ Obviously such a scheme is not provable in a consistent theory Th. Nevertheless, such soundness assertions are said to provide a natural and justified way of extending ones initial theory.

This perspective is nowadays very fruitfully exploited in the context of formal theories of truth. One of the most basic observations is that strong axioms for the notions of truth follow from formally weak types of axiomatizations modulo reflection principles. In such a way compositional axioms are consequences of the uniform disquotational scheme for for the truth predicate, which is

$\forall x\, \, T(\phi(\dot{x}))\equiv \phi(x).$

The above observation is also used in the recent approach to ordinal analysis of theories of predicative strength by Lev Beklemishev and Fedor Pakhomov. The assignment of ordinal notations to theories proceeds via partial reflection principles (for formulae of a fixed $\Sigma_n$ complexity) over (iterated) disquotational scheme. It becomes important to relate theories of this form to fragments of standard theories of truth, in particular the ones based on induction for restricted classes of formulae such as $\text{CT}_0$ (the theory of compositional truth with $\Delta_0$-induction for the extended language. The theory was discussed at length in Bartek Wcisło's talk). Beklemishev and Pakhomov leave the following open question:

Is $\Sigma_1$-reflection principle over the uniform disquotational scheme provable in $\text{CT}_0$?

The main goal of our talk is to present the proof of the affirmative answer to this question. The result significantly improves the known fact on the provability of Global Reflection over PA in $\text{CT}_0$. During the talk, we explain the theoretical context described above including the information on how the result fits into Beklemishev-Pakhomov project. In the meantime we give a different proof of their characterisation of $\Delta_0$-reflection over the disquotational scheme.

Despite the proof-theoretical flavour of these results, our proofs rests on essentially model-theoretical techniques. The important ingredient is the Arithmetized Completeness Theorem.

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The maximality principle (${\rm MP}$) is the assertion that any sentence which can be forced in such a way that after any further forcing the sentence remains true, must already be true. In modal terms, ${\rm MP}$ states that forceably necessary sentences are true. The resurrection axiom (${\rm RA}$) asserts that the ground model is as existentially closed in its forcing extensions as possible. In particular, ${\rm RA}$ relative to $H_{\mathfrak c}$ states that for every forcing $\mathbb Q$ there is a further forcing $\mathbb R$ such that $H_{\mathfrak c}^V \prec H_{\mathfrak c}^{V[G][H]}$, for $G*H \subseteq \mathbb Q *\dot{\mathbb R}$ generic.

It is reasonable to ask whether ${\rm MP}$ and ${\rm RA}$ can consistently both hold. I showed that indeed they can, and that ${\rm RA}+{\rm MP}$ is equiconsistent with a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals used to force the principles separately. In this talk I give a sketch of the equiconsistency result.

Cofinal extensions generally preserve many more properties of a model of arithmetic than their sisters, end extensions. Exactly how much must or can they preserve? The answer is intimately related to how much arithmetic the model can do. I will survey what is known and what is not known about this question, and report on some recent work on this line.

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An old result of Isbell characterises measurable cardinals in terms of certain canonical limits in the category of sets. After introducing this characterisation, I will talk about recent work with Adamek, Campion, Positselski and Rosicky teasing out the importance of the canonicity for this and related results. The language will be category-theoretic but the proofs will be quite hands-on combinatorial constructions with sets.

Recall that if $M \subseteq N$ are models of set theory then $N$ end-extends $M$ if $N$ does not have new elements for sets in $M$. In this talk I will discuss a $\Sigma_1$-definable finite sequence which is universal for end extensions in the following sense. Consider a computably axiomatizable extension $\overline{\mathsf{ZF}}$ of $\mathsf{ZF}$. There is a $\Sigma_1$-definable finite sequence $$a_0, a_1, \ldots, a_n$$ with the following properties.

* $\mathsf{ZF}$ proves that the sequence is finite.
* In any transitive model of $\overline{\mathsf{ZF}}$ the sequence is empty.
* If $M$ is a countable model of $\overline{\mathsf{ZF}}$ in which the sequence is $s$ and $t \in M$ is a finite sequence extending $s$ then there is an end-extension $N \models \overline{\mathsf{ZF}}$ of $M$ in which the sequence is exactly $t$.
* Indeed, for the previous statements it suffices that $M \models \mathsf{ZF}$ and end-extends a submodel $W \models \overline{\mathsf{ZF}}$ of height at least $(\omega_1^{\mathrm{L}})^M$.

This universal finite sequence can be used to determine the modal validities of end-extensional set-theoretic potentialism, namely to be exactly the modal theory $\mathsf{S4}$. The sequence can also be used to show that every countable model of set theory extends to a model satisfying the end-extensional maximality principle, asserting that any possibly necessary sentence is already true.

This talk is about joint work with Joel David Hamkins. The $\Sigma_1$ universal finite sequence is a sister to the $\Sigma_2$ universal finite sequence for rank-extensions of Hamkins and Woodin, and both are cousins of Woodin's universal algorithm for arithmetic.

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This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from $\omega^\omega$ to $\omega^\omega$. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen's subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height $\omega_1$ with no branch can be embedded into an $\omega_1$ tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails.

To what extent can formulas from infinitary logics be used in set-theoretic reflection arguments? If $\kappa$ is a measurable cardinal, any $L_{\kappa,\kappa}$ sentence which is true in $(\kappa,\in)$, must be true about some strictly smaller cardinal. Whereas, there are $L_{\kappa^+,\kappa^+}$ sentences of length $\kappa$ which are true in $(\kappa,\in)$ and which are not true about any smaller cardinal. However, if $\kappa$ is a measurable cardinal and some $L_{\kappa^+,\kappa^+}$ sentence $\varphi$ is true in $(\kappa,\in)$, then there must be some strictly smaller cardinal $\alpha<\kappa$ such that a canonically restricted version of $\varphi$ holds about $\alpha$. Building on work of Bagaria and Sharpe-Welch, we use canonical restriction of formulas to define notions of $\Pi^1_\xi$-indescribability of a cardinal $\kappa$ for all $\xi<\kappa^+$. In this context we show that such higher indescribability hypotheses are strictly weaker than measurability, we prove the existence of universal $\Pi^1_\xi$-formulas, study the associated normal ideals and notions of $\xi$-clubs and prove a hierarchy result. Time permitting we will discuss some applications.

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In a paper with Kossak in 2018, we studied the notion of neutrality: a subset X of a model M of PA is called neutral if the definable closure relation in (M, X) coincides with that in M. This notion was suggested by Dolich. motivated by work by Chatzidakis-Pillay on generic expansions of theories. In this talk, we will look at a more direct translation of the Chatzidakis-Pillay notion of genericity, which we call 'CP-genericity', and discuss its relation to neutrality. The main result shows that for recursively saturated models, CP-generics are always neutral; previously we had known that not all neutral sets are CP-generic.

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I have been working over the past few years on the project of trying to improve our understanding of the forcing axiom for subcomplete forcing. The most compelling feature of this axiom is its consistency with the continuum hypothesis. On the other hand, it captures many of the major consequences of Martin's Maximum. It is a compelling feature of Martin's Maximum that many of its consequences filter through Todorcevic's Strong Reflection Principle SRP. SRP has some consequences that the subcomplete forcing axiom does not have, like the failure of CH and the saturation of the nonstationary ideal. It has been unclear until recently whether there is a version of SRP that relates to the subcomplete forcing axiom as the full SRP relates to Martin's Maximum, but it turned out that there is: I will detail how to associate in a canonical way to an arbitrary forcing class its corresponding fragment of SRP in such a way that (1) the forcing axiom for the forcing class implies its fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. I will describe how this association works, describe some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity, and say a little more about the extent of (3).

