arXiv subjects

Victoria Gitman

Publications and source records attributed to Victoria Gitman.

At least 19 recordsLinked to original sources

arXiv2026

Class choice and the surprising weakness of Kelley-Morse set theory

Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM includes the global choice principle, for example, (i) KM does not prove the class choice scheme, asserting that whenever every set $x$ admits a class $X$ with $\varphi(x,X)$, then there is a class $Z\subseteq V\times V$ for which $\varphi(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $\varphi$ and even when only a set of indices $x$ are relevant. For closely related reasons, (ii) the theory KM does not prove the {\L}o\'s theorem scheme for internal second-order ultrapowers, even for large cardinal ultrapowers, such as the ultrapower by a normal measure on a measurable cardinal. Indeed, the theory KM itself is not generally preserved by internal ultrapowers. Finally, (iii) KM does not prove that the $\Sigma^1_n$ logical complexity is invariant under first-order quantifiers, even bounded first-order quantifiers. For example, $\forall \alpha{<}\delta\ \psi(\alpha,X)$ is not always provably equivalent to a $\Sigma^1_1$ assertion when $\psi$ is. Nevertheless, these various weaknesses in KM are addressed by augmenting it with the class choice scheme, thereby forming the theory KM+, which we propose as a robust KM alternative for the foundations of second-order set theory.

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arXiv2025

Cardinals of the $P_\kappa(\lambda)$-Filter Games

We investigate forms of filter extension properties in the two-cardinal setting involving filters on $P_\kappa(\lambda)$. We generalize the filter games introduced by Holy and Schlicht in \cite{HolySchlicht:HierarchyRamseyLikeCardinals} to filters on $P_\kappa(\lambda)$ and show that the existence of a winning strategy for Player II in a game of a certain length can be used to characterize several large cardinal notions such as: $\lambda$-super/strongly compact cardinals, $\lambda$-completely ineffable cardinals, nearly $\lambda$-super/strongly compact cardinals, and various notions of generic super and strong compactness. We generalize a result of Nielson from \cite{NielsenWelch:games_and_Ramsey-like_cardinals} connecting the existence of a winning strategy for Player II in a game of finite length and two-cardinal indescribability. We generalize the result of \cite{ForMagZem} to construct a fine $\kappa$-complete precipitous ideal on $P_\kappa(\lambda)$ from a winning strategy for Player II in a game of length $\omega$. Finally, we improve Theorems 1.2 and 1.4 from \cite{ForMagZem} and partially answer questions Q.1 and Q.2 from \cite{ForMagZem}.

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arXiv2024

Upward L\"owenheim-Skolem-Tarski Numbers for Abstract Logics

Galeotti, Khomskii and V\"a\"an\"aanen recently introduced the notion of the upward L\"owenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal $\kappa$ is the \emph{upward L\"owenheim-Skolem-Tarski number} (ULST) of a logic $\mathcal L$ if it is the least cardinal with the property that whenever $M$ is a model of size at least $\kappa$ satisfying a sentence $\varphi$ in $\mathcal L$, then there are arbitrarily large models satisfying $\varphi$ and having $M$ as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and V\"a\"an\"anen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal. We define the strong ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.

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arXiv2023

Between Ramsey and measurable cardinals

We study several intertwined hierarchies between $\kappa$-Ramsey cardinals and measurable cardinals to illuminate the structure of the large cardinal hierarchy in this region. In particular, we study baby versions of measurability introduced by Bovykin and McKenzie and some variants by locating these notions in the large cardinal hierarchy and providing characterisations via filter games. As an application, we determine the theory of the universe up to a measurable cardinal.

