Set Theory Seminar

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.

Preprint

Slides

Video

Given a set $S$, an $S$-predictor $\mathcal{P}$ is a function that takes as inputs functions of the form $f:(-\infty,t)\rightarrow S$, where $t\in\mathbb{R}$, and outputs a guess $\mathcal{P}(f)$ for what $f(t)$ 'should be.' An $S$-predictor is good if for all total functions $F:\mathbb{R}\to S$ the set of $t\in\mathbb{R}$ for which the guess $\mathcal{P}(F\upharpoonright(-\infty,t))$ is not equal to $F(t)$ has measure zero. Hardin and Taylor proved that every set $S$ has a good $S$-predictor and they raised various questions asking about the extent to which the prediction $\mathcal{P}(f)$ made by a good predictor might be invariant after precomposing $f$ with various well-behaved functions - this leads to the notion of 'anonymity' of good predictors under various classes of functions. Bajpai and Velleman answered several of Hardin and Taylor's questions and asked: Does there exist, for every set $S$, a good $S$-predictor that is anonymous with respect to the strictly increasing analytic homeomorphisms of $\mathbb{R}$? We provide a consistently negative answer to this question by strengthening a result of Erdős, which states that the Continuum Hypothesis is equivalent to the existence of an uncountable family $F$ of (real or complex) analytic functions, such that $\{f(x):f\in F\}$ is countable for every $x$. We strengthen Erdős' result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. This is joint work with Sean Cox and Kayla Lee.

Video

We consider the following two player game of infinite length: We are given a starting set X, and the players go by the names 'Cut' and 'Choose'. They take turns making moves, and in each step, Cut partitions a given set into two disjoint pieces, starting from the set X in their first move, and then Choose gets to pick one of the pieces, which is then partitioned into two pieces by Cut in their next move etc. In the end, Choose wins in case the intersection of all of their choices has at least two (distinct) elements.

We will investigate some of the properties of this game — in particular, we will discuss some classic results on when it is possible for one of the players to have a strategy for winning the game. We will then continue to discuss some variations of this game and their relevance to set theory — many central set theoretic notions, such as certain large cardinal properties, notions of distributivity, precipitousness and strategic closure were either known or turned out to be closely connected and often equivalent to the (non-)existence of winning strategies in certain cut and choose games.

This is joint work with Philipp Schlicht, Christopher Turner and Philip Welch (all University of Bristol).

Video

A tree of height $\omega_1$ with no cofinal branch is called special if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function $f:T \to \omega$ so that if $f(s) = f(t)$ then $s$ and $t$ are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, $\mathfrak{st}$, the least size of a tree of height $\omega_1$ which is neither non-special nor has a cofinal branch. Classical facts imply that $\mathfrak{st}$ can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that $\mathfrak{st}$ can be larger than $\mathfrak{a}$, $\mathfrak{g}$, and both the left hand side and bottom row of the Cichon diagram. Thus $\mathfrak{st}$ is independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.

Video

An ideal $I$ on $\omega_1$ is $\omega_1$-dense if $(\mathcal{P}(\omega_1)/I)^+$ has a dense subset of size $\omega_1$. We prove, assuming large cardinals, that there is a semiproper forcing $\mathbb{P}$ so that $$V^\mathbb{P}\models`\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}\textrm '.$$ This answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of $\mathbb{P}_{\mathrm{max}}$ in Woodin's axiom $(*)$ by $\mathbb{Q}_{\mathrm{max}}$ results in an axiom $\mathbb{Q}_{\mathrm{max}}-(*)$ which implies $`\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}\textrm '$.
We proceed in three steps: First we define and motivate a new forcing axiom $\mathrm{QM}$ and then modify the Asperó-Schindler proof of $`\mathrm{MM}^{++}\Rightarrow(*)\textrm '$ to show $`\mathrm{QM}\Rightarrow\mathbb{Q}_{\mathrm{max}}-(*)\textrm '$. Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing $\mathrm{QM}$. This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