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The usual base theory used in reverse mathematics, $\mathrm{RCA}_0$, is the fragment of second-order arithmetic axiomatized by $\Delta^0_1$ comprehension and $\Sigma^0_1$ induction. The weaker base theory $\mathrm{RCA}^*_0$ is obtained by replacing $\Sigma^0_1$ induction with $\Delta^0_1$ induction (and adding the well-known axiom $\exp$ in order to ensure totality of the exponential function). In first-order terms, $\mathrm{RCA}_0$ is conservative over $\mathrm{I}\Sigma_1$ and $\mathrm{RCA}^*_0$ is conservative over $\mathrm{B}\Sigma_1 + \exp$.

Some of the most interesting open problems in reverse mathematics concern the first-order strength of statements from Ramsey Theory, in particular Ramsey's Theorem for pairs and two colours. In this talk, I will discuss joint work with Kasia Kowalik, Tin Lok Wong, and Keita Yokoyama concerning the strength of Ramsey's Theorem over $\mathrm{RCA}^*_0$.

Given standard natural numbers $n,k \ge 2$, let $\mathrm{RT}^n_k$ stand for Ramsey's Theorem for $k$-colourings of $n$-tuples. We first show that assuming the failure of $\Sigma^0_1$ induction, $\mathrm{RT}^n_k$ is equivalent to its own relativization to an arbitrary $\Sigma^0_1$-definable cut. Using this, we give a complete axiomatization of the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^n_k$ for $n \ge 3$ (this turns out to be a rather peculiar fragment of PA) and obtain some nontrivial information about the first-order consequences of $\mathrm{RT}^2_k$. Time permitting, we will also discuss the question whether our results have any relevance for the well-known open problem of characterizing the first-order consequences of $\mathrm{RT}^2_2$ over the traditional base theory $\mathrm{RCA}_0$.

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Let $\Phi$ be the statement that for any function $f: \omega_1 \times \omega_1 \to \omega$, there are functions $g_1, g_2 : \omega_1 \to \omega$ such that for all $(x,y) \in \omega_1 \times \omega_1$, we have $f(x,y) \le \text{max }\{g_1(x), g_2(y)\}$. We will show that $\Phi$ follows from ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$. On the other hand using core models, we will show that $\Phi + `\text{the club filter on }\omega_1\text{ is normal'}$ implies there are inner models with many measurable cardinals. We conjecture that $\Phi$ and ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$ have the same consistency strength. The research is joint with Francois Dorais at the University of Vermont.

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In bounded arithmetic, we study weak fragments of arithmetic that often correspond in a certain sense to computational complexity classes (e.g., polynomial time). Questions about provability in such theories can be thought of as a form of feasible reasoning: considering a natural object of interest from a complexity class $C$, can we prove its fundamental properties using only concepts from $C$?

Our objects of interest in this talk will be the elementary integer arithmetic operations $+,-,\times,/$, whose complexity class is (uniform) $\mathrm{TC}^0$, a small subclass of $\mathrm{P}$. The corresponding arithmetical theory is $\mathit{VTC}^0$. Since we do not know yet if the theory can prove the totality of division and iterated multiplication $\prod_{i<n}X_i$ which are in $\mathrm{TC}^0$ by an intricate result of Hesse, Allender, and Barrington, we will also consider an extension of the theory $\mathit{VTC}^0+\mathit{IMUL}$.

Our main question is what can $\mathit{VTC}^0\pm\mathit{IMUL}$ prove about the elementary arithmetic operations. The answer is that more than one might expect: $\mathit{VTC}^0+\mathit{IMUL}$ proves induction for quantifier-free formulas in the basic language of arithmetic ($\mathit{IOpen}$), and even induction and minimization for $\Sigma^b_0$ (sharply bounded) formulas in Buss’s language. This result is connected to the existence of $\mathrm{TC}^0$ constant-degree root-finding algorithms; the proof relies on a formalization of a form of the Lagrange inversion formula in $\mathit{VTC}^0+\mathit{IMUL}$, and on model-theoretic abstract nonsense involving valued fields.

The remaining problem is if $\mathit{VTC}^0$ proves $\mathit{IMUL}$. We will discuss issues with formalization of the Hesse–Allender–Barrington construction in $\mathit{VTC}^0$, and some partial results (this is a work in progress).

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Any ultrapower $M$ of the universe by a normal measure on a cardinal $\kappa$ is quite far from $V$ in the sense that it computes $V_{\kappa+2}$ incorrectly. If GCH holds, this amounts to saying that $M$ is missing a subset of $\kappa^+$. Steel asked whether, even in the absence of GCH, normal ultrapowers at $\kappa$ must miss a subset of $\kappa^+$. In the early 90s Cummings gave a negative answer, building a model with a normal measure on $\kappa$ whose ultrapower captures the entire powerset of $\kappa^+$. I will present some joint work with Radek Honzík in which we improved Cummings’ result to get this capturing property to hold at the least measurable cardinal.

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In the theory of nonstandard methods (traditionally known as nonstandard analysis), each mathematical object (a set) $x$ has a uniquely determined so called nonstandard extension ${}^*x$. In general, ${}^*x \supsetneq \{{}^*y; y\in x\}$ - that is, besides the original 'standard' elements ${}^*y$ for $y\in x$, the set ${}^*x$ contains some new 'nonstandard' elements.

For instance, some of the nonstandard elements of ${}^*\mathbb{R}$ can be interpreted as infinitesimals (there is $\varepsilon\in {}^*\mathbb{R}$ such that $0<\varepsilon<1/n$ for all $n\in\mathbb{N}$) allowing for nonstandard analysis to be developed in ${}^*\mathbb{R}$, while ${}^*\mathbb{N}$ turns out to be an (at least $\aleph_1$-saturated) nonstandard elementary extension of $\mathbb{N}$ (in the language of arithmetic).

While the whole nonstandard real analysis is most naturally developed in ${}^*\mathbb{R}$ (with just a few advanced topics where using the second extension ${}^{**}\mathbb{R}$ is convenient, though far from necessary), recent successful applications of nonstandard methods in combinatorics on $\mathbb{N}$ have utilized also higher order extensions ${}^{(n)*}\mathbb{N} = {}^{***\cdots *}\mathbb{N}$ with the chain $***\cdots *$ of length $n>2$.

In this talk we are going to study the structure of the $\omega$-iterated nonstandard extension ${}^{\cdot}\mathbb{N} = \bigcup_{n\in\omega} {}^{(n)*}\mathbb{N}$ of $\mathbb{N}$ and show how the obtained results shed new light on the complexities of Ramsey combinatorics on $\mathbb{N}$ and allow us to drastically simplify proofs of many advanced Ramsey type theorems such as Hindmann's or Milliken's and Taylor's.

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We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence.

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In 1973, Harvey Friedman proved his striking result on initial self-embeddings of countable nonstandard models of set theory and Peano arithmetic. In this talk, I will discuss my joint work with Ali Enayat focused on the fixed point set of initial self-embeddings of countable nonstandard models of arithmetic. Especially, I will survey the proof of some generalizations of well-known results on the fixed point set of automorphisms of countable recursively saturated models of $ \mathrm{PA} $, to results about the fixed point set of initial self-embeddings of countable nonstandard models of $ \mathrm{I}\Sigma_{1} $.

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In 2009 Roman Kossak and I showed that the classification of countable models of PA is Borel complete, which means it is as complex as possible. The proof is a straightforward application of Gaifman’s canonical I-models. In 2017 Sam Dworetzky, John Clemens, and I showed that the argument may also be used to show the classification of countable models of ZFC is Borel complete too. In this talk I'll outline the original argument for models of PA, the adaptation for models of ZFC, and briefly consider several subclasses of countable models of ZFC.

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The Ultrapower Axiom UA, introduced by Goldberg and Woodin, is known to have many striking consequences. In particular, Goldberg has shown that assuming UA, the Mitchell ordering of normal measures over a measurable cardinal is linear. I will discuss how this result may be used to construct choiceless models of ZF in which the number of normal measures at successors of singular cardinals can be precisely controlled.

