NYLogic

Given a model M of an axiomatic theory A, and a model N of an axiomatic theory B, we say that they are bi-interpretable if, roughly speaking, they have the same definable sets: that is, there are definable maps that move definable sets in one to definable sets in the other. One interesting question we might ask, given an axiomatic theory A, is which of its models are bi-interpretable with the integers (seen as a model of the first-order theory of rings)? As self-interpretations of the integers are particularly simple, this gives a lot of information about properties of the model. In this talk, we will consider arithmetic groups like SL(n, Z) and discuss recent progress in understanding when such groups are bi-interpretable with arithmetic and what consequences this has when it occurs.

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Nonstandard methods have been successfully applied to standard problems in number theory by R. Jin, T. Tao and others. A. Boudaoud and D. Bellaouar are pursuing the opposite direction: they are formulating number-theoretic problems in the language of nonstandard analysis and solving them by standard methods. Two examples of the kind of questions they consider are:
(1) Can every unlimited natural number n be represented in the form n = s + w_1w_2 where s is a limited integer and w_1, w_2 are unlimited?
(2) Can every unlimited natural number n be represented in the form n = w_1w_2 + w_3w_4 so that each ratio w_i / w_j is appreciable (ie, neither infinitesimal nor unlimited)?
I give a negative answer to question (1) (assuming Dickson’s Conjecture) and a positive answer to question (2).
A. Boudaoud, D. Bellaouar, Representation of integers: A nonclassical point of view, Journal of Logic & Analysis. 12:4 (2020) 1{31; K. Hrbacek, Journal of Logic & Analysis 12:5 (2020) 1–6.

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It is a well-known fact that non-standard models of Peano Arithmetic (PA) have order type N + Z · D, where D is a dense linear order. The question of which dense linear orders D can occur in such order types is non-trivial and widely studied. In this context Friedman asked the following question:

Are there consistent extensions of Peano Arithmetic T and T′ such that the class of order types of models of T and the class of order types of models of T′ differ?

Friedman’s question is very complex and still wide open. In this talk we will go in the opposite direction and consider a version of Friedman’s question for syntactic fragments of PA. We will present results from a joint work with Benedikt Löwe on order types of non-standard models of syntactic subsystems of arithmetic obtained by restricting the language to subsets of the operations. We will put particular emphasis on models of syntactic subsystems of Peano Arithmetic obtained by dropping the schema of induction.

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I call a subset of the domain of a countable model absolutely undefinable if the set of its images under automorphisms of the model is uncountable. By the Kueker-Reyes theorem, all sets that are not absolutely undefinable are parametrically definable in $L_{\omega_1 \omega}$. I will survey classical results about first-order undefinability in the standard model of arithmetic, and I will contrast them with some old and some new results about absolute undefinability in nonstandard models of PA.​

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A typical Ramsey statement is the following: given a coloring of a complete graph, we aim to find a 'large' complete subgraph that is monochromatic. The weaker variation we are considering here (introduced by Bergfalk-Hrusak-Shelah) is to relax the 'complete subgraph' in the goal. More precisely, we aim to find a certain 'large' connected monochromatic subgraph. We will discuss the motivation and the connections with other combinatorial and algebraic problems. We demonstrate consistently, such partition relations can hold at small uncountable cardinals like aleph_2, and successors of singular cardinals like aleph_{omega+1}. Joint work with Hrusak and Shelah.

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We survey the effective foundations for analysis with infinitesimals recently developed by Hrbacek and Katz, and detail some applications. Theories SPOT and SCOT illustrate the fact that analysis with infinitesimals requires no more choice than traditional analysis. The theory SCOT incorporates in particular all the axioms of Nelson's Radically Elementary Probability Theory, which is therefore conservative over ZF+ADC.

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It is well known that a countable model of PA has a truth predicate if and only if it is recursively saturated. It is also well known that not all countable recursively saturated models of PA have *inductive* or even $\Delta_0$-inductive truth predicates: indeed, such models must satisfy Con(PA), for example. Recent work by Enayat-Pakhomov and Cieśliński-Łełyk-Wcisło explored the principle of 'disjunctive correctness', asserting that every disjunction is true if and only if it has a true disjunct. In particular, one can show that every countable model of PA has a 'disjunctively trivial' elementary extension: that is, an elementary extension with a truth predicate in which all nonstandard length disjunctions are evaluated as true. In this talk, we will see that such 'disjunctively trivial' models are necessarily arithmetically saturated; indeed, we will see that a countable model of PA is arithmetically saturated if and only if it has a disjunctively trivial truth predicate. We will explore related pathologies in truth predicates, and classify the sets which can be defined using such pathologies. We find other surprising connections between arithmetic saturation and these questions of definability. This is joint work with Mateusz Łełyk, based heavily on unpublished work by Jim Schmerl.