Video

An ideal $I$ on $\omega_1$ is $\omega_1$-dense if $(\mathcal{P}(\omega_1)/I)^+$ has a dense subset of size $\omega_1$. We prove, assuming large cardinals, that there is a semiproper forcing $\mathbb{P}$ so that $$V^\mathbb{P}\models`\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}\textrm '.$$ This answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of $\mathbb{P}_{\mathrm{max}}$ in Woodin's axiom $(*)$ by $\mathbb{Q}_{\mathrm{max}}$ results in an axiom $\mathbb{Q}_{\mathrm{max}}-(*)$ which implies $`\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}\textrm '$.
We proceed in three steps: First we define and motivate a new forcing axiom $\mathrm{QM}$ and then modify the Asperó-Schindler proof of $`\mathrm{MM}^{++}\Rightarrow(*)\textrm '$ to show $`\mathrm{QM}\Rightarrow\mathbb{Q}_{\mathrm{max}}-(*)\textrm '$. Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing $\mathrm{QM}$. This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

Video

Various results establish deep connections between the existence of large cardinals, regularity properties of strong logics and the validity of set-theoretic reflection principles. In particular, several compactness properties of strong logics were proven to be equivalent to large cardinal axioms. An important example of such an equivalence is given by a theorem of Makowsky that shows that Vopěnka's Principle is equivalent to the existence of strong compactness cardinals for all abstract logics. Motivated by work of Boney, Dimopoulos, Gitman and Magidor, I recently proved an analogous combinatorial characterization of the existence of weak compactness cardinals for all abstract logics that is closely connected to the notion of subtle cardinals, introduced by Kunen and Jensen in their studies of strong diamond principles, and the concept of shrewd cardinals, defined by Rathjen in proof-theoretic work. In my talk, I want to first discuss the details of this characterization and then present connections to recent joint work with Joan Bagaria (Barcelona) on recurring patterns in the large cardinal hierarchy.

Video

For a class ${\mathcal{P}}$ of posets, a cardinal $\kappa$ is said to be generically supercompact by ${\mathcal{P}}$ (or ${\mathcal{P}}$-gen. supercompact for short) if, for any $\lambda\geq\kappa$, there are $P\in{\mathcal{P}}$ such that, for all $({\sf V},P)$-generic $G$ there are $j$, $M\subseteq{\sf V}[G]$ with $j:{{\sf V}}\stackrel{\prec}{\rightarrow}_\kappa{M}$, $j(\kappa)\gt\lambda$, and $j\mbox{''}{\lambda}\in M$.

A cardinal $\kappa$ is Laver-generically supercompact for ${\mathcal{P}}$ (or ${\mathcal{P}}$-Laver-gen. supercompact for short) if, for any $\lambda\geq\kappa$, $P\in{\mathcal{P}}$ and $({\sf V},P)$-generic $G$, there are ${\mathcal{P}}$-name $\dot{Q}$ with $\Vdash_{P}\,''\dot{Q}\in{\mathcal{P}}''$ such that, for all $({\sf V},P*\dot{Q})$-generic $H \supseteq G$, there are $j$, $M\subseteq{\sf V}[H]$ such that $j:{{\sf V}}\stackrel{\prec}{\rightarrow}_\kappa{M}$, $j(\kappa)\gt\lambda$, and $P*\dot{Q}$, $H $, $j\mbox{''}\lambda\in M$.

$\mathcal P$-gen. superhuge, and $\mathcal P$-Laver-gen. superhuge cardinals are defined if the condition $j\mbox{''}\lambda\in M$ is replaced with $j\mbox{''}j(\kappa)\in M$.

Perhaps it is not apparent at first sight in the formulation the definitions above but these notions of generic large cardinals are first-order definable (S.F, and H. Sakai [1]).

While the generic supercompactness does not determine the size of the cardinal. Laver-generic supercompactness determines the size of the cardinal and that of the continuum in most of the natural settings of ${\mathcal{P}}$ (see S.F., A.Ottenbreit Maschio Rodrigues, and H. Sakai [0] for a proof):

(A) If $\kappa$ is ${\mathcal{P}}$-Laver-gen. supercompact for a class ${\mathcal{P}}$ of posets such that (1) all $P\in{\mathcal{P}}$ are $\omega_1$-preserving, (2) all $P\in{\mathcal{P}}$ do not add reals, and (3) there is a $P_1\in{\mathcal{P}}$ which collapses $\omega_2$, then $\kappa=\aleph_2$ and CH holds.