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Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and 'consistent' needs to mean 'consistent in a strong sense.' It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler's 'consistency properties'.

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The usual base theory used in reverse mathematics, $\mathrm{RCA}_0$, is the fragment of second-order arithmetic axiomatized by $\Delta^0_1$ comprehension and $\Sigma^0_1$ induction. The weaker base theory $\mathrm{RCA}^*_0$ is obtained by replacing $\Sigma^0_1$ induction with $\Delta^0_1$ induction (and adding the well-known axiom $\exp$ in order to ensure totality of the exponential function). In first-order terms, $\mathrm{RCA}_0$ is conservative over $\mathrm{I}\Sigma_1$ and $\mathrm{RCA}^*_0$ is conservative over $\mathrm{B}\Sigma_1 + \exp$.

Some of the most interesting open problems in reverse mathematics concern the first-order strength of statements from Ramsey Theory, in particular Ramsey's Theorem for pairs and two colours. In this talk, I will discuss joint work with Kasia Kowalik, Tin Lok Wong, and Keita Yokoyama concerning the strength of Ramsey's Theorem over $\mathrm{RCA}^*_0$.

Given standard natural numbers $n,k \ge 2$, let $\mathrm{RT}^n_k$ stand for Ramsey's Theorem for $k$-colourings of $n$-tuples. We first show that assuming the failure of $\Sigma^0_1$ induction, $\mathrm{RT}^n_k$ is equivalent to its own relativization to an arbitrary $\Sigma^0_1$-definable cut. Using this, we give a complete axiomatization of the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^n_k$ for $n \ge 3$ (this turns out to be a rather peculiar fragment of PA) and obtain some nontrivial information about the first-order consequences of $\mathrm{RT}^2_k$. Time permitting, we will also discuss the question whether our results have any relevance for the well-known open problem of characterizing the first-order consequences of $\mathrm{RT}^2_2$ over the traditional base theory $\mathrm{RCA}_0$.

In the first part of the talk, we concentrated on Ramsey's Theorem for $n$-tuples where $n \ge 3$. In this second part, the focus will be on $\mathrm{RT}^2_2$.

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This will be a sequel to Ralf Schindler’s talk on 9/25. My plan is to give a reasonably detailed account of the proof of the result in the title.

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The talk will be a survey of results on first-order theories of pairs (N,M), where M is a model of PA and N is its elementary extension, under various assumptions on the models and on the type of extension. In particular, I will discuss in detail the results on countable recursively saturated models and their cofinal submodels from a joint paper with Jim Schmerl.

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Guzmán and Kalajdzievski introduced a variant of Miller forcing $P(F)$ that diagonalises a given filter $F$ over $\omega$ and has Axiom A. We investigate the effect of $P(F)$ for particularly chosen Canjar filters $F$. This is joint work with Christian Bräuninger.

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The talk will be a survey of results on first-order theories of pairs (N,M), where M is a model of PA and N is its elementary extension, under various assumptions on the models and on the type of extension. In particular, I will discuss in detail the results on countable recursively saturated models and their cofinal submodels from a joint paper with Jim Schmerl.

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Many large cardinals can be defined through elementary embeddings from the set-theoretic universe to some inner model, with the guiding principle being that the closer the inner model is to the universe the stronger the resulting theory. Under ZFC, the Kunen Inconsistency places a hard limit on how close this can be. One is then naturally led to the question of what theory is necessary to derive this inconsistency with the primary focus having historically been embeddings in ZF without Choice.

In this talk we take a different approach to weakening the required theory, which is to study elementary embeddings from the universe into itself in ZFC without Power Set. We shall see that I1, one of the largest large cardinal axioms not known to be inconsistent with ZFC, gives an upper bound to the naive version of this question. However, under reasonable assumptions, we can reobtain this inconsistency in our weaker theory.

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The talk will be a survey of results on first-order theories of pairs (N,M), where M is a model of PA and N is its elementary extension, under various assumptions on the models and on the type of extension. In particular, I will discuss in detail the results on countable recursively saturated models and their cofinal submodels from a joint paper with Jim Schmerl.

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Countably complete ultrafilters are the combinatorial manifestation of strong large cardinal axioms, but many of their basic properties are undecidable no matter the large cardinal axioms one is willing to adopt. The Ultrapower Axiom (UA) is a set theoretic principle that permits the development of a much clearer picture of countably complete ultrafilters and, consequently, the large cardinals from which they derive. It is not known whether UA is (relatively) consistent with very large cardinals, but assuming there is a canonical inner model with a supercompact cardinal, the answer should be yes: this inner model should satisfy UA and yet inherit all large cardinals present in the universe of sets. The predicted resemblance between the large cardinal structure of this model and that of the universe itself is so extreme as to suggest that certain consequences of UA must in fact be provable outright from large cardinal axioms. While the inner model theory of supercompact cardinals remains a major open problem, this talk will describe a technique that already permits a number of consequences of UA to be replicated from large cardinals alone. Still, the technique rests on the existence of inner models that absorb large cardinals, but instead of building canonical inner models, one takes ultrapowers.

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One of the strongest second-order arithmetic systems is full second-order arithmetic ${\rm Z}_2$ which asserts that every second-order formula (with any number of set quantifiers) defines a set. We can augment ${\rm Z}_2$ with choice principles such as the choice scheme and the dependent choice scheme. The $\Sigma^1_n$-choice scheme asserts for every $\Sigma^1_n$-formula $\varphi(n,X)$ that if for every $n$, there is a set $X$ witnessing $\varphi(n,X)$, then there is a single set $Z$ whose $n$-th slice $Z_n$ is a witness for $\varphi(n,X)$. The $\Sigma^1_n$-dependent choice scheme asserts that every $\Sigma^1_n$-relation $\varphi(X,Y)$ without terminal nodes has an infinite branch: there is a set $Z$ such that $\varphi(Z_n,Z_{n+1})$ holds for all $n$. The system ${\rm Z}_2$ proves the $\Sigma^1_2$-choice scheme and the $\Sigma^1_2$-dependent choice scheme. The independence of $\Pi^1_2$-choice scheme from ${\rm Z}_2$ follows by taking a model of ${\rm Z}_2$ whose sets are the reals of the Feferman-Levy model of ${\rm ZF}$ in which every $\aleph_n^L$ is countable and $\aleph_\omega^L$ is the first uncountable cardinal.

We construct a model of ${\rm ZF}+{\rm AC}_\omega$ whose reals give a model of ${\rm Z}_2$ together with the full choice scheme in which $\Pi^1_2$-dependent choice fails. This result was first proved by Kanovei in 1979 and published in Russian. It was rediscovered by Sy Friedman and myself with a slightly simplified proof.

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The real numbers are up to isomorphism the only completely ordered field with a countable dense subset. We consider non-Archimedean ordered fields whose smallest dense subset has cardinality kappa and investigate whether anything resembling ordinary analysis works on these fields.

In particular, we look at generalisations of the intermediate value theorem and the Bolzano-Weierstrass theorem, and realise that there is some mathematical tension between these theorems: the intermediate value theorem requires some saturation whereas Bolzano-Weierstrass fails if the field is saturated. We consider weakenings of Bolzano-Weierstrass compatible with saturation and realise that these are equivalent to the weak compactness of kappa.

This is joint work with Merlin Carl, Lorenzo Galeotti, and Aymane Hanafi.

