Large cardinals are such cardinals whose existence cannot be proved within ZFC, and requires stronger axioms to be added to ZFC, they are often used to measure the consistency strength of a certain statement in the language of set theory.

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Breaking the AC barrier using Kelley-Morse set theory

In her blog post Variants of Kelley-Morse set theory, Prof. Gitman proves that every model of the the common version of Kelley-Morse set theory ($\mathsf{KM}$) is a model of the Wikipedia version of ...
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Why does a nontrivial $V \to V$ have a critical point?

Let $V$ denote the von Neumann universe, and let $j: V \to V$ be a nontrivial (non-identity) elementary embedding. The critical point is the smallest $\kappa$ such that $j(\kappa) > \kappa$. The ...
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Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, ...
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Replacing an ordinal with its cardinality in a partition relation

In The Higher Infinite, Kanamori claims that if $\alpha$ is a cardinal, and $\beta \to (\alpha)^\gamma_\delta$ for some $\beta$, then the least such $\beta$ is a cardinal. I can't seem to think of a ...
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Choice-less Set Theory for Dummies

In almost every graduate set theory text there are some parts about equivalences of $AC$, its consequences, some axioms like $AD$ which imply $\neg AC$, some well-known axiomatic systems which $AC$ ...
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Trouble understanding elementary embedding proofs

Here are two pretty standard results about elementary embeddings that I don't understand. (1) Let $\pi:V \to M$ be a non-trivial elementary embedding with $M$ a transitive class. Let ...
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How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam's original argument about measure theory and measurable cardinals. Here is the result I am looking for: ...
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Choice-like axioms close to AC & rank-into-rank hypothesizes

Consider the following folklore theorems within ZF, Theorem 1: $V=L$ implies "$0^{\sharp}$ doesn't exist". Theorem 2: $V=L[U]$ implies "$0^{\dagger}$ doesn't exist". Theorem 3: $AC$ ...
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A question regarding a paper of M. Magidor [closed]

I am interested in the following paper of M. Magidor: "On the role of supercompact and extendible cardinals in logic", Israel Journal of Mathematics, 05/1971; 10(2): 147-157. The abstract (which I ...
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2 Questions regarding Relative Consistency Proofs

First Question: Let IC be the statement "There is an inaccessible cardinal." I have read that one cannot prove (in ZFC) the relative consistency of ZFC + IC w.r.t. ZFC. i.e. $ Con(ZFC) \rightarrow ...
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On the contradictory nature of large cardinals & choice-like axioms

Compare these two results: Theorem (Scott): $ZFC+V=L\vdash \nexists~\text{Measurable cardinal}$ Theorem (Kunen): $ZFC+AC\vdash \nexists~\text{Reinhardt cardinal}$ Now compare these two ...
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Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?

A bit of philosophy: under the usual definition of the aleph numbers, ZFC proves the sentence "$\aleph_1$ is an ordinal." However, in some sense $\aleph_1$ isn't really an ordinal (in my opinion), ...
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Do large cardinal properties tend to be semiabsolute?

I don't know much about large cardinals; so, I want to get a feeling of the landscape. Hence this question. Definition. Whenever $C$ is a unary predicate in the language of ZFC, let us call $C$ ...
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Inaccessible Cardinals and Grothendieck Universes

I'm trying to prove that the following statements are equivalent: 1.$\forall\alpha\in\mathbb{ON}\ \exists\ \kappa>\alpha$ is an inaccessible cardinal. 2.$\forall x\ \exists\ U\ x\in U$ and $U$ is ...
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Applications of Infinitary Matrices in Set Theory

Matrices have a natural generalization to infinitary context. There are few known applications of such matrices in set theory. For example one may use Ulam matrices to show that real-valued measurable ...
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Is the following notion equivalent to subtle cardinals?

Let $\kappa$ be a regular, uncountable cardinal. We call $\kappa$ $\dagger$ if for every sequence $(A_\alpha \colon \alpha < \kappa)$, $A_\alpha \subseteq \alpha$ and every $\xi_0 < \kappa$ ...
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Is there any inconsistent large cardinal axiom which its inconsistency proof is essentially different from proof of Kunen inconsisteny theorem?

There is a long list of large cardinal axioms. Most of them deemed to be consistent with ZFC but there are also some axioms like existence of Reinhardt or $\omega$-huge cardinals which are natural ...
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How large is an uncountable regular cardinal which is closed under arbitrary fast operators?

Let $Card$ be the proper class of all cardinals, define an infinite set of operators like $\otimes_{n}:(Card\setminus \omega)\times (Card\setminus\{0\})\longrightarrow Card$ which are defined for each ...
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$\kappa$ ineffable $\Rightarrow$ $\kappa$ tree-property

Let $\kappa$ be an uncountable, regular cardinal. We call $\kappa$ ineffable iff for every sequence $(A_\xi \colon \xi < \kappa)$ of subsets $A_\xi \subseteq \xi$ there is a stationary subset $S ...
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Is there any filter space characterization for strong cardinals?