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The popular game of Hex can be extended to the infinite hexagonal lattice, defining a winning condition which formalises the idea of a chain of colored stones stretching towards infinity. The descriptive-set-theoretic complexity of the set of winning positions is unknown, although it is at most Σ^1_1, and it is conjectured to be Borel; this has implications on whether games of infinite Hex are determined from all initial positions as either first-player wins or draws.

I will show that, unlike the finite game, infinite Hex with an initially empty board is a draw. But is the game still a draw when starting from a non-empty board? This open question can be partially answered in the positive by assuming the existence of certain local strategies, and in the negative by giving the advantage of placing two stones at each turn to one of the players. This is joint work with Joel David Hamkins.

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Parameters are free variables in various axiom schemata in PA, ZFC, and other similar theories. Given an axiom schema S, we let S* be the parameter-free sub-schema.

Kreisel (A survey of proof theory, JSL 1968) was one of those who paid attention to the comparison of some schemata in second-order PA and their parameter-free versions. In particular, Kreisel noted that

[...] if one is convinced of the significance of something like a given axiom schema, it is natural to study details, such as the effect of parameters.

This talk is devoted to the effect of parameters in the schemata of Comprehension and Choice in second-order arithmetic.

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Fix a computable presentation $\overline{\mathbb Q}$ of the algebraic closure of the rational numbers. The absolute Galois group of the rational numbers, which is precisely the automorphism group of the field $\overline{\mathbb Q}$, may then be viewed as a collection of paths through a finite-branching tree. Each individual automorphism has a Turing degree. We will use known results in computability to try to build natural countable elementary subgroups of the absolute Galois group. Several intriguing questions in number theory will appear as we measure the extent to which we succeed in doing so.

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In this talk I will discuss some recent advances in the first-order classification problem. I will touch on first-order rigidity and quasi finite axiomatization. However, the main point of the presentation is on how, in principle, one can describe all structures which are first-order equivalent to a given one. This leads to non-standard models of algebraic structures (aka non-standard analysis or non-standard arithmetic), which are interesting in their own right.

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When Gromov asked 'What is a typical group?', he was thinking of finitely presented groups, and he proposed an approach involving limiting density. Here, we reframe this question in the context of universal algebra and discuss some examples illustrating the behaviors of some of these algebraic varieties and then general conditions that imply some of these behaviors. Our primary general result states that for a commutative generalized bijective variety and presentations with a single generator and single identity, the zero-one law holds and, furthermore, that the sentences with density 1 are those true in the free structure. The proof of this result requires a specialized version of Gaifman's Locality Theorem that enables us to get a better bound on the complexity of the formulas of interest to us.

This work is joint with Meng-Che 'Turbo' Ho and Julia Knight.

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Vopěnka first presented his Alternative Set Theory (AST) in the monograph 'Mathematics in the Alternative Set Theory' published by Teubner, Leipzig in 1979. Another book presenting the theory, 'Introduction to Mathematics in the Alternative Set Theory', was published in 1989 in Slovak by Alfa, Bratislava. In addition there are numerous journal papers on the AST by members of the research group established by Vopěnka, and the proceedings of a conference dedicated to the AST, also from 1989. In several essays, Vopěnka sought to lay out the motivation and philosophical import of the AST and some of his subsequent work. As one consequence of the emphasis on his philosophy, the mathematical inspiration for the AST has been somewhat obliterated. The aim of the talk is to discuss the design choices Vopěnka made for the AST in relation to pertinent mathematical developments of the 20th century, such as Skolem's work on nonstandard models of arithmetic, Robinson's nonstandard analysis, Rieger's nonstandard models of arithmetic, Vopěnka's nonstandard model of set theory, Vopěnka and Hájek's theory of semisets, or Parikh's almost consistent theories. The presentation will include an outline of the AST following the works of Vopěnka and Sochor. This is a historical talk; no new mathematical results on the AST will be presented.