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One of the strongest second-order arithmetic systems is full second-order arithmetic ${\rm Z}_2$ which asserts that every second-order formula (with any number of set quantifiers) defines a set. We can augment ${\rm Z}_2$ with choice principles such as the choice scheme and the dependent choice scheme. The $\Sigma^1_n$-choice scheme asserts for every $\Sigma^1_n$-formula $\varphi(n,X)$ that if for every $n$, there is a set $X$ witnessing $\varphi(n,X)$, then there is a single set $Z$ whose $n$-th slice $Z_n$ is a witness for $\varphi(n,X)$. The $\Sigma^1_n$-dependent choice scheme asserts that every $\Sigma^1_n$-relation $\varphi(X,Y)$ without terminal nodes has an infinite branch: there is a set $Z$ such that $\varphi(Z_n,Z_{n+1})$ holds for all $n$. The system ${\rm Z}_2$ proves the $\Sigma^1_2$-choice scheme and the $\Sigma^1_2$-dependent choice scheme. The independence of $\Pi^1_2$-choice scheme from ${\rm Z}_2$ follows by taking a model of ${\rm Z}_2$ whose sets are the reals of the Feferman-Levy model of ${\rm ZF}$ in which every $\aleph_n^L$ is countable and $\aleph_\omega^L$ is the first uncountable cardinal.

We construct a model of ${\rm ZF}+{\rm AC}_\omega$ whose reals give a model of ${\rm Z}_2$ together with the full choice scheme in which $\Pi^1_2$-dependent choice fails. This result was first proved by Kanovei in 1979 and published in Russian. It was rediscovered by Sy Friedman and myself with a slightly simplified proof.

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Theorem (Mitchell and Schimmerling, submitted for publication) Assume there is no transitive class model of ZFC with a Woodin cardinal. Let $\nu$ be a singular ordinal such that $\nu > \omega_2$ and $\mathrm{cf}(\nu) < | \nu |$. Suppose $\nu$ is a regular cardinal in K. Then $\nu$ is a measurable cardinal in K. Moreover, if $\mathrm{cf}(\nu) > \omega$, then $o^\mathrm{K}(\nu) \ge \mathrm{cf}(\nu)$.

I will say something intuitive and wildly incomplete but not misleading about the meaning of the theorem, how it is proved, and the history of results behind it.

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Ali Enayat had asked whether there is a model of Peano arithmetic (PA) that can be represented as $\langle\mathbb Q,\oplus,\otimes\rangle$, where $\oplus$ and $\otimes$ are continuous functions on the rationals $\mathbb Q$. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. The reals $\mathbb{R}$, the reals in any finite dimension $\mathbb{R}^n$, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open.

This is joint work with Ali Enayat, myself and Bartosz Wcisło.

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The classical concept of independence, first introduced by Fichtenholz and Kantorovic has been of interest within the study of combinatorics of the subsets of the real line. In particular the study of the cardinal characteristic $\mathfrak{i}$ defined as the minimum size of a maximal independent family of subsets of $\omega.$ In the first part of the talk, we will review the basic theory, as well as the most important results regarding the independence number. We will also point out our construction of a poset $\mathbb{P}$ forcing a maximal independent family of minimal size which turns out to be indestructible after forcing with a countable support iteration of Sacks forcing.

In the second part, we will talk about the generalization (or possible generalizations) of the concept of independence in the generalized Baire spaces, i.e. within the space $\kappa^\kappa$ when $\kappa$ is a regular uncountable cardinal and the new challenges this generalization entails. Moreover, for a specific version of generalized independence, we can have an analogous result to the one mentioned in the paragraph above.

This is joint work with Vera Fischer.

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A set x of ordinals is called recognisable if it is defined, as a singleton, by a formula phi(y) with ordinal parameters that is evaluated in L[y]. The evaluation is always forcing absolute, in contrast to even Sigma_1-formulas with ordinal parameters evaluated in V. Furthermore, this notion is closely related to similar concepts in infinite computation and Hamkins' and Leahy's implicitly definable sets.

It is conjectured that the recognisable universe generated by all recognisable sets is forcing absolute, given sufficient large cardinals. Our goal is thus to determine the recognisable universe in the presence of large cardinals. The new main result, joint with Philip Welch, is a computation of the recognisable universe within the least inner model with infinitely many measurable cardinals.

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The Triangular Embedding Theorem gives a sort of internal generic absoluteness principle that holds under determinacy or large-cardinal assumptions. It originated in work (joint with Itay Neeman) on mad families and the Ramsey Property under AD^+. I will discuss these origins, some applications, and some questions.

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We consider a bounded arithmetic in Buss's language enriched with a predicate Tr which is assumed to be a truth definition for bounded sentences. Among other things we assume polynomial induction for $\Sigma^b_1(\text{Tr})$ formulas. We show that such an arithmetic captures PSPACE. We prove a witnessing theorem for such an arithmetic by an interpretation of free-cuts free proofs of strict $\Sigma^{1,b}_1$ in $U^{1,*}_2$, a canonical second order arithmetic capturing PSPACE. It follows that the problem of the existence of a truth definition for $\Delta_0$ sentences without the totality of $\exp$ might be more about separating subexponential time alternation hierarchies from PSPACE.

The presentation is based on the following article: Konrad Zdanowski, Truth definition for $\Delta_0$ formulas and PSPACE computations, Fundamenta Mathematicae 252(2021) , 1-38.

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There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness.

Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and
(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_\omega$ by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot.

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We are going to talk about some inequalities between cardinal characteristics of the continuum. In particular we are going to relate cardinal characteristics pertaining to the convergenve of series, recently isolated by Blass, Brendle, Brian and Hamkins, other characteristcs concerning equitable splitting defined comparatatively recently by Brendle, Halbeisen, Klausner, Lischka and Shelah and some characteristics defined less recently by Miller, Blass, Laflamme and Minami. All proofs in question are elementary.

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Vopenka's Principle (VP) says that for every proper class of structures with the same signature, there is an elementary embedding from one structure in the class to another. An equivalent form of VP says that for every proper class of graphs, there is an embedding from one graph in the class to another; let us denote this form by VP(graphs, embeddings) with the obvious meaning. We can obtain weaker instances of VP by restricting to particular kinds of graphs such as trees, which are connected acyclic graphs, and rayless trees, which are trees with no infinite path. We will show that VP(trees, embeddings) and VP(rayless trees, embeddings) occupy two different places in the large cardinal hierarchy below VP, and that each is equivalent to the existence of certain virtual large cardinals. Namely, we will show that VP(trees, embeddings) is equivalent to the existence of a weakly virtually A-extendible cardinal (as defined by Gitman and Hamkins) for every class A, and VP(rayless trees, embeddings) is equivalent to the existence of what we will call a weakly virtually A-strong cardinal for every class A. For a better-known point of comparison: the former large cardinal hypothesis is stronger than the existence of a remarkable cardinal, whereas the latter is weaker. We will also relate these two instances of VP to other variants of VP such as generic Vopenka's Principle (as defined by Bagaria, Gitman, and Schindler) and generic Weak Vopenka's Principle.

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This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor's identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, and where $\kappa_1$ is least with Mitchell order 1, and $\kappa_2$ is the least with Mitchell order 2. Another main theorem is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, with $\kappa_1$ the least measurable cardinal such that $o(\kappa_1) = 2$ and $\kappa_2$ the least measurable cardinal above $\kappa_1$. This is a joint work in progress with Victoria Gitman and Arthur Apter.

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This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor's identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, and where $\kappa_1$ is least with Mitchell order 1, and $\kappa_2$ is the least with Mitchell order 2. Another main theorem is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, with $\kappa_1$ the least measurable cardinal such that $o(\kappa_1) = 2$ and $\kappa_2$ the least measurable cardinal above $\kappa_1$. This is a joint work in progress with Victoria Gitman and Arthur Apter.

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Two types of principles are commonly called “reflection principles” in reverse mathematics. According to syntactic reflection principles for T, every theorem of T (from some complexity class) is true. According to semantic reflection principles, every set belongs to some (sufficiently correct) model of T. We will present a connection between syntactic reflection and semantic reflection in second-order arithmetic: for any Pi^1_2 axiomatized theory T, every set is contained in an omega model of T if and only if every iteration of Pi^1_1 reflection for T along a well-ordering is Pi^1_1 sound. There is a thorough proof-theoretic understanding of the latter in terms of ordinal analysis. Accordingly, these reductions yield proof-theoretic analyses of omega-model reflection principles. This is joint work with Fedor Pakhomov.