(B) If $\kappa$ is ${\mathcal{P}}$-Laver-gen. supercompact for a class ${\mathcal{P}}$ of posets such that (1) all $P\in{\mathcal{P}}$ are $\omega_1$-preserving, (2)' there is an a $P_0\in{\mathcal{P}}$ which add a real, and (3) there is a $P_1$ which collapses $\omega_2$, then $\kappa=\aleph_2=2^{\aleph_0}$.

(C) If $\kappa$ is ${\mathcal{P}}$-Laver-gen. supercompact for a class ${\mathcal{P}}$ of posets such that (1)' all $P\in{\mathcal{P}}$ preserve cardinals, and (2)' there is a $P_0\in{\mathcal{P}}$ which adds a real, then $\kappa$ is very large and $\kappa\leq 2^{\aleph_0}$.

The case (C) can be still improved ([0]):

(C') If $\kappa$ is tightly ${\mathcal{P}}$-Laver-gen. superhuge for a class ${\mathcal{P}}$ of posets such that (1)' all $P\in{\mathcal{P}}$ preserve cardinals, and (2)' there is a $P_0\in{\mathcal{P}}$ which adds a real, then $\kappa$ is very large and $\kappa=2^{\aleph_0}$.

(A ${\mathcal{P}}$-Laver-gen. superhuge cardinal $\kappa$ is tightly ${\mathcal{P}}$-Laver-gen. superhuge, if $\dot{Q}$ in the definition of Laver-gen. superhugeness can always be chosen to be small enough --- see [0] for a precise definition.)

In this talk, we are going to give a sketch of the proof of definability and discuss about a theorem which assesses the largeness of $\kappa$ in (C) under the additional assumption that elements of ${\mathcal{P}}$ satisfy certain chain conditions.

　[0] S.F., A.Ottenbreit Maschio Rodrigues, and H.Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, II --- reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495--523. https://fuchino.ddo.jp/papers/SDLS-x.pdf
　[1] S.F., and H.Sakai, Generically supercompact cardinals by forcing with chain conditions RIMS Kôkûroku, No.2213 (2022). https://fuchino.ddo.jp/papers/RIMS2021-ccc-gen-supercompact-x.pdf
　[2] S.F., and H.Sakai, The first-order definability of generic large cardinals, to appear. https://fuchino.ddo.jp/papers/definability-of-glc-x.pdf

Video

Jensen constructed in $L$, using $\diamondsuit$, a subposet of the Sacks forcing with the ccc and the property that it adds a unique generic real over $L$ (in contrast to, say, Cohen forcing which adds continuum many generic reals). He used what came to be known as Jensen's forcing to show that, consistently, there can be a $\Pi^1_2$-definable non-constructible real. The 'uniqueness of generic reals' property of Jensen's forcing extends to products of Jensen's forcing and to finite iterations, when forcing over $L$. Indeed, a Jensen-like forcing with the same uniqueness properties can be constructed in any universe with a $\diamondsuit$-sequence. In a joint work with Friedman and Kanovei, we used a tree iteration of Jensen's forcing to construct (in a symmetric submodel of the forcing extension) a model of full second-order arithmetic ${\rm Z}_2$ with the choice scheme in which the dependent choice scheme failed for a $\Pi^1_2$-assertion (this is optimal because ${\rm Z}_2$ with the choice scheme implies dependent choice for $\Sigma^1_2$-assertions). In this talk, I will present a generalization of Jensen's forcing to forcing with perfect $\kappa$-trees for an inaccessible cardinal $\kappa$. I will show that Jensen's forcing at an inaccessible has the same 'uniqueness of generics' properties as Jensen's forcing. One of the goals of this work is to prove an analogue of the second-order arithmetic result for second-order set theory by showing that the dependent choice scheme is independent of the second-order Kelley-Morse set theory with the choice scheme. This is joint work with Sy-David Friedman.

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