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arXiv2022

ZFC without power set II: Reflection strikes back

The theory ZFC implies the scheme that for every cardinal $\delta$ we can make $\delta$ many dependent choices over any definable relation without terminal nodes. Friedman, the first author, and Kanovei constructed a model of ZFC$^-$ (ZFC without power set) with largest cardinal $\omega$ in which this principle fails for $\omega$ many choices. In this article we study failures of dependent choice principles over ZFC$^-$ by considering the notion of big proper classes. A proper class is said to be big if it surjects onto every non-zero ordinal. We shall see that if one assumes the scheme of dependent choices of any arbitrary set length then every proper class is indeed big. However, by building on work of Zarach, we provide a general framework for separating dependent choice schemes of various lengths by producing models of ZFC$^-$ with proper classes that are not big. Using a similar idea, we then extend the earlier result by producing a model of ZFC$^-$ in which there are unboundedly many cardinals but the scheme of dependent choices of length $\omega$ still fails. Finally, the second author has proven that a model of ZFC$^-$ cannot have a non-trivial, cofinal, elementary self-embedding for which the von-Neumann hierarchy exists up to its critical point. We answer a related question posed by the second author by showing that the existence of such an embedding need not imply the existence of any non-trivial fragment of the von-Neumann hierarchy. In particular, that in such a situation $\mathcal{P}(\omega)$ can be a proper class.

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arXiv2022

Model Theoretic Characterizations of Large Cardinals Revisited

In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.

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arXiv2021

The Virtual Large Cardinal Hierarchy

We continue the study of the virtual large cardinal hierarchy by analysing virtual versions of superstrong, Woodin, and Berkeley cardinals. Gitman and Schindler showed that virtualizations of strong and supercompact cardinals yield the same large cardinal notion. We provide various equivalent characterizations of virtually Woodin cardinals, including showing that On is virtually Woodin if and only if for every class A, there is a proper class of virtually A-extendible cardinals. We introduce the virtual Vopenka principle for finite languages and show that it is not equivalent to the virtual Vopenka principle (although the two principles are equiconsistent), but is equivalent to the assertion that On is virtually pre-Woodin, a weakening of virtually Woodin, which is equivalent to having for every class A, a weakly virtually A-extendible cardinal. We show that if there are no virtually Berkeley cardinals, then On is virtually Woodin if and only if On is virtually pre-Woodin (if and only if the virtual Vopenka principle for finite languages holds). In particular, if the virtual Vopenka principle holds and On is not Mahlo, then On is not virtually Woodin, and hence there is a virtually Berkeley cardinal.

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arXiv2019

Structural Properties of the Stable Core

The stable core, an inner model of the form $\langle L[S],\in, S\rangle$ for a simply definable predicate $S$, was introduced by the first author in [Fri12], where he showed that $V$ is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname{GCH}$ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in $V$, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots, C_n]$ for specially nested class clubs $C_1, \dots, C_n$, like those arising in the stable core, generalizing recent results of Welch [Wel19].

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arXiv2019

Kelley-Morse set theory does not prove the class Fodor principle

We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\text{Ord}$ that there is a class function $F:\text{Ord}\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteq\omega\times\text{Ord}$, such that each section $A_n=\{\alpha\mid (n,\alpha)\in A\}$ contains a class club, but $\bigcap_n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.

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arXiv2019

Forcing a $\square(\kappa)$-like principle to hold at a weakly compact cardinal

Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of $\Pi^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. By replacing clubs with $n$-clubs in the definition of $\Box(\kappa)$, one obtains a $\Box(\kappa)$-like principle $\Box_n(\kappa)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(\kappa)$ is consistent with the $\Pi^1_n$-indescribability of $\kappa$ but inconsistent with the $\Pi^1_{n+1}$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-weakly compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-weakly compact, $\Box_1(\kappa)$ holds and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable cardinal, it is consistent that $\kappa$ is $\kappa^+$-weakly compact, every weakly compact subset of $\kappa$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta<\kappa$ for which both $S^0\cap\beta$ and $S^1\cap\beta$ are weakly compact.

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arXiv2018

A model of second-order arithmetic satisfying AC but not DC

We show that there is a $β$-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a $Π^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ${\rm ZFC}^-$. This work is a rediscovery by the first two authors of a result obtained by the third author.

