Set Theory Seminar

Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results. The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH). These flavors are apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda). They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa. In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage. These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly. They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.

Gödel–Bernays set theory GB and Kelley–Morse set theory KM are two formal theories for second-order set theory, allowing both sets and proper classes as objects. GB is the weaker of the two theories, being conservative over ZF, while KM is the stronger. Set theorists have used KM in applications where GB is not strong enough; for instance, Kunen formulated his celebrated inconsistency result in the context of KM, as KM has the resources to directly allow talk of elementary embeddings of the universe of sets. But weaker theories than KM suffice for many of these applications. Between GB and KM there is a hierarchy of intermediate theories based upon restricting the logical complexity allowed in the comprehension axiom.

In this talk I will present a hierarchy of second-order set theories which refines the comprehension-based hierarchy. This hierarchy is based upon transfinite recursion principles, ordered first by the logical complexity of the properties allowed and second by the lengths of well-orders on which we may carry out the recursions. Theories in this hierarchy are separated in terms of consistency strength. The substantive new result to establish this hierarchy is the following: Let $k$ be a natural number. Suppose $(M,\mathcal{X})$ satisfies GB and that $\Gamma \in \mathcal{X}$ is a class well-order which is closed under addition. In case $k = 0$ further assume $\Gamma \ge \omega^\omega$. Then, if $(M,\mathcal{X})$ satisfies $\Pi^1_k$-Transfinite Recursion for recursions along $\Gamma$, there is $\mathcal{Y} \subseteq \mathcal{X}$ coded in $\mathcal{X}$ so that $(M,\mathcal{Y})$ satisfies GB plus the principle of $\Pi^1_k$-Transfinite Recursion for recursions along well-orders of length $< \Gamma$.

From non-trivial elementary embeddings $j_{1}\dots j_{r}:V_{\lambda}\rightarrow V_{\lambda}$, we obtain a sequence of polynomials $(p_{n}(x_{1},\dots,x_{r}))_{n\in\omega}$ that satisfies the infinite product $$\prod_{k=0}^{\infty}p_{k}(x_{1},\dots,x_{r})=\frac{1}{1-(x_{1}+\dots+x_{r})}.$$ From this infinite product, we deduce lower bounds of the cardinality of $|\langle j_{1},...,j_{r}\rangle/\equiv^{\alpha}|$ using analysis and analytic number theoretic techniques. Computer calculations that search for Laver-like algebras give some empirical evidence that these lower bounds cannot be greatly improved.

In a recent exciting paper Learnability can be undecidable by S. Ben-David et. al. published in Nature Machine Intelligence the authors argue that certain abstract learnability questions are undecidable by ZFC axioms. The general learning problem considered there is to find a way of choosing a finite set that maximizes a particular expected value (within a certain range of error) with an obstacle that the probability distribution is unknown, or more formally:
given a family of functions $\mathcal{F}$ from some fixed domain $X$ to the real numbers and an unknown probability distribution $\mu$ over $X$, find, based on a finite sample generated by $\mu$, a function in $\mathcal{F}$ whose expectation with respect to $\mu$ is (close to) maximal.

The authors then provide a translation from this statistical framework to infinite comibnatorics: namely, they prove that existence of certain learning functions corresponding to the problem above (the so-called estimating the maximum learners, or EMX-learners) translates into the existence of the so-called monotone compression schemes, which in turn is equivalent to a statement in cardinal arithmetic that is indeed independent of ZFC. Specifically, let $X$ be an infinite set, $Fin(X)$ be the family of its finite subsets, and let $m > k$ be natural nubers. A monotone compressions scheme for $(X, m, k)$ is a function $f: [X]^k \rightarrow Fin(X)$ such that $$\forall A \in [X]^m \exists B \in [X]^k \: (B \subseteq A \subseteq f(B)).$$

The main result of the paper then is that there exists a monotone compressions scheme for $([0,1], m+1, m)$ for some $m$ if and only if $2^{\aleph_0} < \aleph_\omega$.

K.P. Hart immediately observed that the main combinatorial content of the results in the paper is related to Kuratowski's theorem on decompositions of finite powers of sets and that the monotone compression functions on the unit interval cannot, in a certain sense, be constructive or descriptively nice - namely, they cannot be Borel measurable. During the talk I will introduce the subject of the paper in question, and present the set-theoretic aspects of the main results.

Can we define powerset(kappa) using the uB sets? Woodin showed that this cannot be done once there are supercompacts in the universe. Trang and the speaker recently showed that this cannot be done even much below than supercompacts, at the level of a Woodin cardinal that is a limit of Woodin cardinals. We will then discuss what powerset(kappa) could be.

I will discuss realizing Easton functions in the presence of non-supercompact strongly compact cardinals and connections with indestructibility. This is joint work with Stamatis Dimopoulos and Toshimichi Usuba.

I shall discuss recent joint work with Victoria Gitman and Asaf Karagila, in which we proved that Kelley-Morse set theory (which includes the global choice principle) does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to {\rm Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq{\rm Ord}$ that there is a class function $F:{\rm Ord}\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a sequence of classes $A_n$, each containing a class club, but the intersection of all $A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.

