21 citations · 88 across the 30 of their papers we have counts for
31 papers
Short proof that Vopěnka's Principle is symmetrically stable
Asaf Karagila
We show that Vopěnka's Principle is stable under symmetric extensions. Recently, de Groot and Woloszyn [1] proved this fact. We found the proof of this theorem unnecessarily compli…
Comments on Choiceless Chain Conditions
Constance Bromham, Asaf Karagila
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 tuto…
Towards a theory of symmetric extensions
Asaf Karagila, Jonathan Schilhan
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 su…
Intermediate models and Kinna--Wagner Principles
Asaf Karagila, Jonathan Schilhan
Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write to denote that there is some for which…
Critical embeddings
Asaf Karagila, Jiachen Yuan
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…
The first measurable can be the first inaccessible cardinal
Moti Gitik, Yair Hayut, Asaf Karagila
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 . In this paper we…
Which Pairs of Cardinals Can Be Hartogs and Lindenbaum Numbers of a Set?
Asaf Karagila, Calliope Ryan-Smith
Given any , we construct a symmetric extension in which there is a set such that and . Consequently, we show that "For all pa…
Hilbert Spaces Without Countable AC
Bruce Blackadar, Ilijas Farah, Asaf Karagila
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 wha…
Geometric condition for Dependent Choice
Asaf Karagila, Jonathan Schilhan
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 we…
Small measurable cardinals
Yair Hayut, Asaf Karagila
We continue the work from [8] and make a small -- but significant -- improvement to the definition of -decomposable system. This provides us with a better lifting of elementary…
Sequential and distributive forcings without choice
Asaf Karagila, Jonathan Schilhan
In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "-distributive" if and only if it is "-sequential". We show that without the Axiom of Choice…
Choiceless Chain Conditions
Asaf Karagila, Noah Schweber
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…
Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice
Asaf Karagila
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…
Approaching a Bristol model
Asaf Karagila
The Bristol model is an inner model of , where is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristo…
Methods in Higher Forcing Axioms (Workshop Notes)
David Asperó, Asaf Karagila
Methods of Higher Forcing Axioms was a small workshop in Norwich, taking place between 10--12 of September, 2019. The goal was to encourage future collaborations, and create more f…
The -Strongly Proper Forcing Axiom
David Asperó, Sean Cox, Asaf Karagila +1
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 to get the cons…
Iterated failures of choice
Asaf Karagila
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…
How to have more things by forgetting how to count them
Asaf Karagila, Philipp Schlicht
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 C…
Realizing realizability results with classical constructions
Asaf Karagila
J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settin…
Kelley-Morse set theory does not prove the class Fodor principle
Victoria Gitman, Joel David Hamkins, Asaf Karagila
We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function defined on a stationary class $…
The Morris model
Asaf Karagila
Douglass B. Morris announced in 1970 that it is consistent with ZF that "For every , there exists a set which is the countable union of countable sets, and $\mathcal P(A_α…
Preserving Dependent Choice
Asaf Karagila
We prove some general theorems for preserving Dependent Choice when taking symmetric extensions, some of which are unwritten folklore results. We apply these to various constructio…
Dependent Choice, Properness, and Generic Absoluteness
David Asperó, Asaf Karagila
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 p…
Critical Cardinals
Yair Hayut, Asaf Karagila
We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equiv…