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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.

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Based on permanence principles of nonstandard analysis and as a continuation of the papers [1-3], we present some notes and questions on the representation of unlimited natural numbers. As a natural generalization, let $A$ be an unlimited $m$ by $n$ matrix with integer entries (i.e one of its integer entries is unlimited). Here we prove that every unlimited matrix $A$ with integer entries can be written as the sum of a limited matrix S with integer entries and the product of two unlimited matrices $W_1$ and $W_2$ with integer entries, that is, $A = S + W_1 \cdot W_2$. For further research, we propose several matrix representation forms.

Finally, we consider the numbers of the form $z = a+bi$ where $a$,$b$ are integers, which are called Gaussian integers. In the case when $a$ or $b$ is unlimited, the number $z = a+bi$ is said to be unlimited. Also, some notes on the representation of unlimited Gaussian integers are given.

[1] A. Boudaoud, La conjecture de Dickson et classes particulière d'entiers, Ann. Math. Blaise Pascal. 13 (2006), 103-109.
[2] A. Boudaoud and D. Bellaouar, Representation of integers: A nonclassical point of view, J. Log. Anal. 12:4 (2020) 1-31.
[3] K. Hrbacek, On Factoring of unlimited integers, J. Log. Anal. 12:5 (2020) 1-6.

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The ordinal $\Theta$ has lots of interesting results in the context of $L(\mathbb R)$. Here, we try to find an analogue of $\Theta$ for the Chang model, and see what assumptions about it are natural. These assumptions come out of the process of forcing more dependent choice over the Chang model.

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Satisfaction classes are subsets of models of Peano arithmetic which satisfy Tarski's compositional clauses. Alternatively, we can view satisfaction or truth classes as the extension of a fresh predicate T(x) (the theory in which compositional clauses are viewed as axioms is called CT^-).

It is easy to see that CT^- extended with a full induction scheme is not conservative over PA, since it can prove, for instance, the uniform reflection over arithmetic. By a nontrivial argument of Kotlarski, Krajewski, and Lachlan, the sole compositional axioms of CT^- in fact form a conservative extension of PA. Moreover, in order to obtain non-conservativity it is enough to add induction axioms for the Delta_0 formulae containing the truth predicate.

Answering a question of Kaye, we will show that the theory of compositional truth, CT^- with the full collection scheme is a conservative extension of Peano Arithmetic. Following the initial suggestion of Kaye, we will in fact show that any countable recursively saturated model M of PA has an elementary omega_1-like end extension M' such that M' carries a full satisfaction class.

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It has been known since the 1980s that Vopěnka's Principle (VP) is equivalent to certain statements about logics, e.g. to the schema 'Every logic has a compactness cardinal.' On the other hand, it was only recently shown by Trevor Wilson that a related statement statement called Weak Vopěnka's Principle (WVP) is strictly weaker than VP. In fact, Joan Bagaria and Wilson showed that WVP is equivalent to the existence of $\Pi_n$-strong cardinals for all natural numbers $n$. We generalize logical characterizations of strong cardinals to achieve a characterization of $\Pi_n$-strong cardinals and therefore of WVP in terms of properties of strong logics. This is partly joint work with Will Boney and partly with Trevor Wilson.

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The theory of countable Borel equivalence relations analyzes the actions of countable groups on Polish spaces. The main question studied is how much information is encoded by the corresponding orbit space. The amount of encoded information reflects the extent to which the action is rigid.

In this talk we will discuss rigidity results for the action of the group of rational rotations. In particular we will analyze the rotation equivalence on spheres in higher dimension. This is connected to superrigidity results of Margulis and to Zimmer’s program about the actions of discrete subgroups of Lie groups on manifolds.

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Satisfaction classes are subsets of models of Peano arithmetic which satisfy Tarski's compositional clauses. Alternatively, we can view satisfaction or truth classes as the extension of a fresh predicate T(x) (the theory in which compositional clauses are viewed as axioms is called CT^-).

It is easy to see that CT^- extended with a full induction scheme is not conservative over PA, since it can prove, for instance, the uniform reflection over arithmetic. By a nontrivial argument of Kotlarski, Krajewski, and Lachlan, the sole compositional axioms of CT^- in fact form a conservative extension of PA. Moreover, in order to obtain non-conservativity it is enough to add induction axioms for the Delta_0 formulae containing the truth predicate.