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Tarski proved (but didn't publish) the theorem that choice from pairs implies choice from four-element sets. Mostowski (1937) began a systematic study of such implications between choice axioms for families of finite sets. Gauntt (1970) completed that study (but didn't publish the results), obtaining equivalent characterizations in terms of fixed points of permutation groups. Truss (1973) extended Gauntt's results (and published this work).

It turns out that these finite choice axioms and their group-theoretic characterizations are instances of the same topos-theoretic statements, interpreted in two very different classes of topoi. My main result is an extension of that observation to the class of all topoi.

Most of my talk will be explaining the background: finite choice axioms, permutation groups, and a little bit about topoi - just enough to make sense of the main result. If time permits, I'll describe some of the ingredients of the proof.

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We prove that any two countable models of the theory ${\rm WKL}^*_0$ sharing the same first-order universe and containing the same counterexample to $\Sigma^0_1$ induction are isomorphic.

This theorem implies that over ${\rm WKL}^*_0 + \neg I\Sigma^0_1$, the analytic hierarchy collapses to the arithmetic hierarchy. It also implies that ${\rm WKL}^*_0$ is the strongest $\Pi^1_2$ statement that is $\Pi^1_1$-conservative over ${\rm RCA}^*_0 + \neg I\Sigma^0_1$. Together with the (slightly subtle) generalizations of the theorem to higher levels of the arithmetic hierarchy, this gives an 'almost negative' answer to a question of Towsner, who asked whether $\Pi^1_1$-conservativity of $\Pi^1_2$ sentences over collection principles is a $\Pi^0_2$-complete computational problem. Our results also have some implications for the reverse mathematics of combinatorial principles: for instance, we get a specific $\Pi^1_1$ sentence that is provable in ${\rm RCA}_0 + B\Sigma^0_2$ exactly if the $\Pi^1_1$ consequences of ${\rm RCA}_0 + {\rm RT}^2_2$ coincide with $B\Sigma^0_2$.

On the side, we also give a positive answer to Towsner's question as originally stated.

Joint work with Marta Fiori Carones, Tin Lok Wong, and Keita Yokoyama.

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Indestructibility results of large cardinals have been an area of interest since Laver's 1978 proof that the supercompactness of $\kappa$ may be made indestructible by any $<\kappa$-directed closed forcing. I will present a continuation of this work, showing that $\alpha$-subcompact cardinals may be made suitably indestructible, but that for C(n)-supercompact cardinals this is largely not possible.

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We study a family of axioms expressing $$`\text{All axioms of PA are true.' (*)}$$ where PA denotes Peano Arithmetic. More precisely, each such axiom states that all axioms from a chosen axiomatization of PA are true. We start with a very natural theory of truth ${\rm CT}^-({\rm PA})$ which is a finite extension of PA in the language of arithmetic augmented with a fresh predicate T to serve as a truth predicate for the language of arithmetic. Additional axioms of this theory are straightforward translations of inductive Tarski truth conditions. To study various possible ways of expressing (*), we investigate extensions of ${\rm CT}^-({\rm PA})$ with axioms of the form $$\forall x \bigl(\delta(x)\rightarrow T(x)\bigr).$$ In the above (and throughout the whole abstract) $\delta(x)$ is an elementary formula which is proof-theoretically equivalent to the standard axiomatization of PA with the induction scheme, i.e. the equivalence $$\forall x \bigl(\text{Prov}_{\delta}(x)\equiv \text{Prov}_{\rm PA}(x)\bigr).$$ is provable in $I\Sigma_1$. For every such $\delta$, the extension of ${\rm CT}^-({\rm PA})$ with the above axiom will be denoted ${\rm CT}^-[\delta]$.

In particular we shall focus on the arithmetical strength of theories ${\rm CT}^-[\delta]$. The 'line' demarcating extensions of ${\rm CT}^-({\rm PA})$ which are conservative over PA from the nonconservative ones is known in the literature as the Tarski Boundary. For some time, there seemed to be the least (in terms of deductive strength) *natural* extension of ${\rm CT}^-({\rm PA})$ on the nonconservative side of the boundary, whose one axiomatization is given by ${\rm CT}^-({\rm PA})$ and $\Delta_0$ induction for the extended language (the theory is called ${\rm CT}_0$). This theory can equivalently be axiomatized by adding to ${\rm CT}^-({\rm PA})$ the natural formal representation of the statement 'All theorems of ${\rm PA}$ are true.'. We show that the situation between the Tarski Boundary and ${\rm CT}_0$ is much more interesting:

Theorem 1: For every r.e. theory Th in the language of arithmetic the following are equivalent:
1) ${\rm CT}_0\vdash$ Th
2) there exists $\delta$ such that ${\rm CT}^-[\delta]$ and Th have the same arithmetical consequences.

Theorem 1 can be seen as a representation theorem for r.e. theories below ${\rm REF}^{\omega}({\rm PA})$ (all finite iterations of uniform reflection over ${\rm PA}$, which is the set of arithmetical consequences of ${\rm CT}_0$): each such theory can be finitely axiomatized by a theory of the form ${\rm CT}^-[\delta]$, where $\delta$ is proof-theoretically reducible to ${\rm PA}$.

Secondly, we use theories ${\rm CT}^-[\delta]$ to investigate the situation below the Tarski Boundary. We shall prove the following result

Theorem 2: There exists a family $\{\delta_f\}_{f\in\omega^{<\omega}}$ such that for all $f,g\in\omega^{<\omega}$
1) ${\rm CT}^-[\delta_f]$ is conservative over ${\rm PA}$;
2) if $f\subsetneq g$, then ${\rm CT}^-[\delta_g]$ properly extends ${\rm CT}^-[\delta_f]$;
3) if $f\perp g$ then ${\rm CT}^-[\delta_g]\cup CT^-[\delta_f]$ is nonconservative over ${\rm PA}$ (but consistent).

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Motivated by a classical theorem of Magidor, I will present results providing characterizations of important objects from the lower end of the large cardinal hierarchy through the existence of elementary embeddings between set-sized models that map their critical point to the large cardinal in question. Focusing on the characterization of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, I will show how these results can be used in the study of the combinatorics of strong chain conditions and the investigation of principles of structural reflection formulated by Bagaria.

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The purpose of this talk is to exposit a method for proving independence over PA of 'mathematical' statements (whatever that means). The method uses the concept of an $(\mathcal L, n)$-model: a finite sequence of finite $\mathcal L$-structures for some first order $\mathcal L$ extending the language of arithmetic. The idea is that this finite sequence is intended to represent increasing approximations of a potentially infinite structure and the machinery developed allows one to translate (meta-mathematical) compactness type statements, which are easily seen to be independent of PA, into statements about finite combinatorics, which have 'mathematical content'. $(\mathcal L, n)$-models were introduced by Shelah in the 70's in his alternative proof of the Paris-Harrington Theorem and also appears (implicitly) in his example of a true, unprovable $\Pi^0_1$ statement of some 'mathematical' content. A similar idea was discovered independently by Kripke (unpublished). In this talk we will flesh out the details of this method and extend the general theory. This will allow us to present, in a fairly systematic fashion, proofs of the Paris-Harrington Theorem and the independence over PA of several, similar, Ramsey Theoretic statements including some which are $\Pi^0_1$.

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Work in ZF and let $j:V_\alpha\to V_\alpha$ be an elementary, or partially elementary, embedding. One can examine the degree of uniqueness, definability or constructibility of $j$. For example, is there $\beta<\alpha$ such that $j$ is the unique (partially) elementary extension of $j\upharpoonright V_\beta$? Is $j$ definable from parameters over $V_\alpha$? We will discuss some results along these lines, illustrating that answers can depend heavily on circumstances. Some of the work is due independently and earlier to Gabriel Goldberg.