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arXiv2018

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a $Δ_2$-definable class containing no regular cardinals. In such a model, there can be no $Σ_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

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arXiv2017

The exact strength of the class forcing theorem

The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the corresponding forcing extensions are forced and forced statements are true -- is equivalent over G\"odel-Bernays set theory GBC to the principle of elementary transfinite recursion $\text{ETR}_{\text{Ord}}$ for class recursions of length $\text{Ord}$. It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal{L}_{\text{Ord},\omega}(\in,A)$, allowing any class parameter $A$; to the existence of truth predicates for the language $\mathcal{L}_{\text{Ord},\text{Ord}}(\in,A)$; to the existence of $\text{Ord}$-iterated truth predicates for first-order set theory $\mathcal{L}_{\omega,\omega}(\in,A)$; to the assertion that every separative class partial order $\mathbb{P}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text{Ord}+1$. Unlike set forcing, if every class forcing notion $\mathbb{P}$ has a forcing relation merely for atomic formulas, then every such $\mathbb{P}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between GBC and Kelley-Morse set theory KM.

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arXiv2016

A Mitchell-like order for Ramsey and Ramsey-like cardinals

Smallish large cardinals $\kappa$ are often characterized by the existence of a collection of filters on $\kappa$, each of which is an ultrafilter on the subsets of $\kappa$ of some transitive $\mathrm{ZFC}^-$-model of size $ \kappa$. We introduce a Mitchell-like order for Ramsey and Ramsey-like cardinals, ordering such collections of small filters. We show that the Mitchell-like order and the resulting notion of rank have all the desirable properties of the Mitchell order on normal measures on a measurable cardinal. The Mitchell-like order behaves robustly with respect to forcing constructions. We show that extensions with cover and approximation properties cannot increase the rank of a Ramsey or Ramsey-like cardinal. We use the results about extensions with cover and approximation properties together with recently developed techniques about soft killing of large-cardinal degrees by forcing to softly kill the ranks of Ramsey and Ramsey-like cardinals.

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arXiv2016

Open determinacy for class games

The principle of open determinacy for class games---two-player games of perfect information with plays of length $ω$, where the moves are chosen from a possibly proper class, such as games on the ordinals---is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth and indeed a transfinite tower of truth predicates for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC+$Π^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.

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arXiv2015

What is the theory ZFC without power set?

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC- in which $ω_1$ is singular, in which every set of reals is countable, yet $ω_1$ exists, in which there are sets of reals of every size $\aleph_n$, but none of size $\aleph_ω$, and therefore, in which the collection axiom sceme fails; there are models of ZFC- for which the Los theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC- for which the Gaifman theorem fails, in that there is an embedding $j:M\to N$ of ZFC- models that is $Σ_1$-elementary and cofinal, but not elementary; there are elementary embeddings $j:M\to N$ of ZFC- models whose cofinal restriction $j:M\to \bigcup j``M$ is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC- to a $Σ_1$-formula or a $Π_1$-formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory $ZFC^-$, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.

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arXiv2015

Indestructibility properties of remarkable cardinals

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the indestructibility properties of remarkable cardinals. We show that if $κ$ is remarkable, then there is a forcing extension in which the remarkability of $κ$ becomes indestructible by all $\ltκ$-closed $\leqκ$-distributive forcing and all two-step iterations of the form ${\rm Add}(κ,θ)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\ltκ$-closed and $\leqκ$-distributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the ${\rm GCH}$.

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arXiv2015

Ehrenfeucht's lemma in set theory

Ehrenfeucht's lemma (1973) asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht's lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht's lemma holds fully for models of set theory satisfying $V=HOD$. We show that the lemma can fail, however, in models of set theory with $V\neq HOD$, and it necessarily fails in the forcing extension to add a generic Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht's lemma, namely, the principles of the form $EL(A,P,Q)$, which asserts that whenever an object $b$ is definable from some $a\in A$ using parameters in $P$, with $b\neq a$, then the types of $a$ and $b$ over $Q$ are different. We also consider various analogues of Ehrenfeucht's lemma obtained by using algebraicity in place of definability, where a set $b$ is algebraic in $a$ if it is a member of a finite set definable from $a$ (as in Hamkins, Leahy arXiv:1305.5953). Ehrenfeucht's lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using similar analysis, we answer two open questions posed by Hamkins and Leahy, by showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.

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