Answering a question of Kaye, we will show that the theory of compositional truth, CT^- with the full collection scheme is a conservative extension of Peano Arithmetic. Following the initial suggestion of Kaye, we will in fact show that any countable recursively saturated model M of PA has an elementary omega_1-like end extension M' such that M' carries a full satisfaction class.

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Second-order arithmetic has two types of objects: numbers and sets of numbers, which we think of as the reals. The second-order arithmetic framework has been used successfully to investigate what kinds of real numbers need to exist to prove various significant results in analysis. One of the strongest second-order arithmetic axiomatizations is the theory ${\rm Z}_2$ consisting of the axioms ${\rm PA}$ (for numbers), the set induction axiom, and comprehension for all second-order formulas with set parameters. How significant is the inclusion of set parameters in the comprehension scheme? Let ${\rm Z}_2^{-p}$ be like ${\rm Z}_2$, but where set parameters are not allowed in the comprehension scheme. Harvey Friedman showed that ${\rm Z}_2$ and ${\rm Z}_2^{-p}$ are equiconsistent because parameter-free comprehension suffices to build a model's version of the constructible universe $L$ inside the model and the 'constructible' reals satisfy ${\rm Z}_2$. Kanovei recently showed that models of ${\rm Z}_2^{-p}$ can be very badly behaved, for example, their sets may not even be closed under complement. Kanovei also showed that there can be nicely behaved models of ${\rm Z}_2^{-p}$ in which $\Sigma^1_2$-comprehension (with set parameters) holds. He constructed his model in a forcing extension by a tree iteration of Sacks forcing. In Kanovei's model, $\Sigma^1_4$-comprehension (with set parameters) fails and he asked whether this can be improved to $\Sigma^1_3$-comprehension. In this talk, I will show how to construct a model of $\Sigma^1_2$-comprehension and ${\rm Z}_2^{-p}$ in which $\Sigma^1_3$-comprehension fails. The model will be constructed in a forcing extension by a tree iteration of Jensen's forcing. Jensen's forcing is a sub-poset of Sacks forcing constructed by Jensen to show that it is consistent to have a non-constructible $\Pi^1_2$-definable singleton real (every $\Sigma^1_2$-definable set of reals is constructible by Shoenfield's Absoluteness).

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Cardinal invariants of the continuum are cardinal numbers which, roughly, measure how 'badly' CH fails in various mathematical contexts such as analysis and topology. For instance the cardinal ${\rm add}(\mathcal N)$ is the least $\kappa$ for which there are $\kappa$ many Lebesgue measure zero sets of reals whose union is not measure zero. Classical facts imply $\aleph_1 \leq {\rm add}(\mathcal N) \leq 2^{\aleph_0}$ but the precise value is undetermined in ZFC and depends heavily on the axioms of set theory. Other numbers follow a similar pattern of 'the least size of a set of reals (Borel sets, etc) lacking a classical smallness property'.

The Cichoń diagram displays cardinal invariants related to Lebesgue measure (the null ideal), Baire category (the meager ideal) as well as the bounding and dominating numbers which concern growth rates of functions. Many surprising ZFC-inequalities exist between these cardinals suggesting a rich world living on the reals in various models of set theory. At the combinatorial heart of every proof of a ZFC inequality derives from a Galois-Tukey reduction: the (ZFC-provable) existence of a pair of continuous maps with simple properties that make sense outside of the context of logic and indeed would be sensible to any analyst or topologist.

In this talk we will discuss some recent work in progress on the descriptive complexity of maps witnessing consistent but non-provable implications. We will show using largely computability theoretic methods that in Gödel's constructible universe there are low level projective reductions between any two cardinal invariants - thus CH holds in a very 'definable' way, while in Solovay's model of 'all sets of reals are Lebesgue measurable' (and therefore the axiom of choice fails) there are no non-ZFC provable implications thus these cardinals are all as different as possible.