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This talk focuses on the theory PAI (I for Indiscernibles), a theory formulated in the language of PA augmented with a unary predicate I(x). Models of PAI are of the form (M,I) where (1) M is a model of PA, (2) I is a proper class of M, i.e., I is unbounded in M and (M,I) satisfies PA*, and (3) I forms a class of indiscernibles over M. The formalizability of the Infinite Ramsey Theorem in PA makes it clear that PAI is a conservative extension of PA. As we will see, nonstandard models of PA (of any cardinality) that have an expansion to a model of PAI are precisely those nonstandard models PA that can carry an inductive partial satisfaction class. The formulation and investigation of PAI was inspired by my work on the set theoretical sibling ZFI of PAI, whose behavior I will also compare and contrast with that of PAI.

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The stationary reflection principle, which is often called the Weak Reflection Principle, is known to have many interesting consequences. As for cardinal arithmetic, it implies that $\lambda^\omega = \lambda$ for all regular cardinal $\lambda \geq \omega_2$. In this talk, we will discuss higher analogues of this stationary reflection principle and their consequences on cardinal arithmetic.

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Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between given $\Pi^1_2$ principles over $\omega$-models of ${\rm RCA}_0$. They also introduced a version of this game that similarly captures provability over (full) ${\rm RCA}_0$. We generalize this game for provability over arbitrary subsystems of second-order arithmetic, and establish a compactness argument that shows that certain winning strategies can always be chosen to win in a number of moves bounded by a number independent of the instance of the principles being considered. Our compactness result also generalizes an old proof-theoretic fact due to H. Wang, and has a number of other applications. This is joint work with Denis Hirschfeldt and Sarah Reitzes.

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Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $\omega_1$. We show it is consistent that $\mathcal{C}$ has minimal elements. This answers an old question due to Galvin.

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This is a continuation of the talk from Feb 16th. This time we shall study different theories of the form ${\rm CT}^-[\delta]$ which are conservative extensions of a ${\rm PA}$. In particular, we prove the following theorem.

Theorem 2 There exists a family $\{\delta_f\}_{f\in\omega^*}$ such that for all $f,g\in\omega^*$
1) ${\rm CT}^-[\delta_f]$ is conservative over ${\rm PA}$;
2) if $f\subsetneq g$, then ${\rm CT}^-[\delta_g]$ properly extends ${\rm CT}^-[\delta_f]$;
3) if $f\perp g$ then ${\rm CT}^-[\delta_g]\cup {\rm CT}^-[\delta_f]$ is nonconservative over ${\rm PA}$ (but consistent).

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Broad Infinity is a new and arguably intuitive axiom scheme in set theory. It states that 'broad numbers', which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal.

Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The talk explains these principles and how they are related under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.

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This is a continuation of the talk from Feb 16th. This time we shall study different theories of the form ${\rm CT}^-[\delta]$ which are conservative extensions of a ${\rm PA}$. In particular, we prove the following theorem.

Theorem 2 There exists a family $\{\delta_f\}_{f\in\omega^*}$ such that for all $f,g\in\omega^*$
1) ${\rm CT}^-[\delta_f]$ is conservative over ${\rm PA}$;
2) if $f\subsetneq g$, then ${\rm CT}^-[\delta_g]$ properly extends ${\rm CT}^-[\delta_f]$;
3) if $f\perp g$ then ${\rm CT}^-[\delta_g]\cup {\rm CT}^-[\delta_f]$ is nonconservative over ${\rm PA}$ (but consistent).

We will finish the proof of the theorem announced in the abstract of part II.

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The distinction between 'algebraic' and 'non-algebraic fields in mathematics, coined by Shapiro (1997), plays an important role in discussions about the status of set theory and connects back to the so-called universe/multiverse debate in the philosophy of set theory. In this talk we will see, that this distinction is not as clear cut as is usually assume when using it in the debate. In particular, we will see that in more recent formulations of this distinction, multiversism seems to split into a a strong and a weaker form. This can be translated to a meta-level, when considering the background theory in which set-theoretic multiversism can take place. This offers a more fine-grained picture of multiversism and allows us to mitigate a standard universist objection based on the conception of a multiversist background theory.

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I will present an axiomatization of a class of residue rings of models of PA. This is obtained using valuation theory and results on models of PA. (Joint work with A. Macintyre)

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The approximation property was introduced by Hamkins for his Gap Forcing Theorem, and it has turned out to be a very useful notion, appearing for example in the partial equiconsistency result of Viale and Weiss on PFA, and in the proof of Woodin's HOD Dichotomy Theorem. In the context of generic embeddings, there can be a useful interplay between elementarity and approximation. We discuss some recent work in this direction: (1) tensions between saturated ideals on $\omega_2$ and the tree property (with Sean Cox), (2) fragility of the strong independence spectra (with Vera Fischer), and (3) mutual inconsistency of Foreman‘s minimal generic hugeness axioms.

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This talk will report on ongoing work that is being done in collaboration with Ali Enayat (University of Gothenburg).

For models of set theory $\mathcal{N}$ and $\mathcal{M}$, $\mathcal{N}$ is a powerset preserving end-extension of $\mathcal{M}$ if $\mathcal{N}$ is an end-extension of $\mathcal{M}$ and $\mathcal{N}$ contains no new subsets of sets in $\mathcal{M}$. A model of Kripke-Platek Set Theory, $\mathcal{N}$, is a rank-extension of a model of Kripke-Platek Set Theory, $\mathcal{M}$, if $\mathcal{N}$ is an end-extension of $\mathcal{M}$ and all of the new sets in $\mathcal{N}$ have rank that exceeds the rank of all of the sets in $\mathcal{M}$. A powerset preserving end-extension (rank-extension) $\mathcal{N}$ of $\mathcal{M}$ is topless if $\mathcal{M} \neq \mathcal{N}$ and there is no set in $\mathcal{N} \backslash \mathcal{M}$ containing only sets from $\mathcal{M}$. If $\mathcal{M}= \langle M, E^\mathcal{M} \rangle$ is a model of set theory, then the admissible cover of $\mathcal{M}$, $\mathbb{C}\mathrm{ov}_\mathcal{M}$, is defined to be the smallest admissible structure with $\mathcal{M}$ forming its urelements and whose language contains a unary function function symbol, $F$, that sends each $m \in M$ to the set $\{x \in M \mid x E^\mathcal{M} m\}$. Barwise has shown that if $\mathcal{M}$ is a model of Kripke-Platek Set Theory, then $\mathbb{C}\mathrm{ov}_{\mathcal{M}}$ exists and its minimality facilitates compactness arguments for infinitary languages coded in $\mathbb{C}\mathrm{ov}_\mathcal{M}$. We extend Barwise's analysis by showing that if $\mathcal{M}$ satisfies enough set theory then the expansion of $\mathbb{C}\mathrm{ov}_\mathcal{M}$ obtained by adding the powerset function remains admissible. This allows us to build powerset preserving end-extensions and rank-extensions of countable models of certain subsystems of $\mathrm{ZFC}$ satisfying any given recursive subtheory of the model being extended. In particular, we show that

1. Every countable model of $\mathrm{KP}^\mathcal{P}$ has a topless rank-extension that satisfies $\mathrm{KP}^\mathcal{P}$.
2. Every countable $\omega$-standard model of $\mathrm{MOST}+\Pi_1\textrm{-collection}$ has a topless powerset preserving end-extension that satisfies $\mathrm{MOST}+\Pi_1\textrm{-collection}$.

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The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin’s derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan’s conjecture.

