arXiv subjects

Asaf Karagila

Publications and source records attributed to Asaf Karagila.

At least 19 recordsLinked to original sources

arXiv2026

Comments on Choiceless Chain Conditions

In [10] it was suggested that the countable chain condition should be defined in the absence of choice as "every predense set contains a countable predense subset". During his tutorial at the "120 Years of Choice" conference in 2024, Philipp Schlicht remarked that this definition does not have an iteration theorem in $\mathsf{ZF}$. We provide a proof of this claim. Specifically, we show that if every finite support iteration of ccc forcings is again ccc, then the Axiom of Choice for countable families of countable sets must hold. However, we prove that under the assumption of the Principle of Dependent Choice, the finite support iteration of ccc forcings is again ccc. This requires correcting some issues with the definition of properness (and therefore Mekler's definition of ccc) from [1]. We then study Knaster principles and their relations to this version of ccc.

math.LO↗

arXiv2026

Critical embeddings

Hayut and the first author isolated the notion of a critical cardinal in [1]. In this work, we answer several questions raised in the original paper. We show that it is consistent for a critical cardinal not to have any ultrapower elementary embeddings, as well as that it is consistent that no target model is closed. We also prove that if $κ$ is a critical point by any ultrapower embedding, then it is the critical point of an ultrapower embedding by a normal measure. The paper concludes by presenting several open questions of interest in the study of critical cardinals.

math.LO↗

arXiv2026

Towards a theory of symmetric extensions

The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique remained fairly limited compared to the theory of forcing. Whereas forcing developed products and iterations, no serious attempts at developing any general framework for iterating symmetric extensions were presented before [10], where only finite support iterations are treated. In this paper we develop the theory of symmetric extensions including different types of iterations, quotients, equivalents, and the structural results that can be described in this language. In particular, we give a modern exposition to some of the important theorems of Grigorieff [3], study Kinna--Wagner Principles in symmetric extensions, and show that it is provable from $\mathsf{ZF}$ that every set lies in a symmetric extension of $\operatorname{HOD}$.

math.LO↗

arXiv2026

Small measurable cardinals

We continue the work from [8] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least $2$.

math.LO↗

arXiv2025

Approaching a Bristol model

The Bristol model is an inner model of $L[c]$, where $c$ is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristol, but was only written in detail by the author in [8]. This paper is a guide for those who want to get a broader view of the construction. We try to provide more intuition that might serve as a jumping board for those interested in this construction and in odd models of $\mathsf{ZF}$. We also correct a few minor issues in the original paper, as well as prove new results. For example, that the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and the ground model is always definable in its Bristol extensions. In addition to this we include a discussion on Kinna--Wagner Principles, which we think may play an important role in understanding the generic multiverse in $\mathsf{ZF}$.

math.LO↗

arXiv2025

Intermediate models and Kinna--Wagner Principles

Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $\mathsf{KWP}$ to denote that there is some $α$ for which this holds. The Kinna--Wagner Conjecture, formulated by the first author in [9], states that if $V$ is a model of $\mathsf{ZF}+\mathsf{KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an intermediate model of $\mathsf{ZF}$, that is $V\subseteq W\subseteq V[G]$, then $W=V(x)$ for some $x$ if and only if $W$ satisfies $\mathsf{KWP}$. In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna--Wagner Principles and new results about Kinna--Wagner Principles in the multiverse of sets.

math.LO↗

arXiv2024

The first measurable can be the first inaccessible cardinal

In [8] the second and third authors showed that if the least inaccessible cardinal is the least measurable cardinal, then there is an inner model with $o(κ)\geq2$. In this paper we improve this to $o(κ)\geqκ+1$ and show that if $κ$ is a $κ^{++}$-supercompact cardinal, then there is a symmetric extension in which it is the least inaccessible and the least measurable cardinal.

math.LO↗

arXiv2024

Which Pairs of Cardinals Can Be Hartogs and Lindenbaum Numbers of a Set?

Given any $λ\leqκ$, we construct a symmetric extension in which there is a set $X$ such that $\aleph(X)=λ$ and $\aleph^*(X)=κ$. Consequently, we show that $\mathsf{ZF}+$"For all pairs of infinite cardinals $λ\leqκ$ there is a set $X$ such that $\aleph(X)=λ\leqκ=\aleph^*(X)$" is consistent.

math.LO↗

arXiv2024

The $κ$-Strongly Proper Forcing Axiom

We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $θ>κ$ to get the consistency of the forcing axiom for $κ$-strongly proper forcing notions which are also $κ$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $κ$-strongly proper forcings. We also produce a model of this forcing axiom with $2^κ$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom.

math.LO↗

arXiv2023

Hilbert Spaces Without Countable AC

This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.

math.LO↗

arXiv2023

Geometric condition for Dependent Choice

We provide a geometric condition which characterises when the Principle of Dependent Choice holds in a Fraenkel--Mostowski--Specker permutation model. This condition is a slight weakening of requiring the filter of groups to be closed under countable intersections. We show that this condition holds nontrivially in a new permutation model we call "the nowhere dense model" and we study its extensions to uncountable cardinals as well.

math.LO↗

arXiv2022

Sequential and distributive forcings without choice

In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$κ$-distributive" if and only if it is "$κ$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for $κ$. Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that while a $κ$-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $κ$. On the other hand, a $κ$-sequential can violate the Axiom of Choice for countable families. We also provide a condition of "quasiproperness" which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential.

math.LO↗

arXiv2022

Choiceless Chain Conditions

Chain conditions are one of the major tools used in the theory of forcing. We say that a partial order has the countable chain condition if every antichain (in the sense of forcing) is countable. Without the axiom of choice antichains tend to be of little use, for various reasons, and in this short note we study a number of conditions which in ZFC are equivalent to the countable chain condition.

math.LO↗

arXiv2021

Iterated failures of choice

We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of counterexamples. For example, the rational numbers have a proper class of non-isomorphic algebraic closures, every partial order embeds into the cardinals of the model, every set is the image of a Dedekind-finite set, every weak choice axiom of the form $\mathsf{AC}_X^Y$ fails with a proper class of counterexamples, every field has a vector space with two linearly independent vectors but without endomorphisms that are not scalar multiplication, etc.

math.LO↗

arXiv2020

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 $λ$ with $ω\leqλ\leq\text{Ord}$ that there is a class function $F:\text{Ord}\toλ$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteqω\times\text{Ord}$, such that each section $A_n=\{α\mid (n,α)\in A\}$ 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 $σ$-closed.

math.LO↗

arXiv2020

Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice

This text is meant for analysis students who want to learn more about the effects of the axiom of choice on functional analysis, and the things that may go wrong in its absence. As this is a text aimed for analysis students, we will not focus on the set theoretic proofs or dwell on any particular set theoretic assumptions that are needed in order to prove certain results.

math.FA↗

arXiv2020

How to have more things by forgetting how to count them

Cohen's first model is a model of Zermelo--Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal $κ$. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding $κ$ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set $A$ which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of "Adding a Cohen subset" by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, $2^A$ is extremally disconnected, or $[A]^{<ω}$ is Dedekind-finite.

math.LO↗

arXiv2020

Dependent Choice, Properness, and Generic Absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to DC-preserving symmetric submodels of forcing extensions. Hence, ZF+DC not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF+DC and ZFC. Our results confirm ZF+DC as a natural foundation for a significant portion of "classical mathematics" and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

math.LO↗