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The standard method of producing a model of a forcing axiom from a supercompact cardinal in fact gives a model of an even stronger principle: that for every small name a and every $\Sigma_2$ formula $arphi$ such that $\varphi(a)$ is forceable by and preserved under further forcing in our forcing class, there is a filter $F$ which meets a desired collection of dense sets and also interprets a such that $\varphi(a^F)$ already holds. I will show how to generalize this result to formulas of higher complexity by starting with slightly stronger large cardinal assumptions, then discuss the bounded versions of these enhanced forcing axioms, their relationships to other similar principles, and their consequences.

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Kunen's theorem that there is no elementary embedding from V to V seems to set an upper bound on the hierarchy of large cardinal axioms. Challenging this, Caicedo asked what happens when V is replaced with an inner model M that is very close to V in the sense that M correctly computes the class of cardinals. Assuming the existence of strongly compact cardinals, we show that there is no elementary embedding from such an inner model M into V or from V into M. The former result (M into V) is joint work with Sebastiano Thei. Without strong compactness assumptions, both questions remain open, but we'll discuss some partial results.

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The chain-antichain principle $\mathsf{CAC}$ is a well-known consequence of Ramsey's theorem for pairs and two colours $\mathsf{RT}^2_2$. It says that for every partial order on $\mathbb{N}$ there exists an infinite chain or antichain with respect to this order. Both of these principles are $\Pi^0_3$-conservative over the weak base theory $\mathsf{RCA}^*_0$. Such conservation results usually prompt to ask about lengths of proofs. Kołodziejczyk, Wong and Yokoyama proved that $\mathsf{RT}^2_2$ has a non-elementary speedup over $\mathsf{RCA}^*_0$ for proofs of $\Sigma_1$ sentences. We show that the behaviour of $\mathsf{CAC}$ is the opposite: it can be polynomially simulated by $\mathsf{RCA}^*_0$ with respect to $\Pi^0_3$ sentences. Our argument uses a technique of forcing interpretation developed by Avigad. In the first step we syntactically simulate a construction of a generic computable ultrapower of a model of $\mathsf{RCA}^*_0$. Then we find a generic cut satisfying $\mathsf{CAC}$ inside the ultrapower.

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We discuss Shelah's memory iteration technique and use it to show that the PFA for posets of size $\aleph_1$ is consistent with large continuum. This is joint work with David Aspero.

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Continuous first-order logic is a generalization of discrete first-order logic suited for studying structures with natural underlying metrics, such as operator algebras and $\mathbb{R}$-trees. While many things from discrete model theory generalize directly to continuous model theory, there are also new subtleties, such as the correct notion of 'definability' for subsets of a structure. Definable sets are conventionally taken to be those that admit relative quantification in an appropriate sense. An easy argument then establishes that the union of definable sets is definable, but in general the intersection of definable sets may fail to be. This raises the question of which semilattices arise as the partial order of definable sets in a continuous theory.

After giving an overview of the basic properties of definable sets in continuous logic, we will give a largely visual proof that any finite semilattice (and therefore any finite lattice) is the partial order of definable sets in some superstable continuous first-order theory. We will then discuss a partial extension of this to certain infinite semilattices.

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This is the second part of the talk given by Athar Abdul-Quader (Pathologically definable subsets of models of CT-), however we will make sure to make it self-contained.

The talk is centered around the following problem: when a subset of a countable and recursively saturated model M can be characterized as the set of the lengths of disjunctions on which a satisfaction class behaves correctly? More precisely: let DC(x) denote a sentence in a language of PA with a fresh binary predicate S which says 'For every disjunction d with at most x disjuncts and every assignment a, S(d,a) iff there is a disjunct d' in d such that S(d',a).' We say that X is a DC-set in (M,S) iff X is precisely the set of those numbers a such that (M,S) satisfies DC(a). We ask: given a countable and recursively saturated model M for which subsets X of M we can find a satisfaction class S such that X is a DC-set in (M,S)?

In the talk we study this problem for idempotent disjunctions, that is: disjunctions which repeat the same sentence. Let IDC(x) be DC(x) restricted to such 'idempotent' disjunctions of length x. The following is one of our core results:

Theorem: For an arbitrary countable and recursively saturated model M of PA the following conditions are equivalent:
(a) M is arithmetically saturated
(b) For every cut I in M there is a satisfaction class S such that I is an IDC-set in (M,S).

We study how this result generalizes to other propositional constructions in the place of disjunctions. The talk is based on a joint work with Athar Abdul-Quader presented in this paper from arxiv: arXiv:2303.18069v1 [math.LO] 31 Mar 2023.