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25 years ago I wrote a paper on four open problems concerning recursively saturated models of PA. The problems are still open. I will talk about two of them: (1) Let M be a countable recursively saturated model of PA. Can every automorphism of M be extended to some recursively saturated elementary end extension of M? (2) Is there a recursively saturated model of PA that has no recursively saturated elementary submodel of the same cardinality as the model? I will present some partial results involving partial inductive satisfaction classes.

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In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah's $\mathbb L^1_\kappa$ (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.

These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].

During the first lecture, I plan to describe two games and a syntax of two logics: Shelah's $\mathbb L^1_\kappa$ and my own logic (joint work with Väänänen) $\mathbb L^{1,c}_\kappa$. I will stress some of the properties of these logics, without any use of large cardinal assumptions.

During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.

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In this talk we shall deal with fragments of first-order Peano Arithmetic obtained by restricting the conclusion of the induction or the collection axiom to elements in a prescribed subclass $D$ of the universe. Fix $n>0$. The schemes of $\Sigma_n$-induction up to $\Sigma_m$-definable elements and the schemes of $\Sigma_n$-collection up to $\Sigma_m$-definable elements form two families of subtheories of $I\Sigma_n$ and $B\Sigma_n$, respectively, obtained in this way.

The properties of $\Sigma_n$-induction up to $\Sigma_m$-definable elements for $n\geq m$ are reasonably well understood and interesting applications of these fragments are known. However, an analysis of the case $n<m$ was pending. In the first part of this talk, we address this problem and show that it is related to the following general question: 'Under which conditions on a model $M$ can we prove that every non-empty $\Sigma_m$-definable subset of $M$ contains some $\Sigma_m$-definable element?'

In the second part of the talk, we show that, for each $n\geq 1$, the scheme of $\Sigma_n$-collection up to $\Sigma_n$-definable elements provides us with an axiomatization of the $\Sigma_{n+1}$-consequences of $B\Sigma_n$. As an application, we obtain that $B\Sigma_n$ is $\Sigma_{n+1}$-conservative over parameter-free $\Delta_n$-minimization (plus $I\Sigma_{n-1}$), thus partially answering a question of R. Kaye.

This is joint work with F.Félix Lara-Martín (University of Seville).

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In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah's $\mathbb L^1_\kappa$ (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.

These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].

During the first lecture, I plan to describe two games and a syntax of two logics: Shelah's $\mathbb L^1_\kappa$ and my own logic (joint work with Väänänen) $\mathbb L^{1,c}_\kappa$. I will stress some of the properties of these logics, without any use of large cardinal assumptions.

During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.

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D'Aquino, Knight and Starchenko showed the real closure of a model of Peano Arithmetic is recursively saturated. Thus any two countable models of PA with the same standard system have isomorphic real closures. Charlie Steinhorn, Jim Schmerl and I showed that even for $\omega_1$-like model of PA the situation is very different. We construct $2^{\aleph_1}$ recursively saturated elementarily equivalent $\omega_1$-like models of PA with the same standard system and non-isomorphic real closures.

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We define a set of real numbers to be perfectly small if it has perfectly many disjoint translates. Such sets have a strong intuitive claim to being probabilistically negligible, yet no non-trivial measure assigns them all a value of 0. We will prove from a moderate amount of choice that any total extension of Lebesgue measure concentrates on a perfectly small set, suggesting that for any such measure, translation-invariance fails 'as badly as possible.' From the ideas of this proof, we will also derive analogues of well-known paradoxes of randomness, specifically Freiling's symmetry paradox and the infinite prisoner hat puzzle, in terms of perfectly small sets. Finally, we discuss how these results constrain what a paradox-free set theory can look like and some related open questions.

A note on paradoxes

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This oral exam talk will present a proof of Woodin's result that every real number is generic over some iterated ultrapower of any model with a Woodin cardinal. No fine structure theory will be used, and there will be a brief introduction to iteration trees.

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Maximal almost disjoint (MAD) families and their relatives have been an important area of combinatorial and descriptive set theory since at least the 60s. In this talk I will discuss some relatives of MAD families, focussing on eventually different families of functions $f:\omega \to \omega$ and eventually different sets of permutations $p \in S(\omega)$. In the context of MAD families it has been fruitful to consider various strengthenings of the maximality condition to obtain several flavors of 'strongly' MAD families. One such strengthening that has proved useful in recent literature is that of tightness. Tight MAD families are Cohen indestructible and come with a properness preservation theorem making them nice to work with in iterated forcing contexts.

I will introduce a version of tightness for maximal eventually different families of functions $f:\omega \to \omega$ and maximal eventually different families of permutations $p \in S(\omega)$ respectively. These tight eventually different families share a lot of the nice, forcing theoretic properties of tight MAD families. Using them, I will construct explicit witnesses to $\mathfrak{a}_e= \mathfrak{a}_p = \aleph_1$ in many known models of set theory where this equality was either not known or only known by less constructive means. Working over $L$ we can moreover have the witnesses be $\Pi^1_1$ which is optimal for objects of size $\aleph_1$ in models where ${\rm CH}$ fails. These results simultaneously strengthen several known results on the existence of definable maximal sets of reals which are indestructible for various definable forcing notions. This is joint work with Vera Fischer.

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In his seminal work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals known as the Magidor iteration. The purpose of this talk is to present a Mathias-type criterion which characterizes when a sequence of omega-sequences is generic for the Magidor iteration. The result extends a theorem of Fuchs, who introduced a Mathias criterion for discrete products of Prikry forcings. We will present the new criterion, discuss several applications, and outline the main ideas of the proof.

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Woodin's HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel's constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin's conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.

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Woodin's HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel's constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin's conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.

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In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.

The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.

Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. I also plan to discuss connections with Hamkins's multiverse theory, and the model of this constructed by Gitman and Hamkins. Throughout the talks I'm keen to discuss both philosophical and mathematical matters with the audience, concerning our Copernican approach to the multiverse of sets.

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In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.

The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.

Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. I also plan to discuss connections with Hamkins's multiverse theory, and the model of this constructed by Gitman and Hamkins. Throughout the talks I'm keen to discuss both philosophical and mathematical matters with the audience, concerning our Copernican approach to the multiverse of sets.

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In a recent Quanta Magazine article discussing difficulties and progress related to Feynman path integrals, Charlie Wood writes, 'No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general.' This statement is arguably refuted by Nonstandard Analysis, but what is perhaps lacking is a constructive approach. We present such an approach based on reduced powers and a class of algebraic structures we call comparison rings. This construction has a nice iteration theory and is able to represent classical integrals via standard parts. We discuss an example of a filter on $\mathbb R^{\lt\omega}$, the direct limit of the $\mathbb R^n$, that respects classical volumes in different dimensions simultaneously, with lower dimensional surfaces being infinitesimal relative to higher dimensional ones. This suggests a corresponding generalization of dimension, which we show under some set-theoretic assumptions may constitute a dense linear order without $(c,c)$-gaps. This is joint work with Emanuele Bottazzi.

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We continue some work on L[Card] (the universe constructed from the predicate for the cardinals) to look at L[Reg] where Reg is the class of uncountable regular cardinals. The latter is also a model of a rich combinatorial structure being, as it turns out, a Magidor iteration of prikry forcings (using recent work of Ben-Neria). But it is limited in size, in fact is a rather 'thin' model. We show, letting O^s = O^sword be the least iterable structure with a measure which concentrates on measurable cardinals:

Theorem (ZFC)

1. Let S be a set, or proper class, of regular cardinals, then O^s is not an element of L[S].
2. (b) This is best possible, in that no smaller mouse M can be substituted for O^s.
3. (c) L[S] is a model of: GCH, Square's, Diamonds, Morasses etc and has Ramsey cardinals, but no measurable cardinals.

Slides

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It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continuum is a proper class. We examine several theories based on maximality considerations in this framework (in particular countabilist analogues of reflection principles) and show how standard set theories (including ZFC with large cardinals added) can be incorporated. We conclude that the systems considered raise questions concerning the foundational purposes of set theory.