Let $M$ be an inner model. $M$`s bedrock $B$, provided it exist, is the $\subseteq$-least inner model such that $M = B[g]$ for some set generic filter $g$.

We will discuss the interplay of (internal) iterability, strong cardinals and the existence of bedrocks in the case that $M$ is an extender model.

Sun proved that, assuming $\kappa$ is a weakly compact cardinal, a subset $W\subseteq\kappa$ is $\Pi^1_1$-indescribable (or equivalently weakly compact) if and only if $W\cap C\neq\emptyset$ for every $1$-club $C\subseteq \kappa$; here, a set $C\subseteq\kappa$ is $1$-club if and only if $C\in\textrm{NS}_\kappa^+$ and whenever $\alpha<\kappa$ is inaccessible and $C\cap \alpha\in\textrm{NS}_\alpha^+$ then $\alpha\in C$. We generalize Sun's characterization to $\Pi^1_1$-indescribable subsets of $P_\kappa\lambda$, which were first defined by Baumgartner by using a natural two-cardinal version of the cumulative hierarchy. Using the minimal strongly normal ideal of non-strongly stationary sets on $P_\kappa\lambda$, which is distinct from $\textrm{NS}_{\kappa,\lambda}$ when $\kappa$ is inaccessible, we formulate a notion of $1$-club subset of $P_\kappa\lambda$ and prove that a set $W\subseteq P_\kappa\lambda$ is $\Pi^1_1$-indescribable if and only if $W\cap C\neq\emptyset$ for every $1$-club $C\subseteq P_\kappa\lambda$. We also show that elementary embeddings considered by Schanker witnessing near supercompactness lead to the definition of a normal ideal on $P_\kappa\lambda$, and indeed, this ideal is equal to Baumgartner's ideal of non-$\Pi^1_1$-indescribable subsets of $P_\kappa\lambda$. Additionally, we will discuss an application which answers a question of Cox-Lücke.

The C-sequence number of an uncountable regular cardinal $\kappa$ is a cardinal invariant that provides a measure of the amount of compactness that holds at $\kappa$. We will begin this talk by introducing the C-sequence number and proving some of its basic properties, linking it to familiar notions including large cardinals and square principles. We will then outline a number of consistency results regarding the C-sequence number at inaccessible cardinals and successors of singular cardinals. We will end by exploring how the C-sequence number interacts with the existence of complicated colorings and the infinite productivity of the $\kappa$-Knaster condition. This is joint work with Assaf Rinot.

A virtual large cardinal is (usually) the critical point of a generic elementary embedding from a rank-initial segment of the universe into a transitive $M\subset V$, as introduced by Gitman and Schindler (2018). A notable feature is that all virtual large cardinals are consistent with $V=L$, and they've proven useful in characterising several properties in descriptive set theory. We'll work with the virtually $\theta$-measurable, $\theta$-strong and $\theta$-supercompact cardinals, where the $\theta$ in particular indicates that the generic embeddings have $H_\theta^V$ as domain, and investigate how these level-by-level virtual large cardinals relate both to each other and to the existence of winning strategies in certain games. This is work in progress and joint with Philipp Schlicht.

Several years ago I introduced the Diagonal Reflection Principle (DRP), a maximal form of simultaneous stationary reflection. Roughly, DRP asserts that for all regular $\theta \ge \omega_2$, there are stationarily many sets $W$ of size $\omega_1$ such that every stationary element of $W$ reflects to $W$ (i.e. if $S \subset [\theta]^\omega$ is stationary and $S \in W$, then $S \cap [W \cap \theta]^\omega$ is stationary). In that paper I showed that that $\text{MM}^{+\omega_1}$--even just the '$+\omega_1$' version of the forcing axiom for $\sigma$-closed forcings--implies DRP. In this talk I will prove that MM, even its technical strengthening $\text{MM}^{+\omega}$, does NOT imply DRP, contrary to my initial expectations. This is joint work with Hiroshi Sakai.

I will talk about connections between three properties of a forcing class $\Gamma$: the most well-known one is the bounded forcing axiom at $(\omega_1,\lambda)$ for this class, introduced by Goldstern and Shelah for proper forcing. The second property is a two cardinal version of $\Sigma^1_1$ absoluteness, at $(\omega_1,\lambda)$, and the third is the preservation of Aronszajn trees of height $\omega_1$ and width $\lambda$. Under the assumption that $\lambda=\lambda^\omega$ and forcings in $\Gamma$ don't add countable subsets of $\lambda$, I will show that these properties are equivalent. The class of forcing notions satisfying Jensen's property of subcompleteness is a canonical class that does not add reals and it is a corollary of the above result that the bounded forcing axiom for subcomplete forcing at $(\omega_1,2^\omega)$ is equivalent to the preservation of Aronszajn trees of height $\omega_1$ and width $2^\omega$. This generalizes a prior result, obtained jointly with Kaethe Minden. I will discuss some further results, as well as some open questions, on the preservation of Aronszajn trees of height $\omega_1$ and other widths.