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Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernible.

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'Classifying' a natural collection of structures is a common goal in mathematics. Providing a classification can mean different things, e.g., identifying a set of invariants that settle the isomorphism problem or creating a list of all structures of a given kind without repetition of isomorphism type. Here we discuss recent work on classifications of the latter kind from the perspective of computable structure theory. We’ll consider natural classes of computable structures such as vector spaces, equivalence relations, algebraic fields, and trees to better understand the nuances of classification via effective lists and its relationship to other forms of classification in this setting.

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The standard set-theoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege's Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every first-order parametrically definable class $F$ with a set object $\varepsilon F$, in such a way that Basic Law V is fulfilled:
$$\varepsilon F =\varepsilon G\leftrightarrow\forall x\, (Fx\leftrightarrow Gx).$$ Russell's elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski's theorem on the nondefinability of truth as a corollary to Russell's argument. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.

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This is the second part of the talk given by Athar Abdul-Quader (Pathologically definable subsets of models of CT-), however we will make sure to make it self-contained.

The talk is centered around the following problem: when a subset of a countable and recursively saturated model M can be characterized as the set of the lengths of disjunctions on which a satisfaction class behaves correctly? More precisely: let DC(x) denote a sentence in a language of PA with a fresh binary predicate S which says 'For every disjunction d with at most x disjuncts and every assignment a, S(d,a) iff there is a disjunct d' in d such that S(d',a).' We say that X is a DC-set in (M,S) iff X is precisely the set of those numbers a such that (M,S) satisfies DC(a). We ask: given a countable and recursively saturated model M for which subsets X of M we can find a satisfaction class S such that X is a DC-set in (M,S)?

In the talk we study this problem for idempotent disjunctions, that is: disjunctions which repeat the same sentence. Let IDC(x) be DC(x) restricted to such 'idempotent' disjunctions of length x. The following is one of our core results:

Theorem: For an arbitrary countable and recursively saturated model M of PA the following conditions are equivalent:
(a) M is arithmetically saturated
(b) For every cut I in M there is a satisfaction class S such that I is an IDC-set in (M,S).

We study how this result generalizes to other propositional constructions in the place of disjunctions. The talk is based on a joint work with Athar Abdul-Quader presented in this paper from arxiv: arXiv:2303.18069v1 [math.LO] 31 Mar 2023.

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An Abelian lattice-ordered group ($\ell$-group) is an Abelian group with a partial ordering, invariant under translations, that is a lattice ordering. A prototypical example of an $\ell$-group is $C(X)$, the continuous real-valued functions on the topological space $X$ with pointwise operations and ordering. Let $\bf{A}$ be the class of $\ell$-groups, viewed as structures for the first-order language $\mathcal{L}=\{+,-, 0, \wedge, \vee \}$. After giving more background on $\ell$-groups, I will survey what is known about the $\ell$-groups existentially closed (e.c.) in $\bf{A}$, including some new examples I constructed using Fraïssé limits. Then I will discuss some recently published work of Scowcroft concerning the $\ell$-groups e.c. in $\bf{W}^+$, the class of nonzero Archimedean $\ell$-groups with distinguished strong order unit (viewed as structures for $\mathcal{L}_1 = \mathcal{L} \cup \{1 \}$). Building on Scowcroft's results, I will present new axioms for the $\ell$-groups e.c. in $\bf{W}^+$ and show how they allow one to characterize those spaces $X$ for which $(C(X), 1_X)$ is e.c. in $\bf{W}^+$.

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Fixing some countable transitive model $M$ of set theory, we can consider its generic multiverse, the family of all models obtainable from $M$ by taking any sequence of forcing extensions and ground models. There is an attractive similarity between the generic multiverse and the Turing degrees, but the multiverse has the drawback (or feature?) that it contains nonamalgamable models, that is, models with no common upper bound, as was observed by several people, going back to at least Mostowski. In joint work with Hamkins, Klausner, Verner, and Williams in 2019, we studied the order-theoretic properties of the generic multiverse and, among other results, gave a characterization of which partial orders embed nicely into the multiverse. I will present our results in the simplest case of Cohen forcing, as well as existing generalizations to wide forcing, and some new results on non-Cohen ccc forcings.

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