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1. I'll begin by reviewing the work I did in 1993-6 on a problem raised by the dynamics group at the Universidad Autonomoa de Barcelona. They were interested in a phenomenon that resembles the Cantor-Bendixson sequence of derivatives, and hoped to prove that the sequence would always stop at a countable stage. Using ideas of Kunen and Martin I showed that it would always stop at or before stage omega_1.

2. In 2002/3, alerted by observations of David Fremlin, to the possibility that the barcelona conjecture was false, I succeeded in constructing an example with recursive initial data where the sequence stops exactly at stage omega_1.

My Réunion colleague Chrstian Delhommé simplified and extended my ideas.

I'll outline the construction, as I think the underlying idea might have applications elsewhere in descriptive set theory.

3. I will outline more recent work using ideas of Blass and Fremlin to to study 'uniform' versions of the results of 1993-96.

I'll end with listing some open problems which I hope will be found interesting.

References

Slides

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1. I'll begin by reviewing the work I did in 1993-6 on a problem raised by the dynamics group at the Universidad Autonomoa de Barcelona. They were interested in a phenomenon that resembles the Cantor-Bendixson sequence of derivatives, and hoped to prove that the sequence would always stop at a countable stage. Using ideas of Kunen and Martin I showed that it would always stop at or before stage omega_1.

2. In 2002/3, alerted by observations of David Fremlin, to the possibility that the barcelona conjecture was false, I succeeded in constructing an example with recursive initial data where the sequence stops exactly at stage omega_1.

My Réunion colleague Chrstian Delhommé simplified and extended my ideas.

I'll outline the construction, as I think the underlying idea might have applications elsewhere in descriptive set theory.

3. I will outline more recent work using ideas of Blass and Fremlin to to study 'uniform' versions of the results of 1993-96.

I'll end with listing some open problems which I hope will be found interesting.

References

Slides

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We shall present some recent results from a joint work with Philipp Lücke on Structural Reflection at the upper ridges of the large-cardinal hierarchy. In particular, we will introduce a natural form of reflection we call 'Exact Reflection', giving upper and lower bounds for its consistency strength. We will also discuss 'sequential' forms of Exact Reflection, which may be viewed as strong forms of Chang's Conjecture, and which, in the case of infinite sequences, their strength goes beyond the strongest large cardinal principles that are not known to be inconsistent with the Axiom of Choice.

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Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. The $\{+,\cdot,0,1\}$-theory $\mathsf{ACF}$ of algebraically closed field is the model companion of the theory of $\mathsf{Fields}$ but not its AMC as $\exists x(x^2+1=0)\in \mathsf{ACF}_{\exists\vee\forall}\setminus \mathsf{Fields}_{\exists\vee\forall}$. Any model complete theory $T$ is the AMC of $T_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the $\in$-theory $\mathsf{ZFC}+$there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}_{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}_\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}_\tau$.

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Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. The $\{+,\cdot,0,1\}$-theory $\mathsf{ACF}$ of algebraically closed field is the model companion of the theory of $\mathsf{Fields}$ but not its AMC as $\exists x(x^2+1=0)\in \mathsf{ACF}_{\exists\vee\forall}\setminus \mathsf{Fields}_{\exists\vee\forall}$. Any model complete theory $T$ is the AMC of $T_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the $\in$-theory $\mathsf{ZFC}+$there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}_{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}_\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}_\tau$.

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By beginning with the order topology on an ordinal $\delta$, and iteratively declaring more and more derived sets to be open, Bagaria defined the derived topologies $\tau_\xi$ on $\delta$, where $\xi$ is an ordinal. He showed that the non-isolated points in the space $(\delta,\tau_\xi)$ can be characterized using a strong form of iterated simultaneous stationary reflection called $\xi$-s-reflection, which is deeply connected to certain transfinite indescribability properties. However, Bagaria's definitions break for $\xi\geq\delta$ because, under his definitions, the $\delta$-th derived topology $\tau_\delta$ is discrete and no ordinal $\alpha$ can be $\alpha+1$-s-stationary. We will discuss some new work in which we use certain diagonal versions of Bagaria's definitions to extend his results. For example, we introduce the notions of diagonal Cantor derivative and use it to obtain a sequence of derived topologies on a regular $\delta$ that is strictly longer than that of Bagaria's, under certain hypotheses.

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Let $n>1$ be a natural number, let $\Gamma_n$ be the hypergraph of isosceles triangles in $\mathbb{R}^n$. Under the axiom of choice, the existence of a countable coloring for $\Gamma_n$ is true for every $n$. Without the axiom of choice, the coloring problems will be hard to answer. We often expect the case that the countable chromatic number of one hypergraph doesn't imply the one for another. With an inaccessible cardinal, there is a model of ZF+DC in which $\Gamma_2$ has countable chromatic number while $\Gamma_3$ has uncountable chromatic number. This result is obtained by a balanced forcing over the symmetric Solovay model.

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A Hardy field is a differential field of germs at infinity of one-variable differentiable real-valued functions defined on half-lines. Hardy fields appear naturally in model theory and its applications to real analytic geometry and dynamical systems, and also have found uses in computer algebra, ergodic theory, and various other fields of mathematics. I will discuss some optimal extension results for Hardy fields obtained in the last few years, which lead to a description of the theory of maximal Hardy fields and applications to ordinary differential equations. (This is joint work with Lou van den Dries and Joris van der Hoeven.)

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Set-theoretic potentialism is the view that the universe of sets is never fully completed but is only given potentially. Tools from modal logic have been applied to understand the mathematics of potentialism. In recent work, Neil Barton and I extended this analysis to class-theoretic potentialism, the view that proper classes are given potentially (while the sets may or may not be fixed).

In this talk, I will survey some results from set-theoretic potentialism. After seeing how the tools apply in that context I will then discuss our work in the class-theoretic context.

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This is a joint work with Albert Visser. We prove that no consistent finitely axiomatized theory one-dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the Second Incompleteness Theorem whose formulation is completely arithmetic-free. Probably the most important novel feature that distinguishes our result from the previous results of this kind is that it is applicable to arbitrary weak theories, rather than to extensions of some base theory. The methods used in the proof of the main result yield a new perspective on the notion of sequential theory, in the setting of forcing-interpretations. https://arxiv.org/abs/2109.02548

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We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form $V[C]$ where $C$ is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are $\kappa^+$-c.c. in the generic extension and therefore do not add fresh subsets to $\kappa^+$. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.

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We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form $V[C]$ where $C$ is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are $\kappa^+$-c.c. in the generic extension and therefore do not add fresh subsets to $\kappa^+$. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.

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Assume that $\mathbb P$ is a forcing notion, $G$ is a generic set for it over the ground model $V$, and a cardinal $\kappa$ is measurable in the generic extension. Let $j$ be an ultrapower embedding, taken in $V[G]$ with a normal measure on $\kappa$. We consider the following questions:

1. Is the restriction of $j$ to $V$ an iterated ultrapower of $V$ (by its measures or extenders)?

2. Is the restriction of $j$ to $V$ definable in $V$?

By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and $V=K$ is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below $\kappa$.

We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal $\kappa$. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of $j$ restricted to $V$ as an iteration of $V$. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.

[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.

[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.

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Assume that $\mathbb P$ is a forcing notion, $G$ is a generic set for it over the ground model $V$, and a cardinal $\kappa$ is measurable in the generic extension. Let $j$ be an ultrapower embedding, taken in $V[G]$ with a normal measure on $\kappa$. We consider the following questions:

1. Is the restriction of $j$ to $V$ an iterated ultrapower of $V$ (by its measures or extenders)?

2. Is the restriction of $j$ to $V$ definable in $V$?

By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and $V=K$ is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below $\kappa$.

We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal $\kappa$. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of $j$ restricted to $V$ as an iteration of $V$. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.

[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.

[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.

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