The theory of filter spaces is introduced by Apter, Diprisco, Henle & Zwicker, in their joint paper: Arthur Apter, Carlos Di Prisco, James Henle, and William Zwicker, Filter spaces: towards a ...
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Consistency strength of the “club ultrafilter”

What are the consistency strengths of $$ZF+``\text{The club filter on $\omega_1$ is an ultrafilter}"$$ and $$ZF + DC + ``\text{The club filter on $\omega_1$ is an ultrafilter}"?$$ I know that the ...
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Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
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Is it true that each large cardinal which is not first order expressible has no extender characterization?

It is well-known that Reinhardt cardinal (i.e. The critical point of a non-trivial self-elementary embedding of the universe in $ZF$) is not first order expressible. Does this imply that Reinhardt ...
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ineffable, weakly compact, and ? cardinal

In the online book, page 312, http://projecteuclid.org/download/pdf_1/euclid.pl/1235419485 what cardinal notion do we get, by requireing that X in the bottom of the page, is not only stationary ...
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Kanamori's proof that Vopenka's principle implies the existence of extendibles.

In The Higher Infinite, Kanamori gives a proof that Vopenka's principle implies the existence of (many) extendibles. The relevant theorem is Proposition 24.15 on p.337. Working in a $V_\kappa$ which ...
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Consistency strength of 0-1 valued Borel measures

The following is an overly fancy way of asking a question suggested in Borel Measures: Atoms vs. Point Masses Let $\phi$ be a property that topological spaces can have (such as "compact", "$T_1$", ...
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Is the notion of “small cardinal” well definable?

When we talk about large cardinals, at least for many of them, we usually isolate a particular property expressing their "relative largeness with respect to cardinals below them". For example being ...
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How to make a large cardinal unique?

A general form of questions regarding large cardinals is the following: Let $A(x)$ be the formula asserting "$x$ is a large cardinal of type $A$" then is the following true? ...
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Large Cardinal Consequences of $\kappa$-Suslin Hypothesis

$\kappa$-Suslin Hypothesis ($\kappa$-SH) for the infinite regular cardinal $\kappa$ says that every tree of height $\kappa$ either has a branch of length $\kappa$ or an antichain of cardinality ...
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$0^\sharp$ and the regularity of $\aleph_\omega$

I'm sure I'm missing something trivial, and the most likely of it is that I'm simply wrong on my understanding of the constructible universe $L$, or maybe one of the Wikipedia entries I'm about to ...
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Are there any large cardinal properties of the critical point of a $j: L \longrightarrow L$?

I've recently been thinking a bit about $L$ and $0 \sharp$. As is well known, the existence of $0 \sharp$ is equivalent to the existence of a non-trivial elementary embedding $j: L \longrightarrow ...
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Is this extension of ZFC known to be outright inconsistent?

Is the following first-order theory known to be outright inconsistent? Adjoin to ZFC a unary function $U$ together with the following axioms. If $\alpha$ is an ordinal, then there exists an ...
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Proof of “If $ZFC$ proves there is an inaccessible cardinal, then $ZFC$ is inconsistent”.

Let $I$ be the statement "there is an inaccessible cardinal". I'm aware of two proofs of "If $ZFC\vdash I$ then $ZFC$ is inconsistent". One proof uses the Second Incompleteness Theorem, which I ...
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Is there “intuition” as to why the Continuum Hypothesis is independent of most large cardinal axioms?

I could not find a question that seemed to answer my specific query, despite lots of material on the Continuum Hypothesis (CH) on MSE and MO. If there is already a question on this, I'd greatly ...
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$\{\alpha < \kappa \mid cf(\alpha) = \lambda\}$ is not ineffable

We call a subset $X \subseteq \kappa$ of a regular cardinal $\kappa$ ineffable, iff for every family $(A_\alpha \mid \alpha \in X)$ of subsets $A_\alpha \subseteq \alpha$, there is a stationary set $S ...
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Is consistency strength order dense?

Definition: Let $\sigma$, $\theta$ be two statement in the language of set theory. We say $\sigma <_{c} \theta$ ($\sigma$ is strictly lower than $\theta$ in consistency strength order) if within ...
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A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
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Expository Papers on Extenders

Extenders are discussed in many set theory text books. Here I am looking for some expository "papers" which are focused on this subject and its connection with forcing and large cardinals. More ...
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$\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
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Proof of PFA from Supercompact

In Jech's proof (in chapter 31) that the consistency of a supercompact cardinal implies the consistency of PFA, he needs the following fact: Let $\mathbb{P}_\kappa$ be the countable support forcing ...
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Dual Constructions for Core Models

Roughly speaking, core models are inner models of ZFC which could contain some large cardinals. e.g. $L$ is the smallest core model and it is possible to have inaccessible, Mahlo, weakly compact, ...
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Is the cardinality of the continuum weakly Mahlo?

Is $2^{\aleph_0}$ a weakly Mahlo cardinal? Can it be? That is, are there conditions (such as the negation of the continuum hypothesis or something) under which it is, and other conditions under which ...
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On the number of countable models of complete theories of models of ZFC [duplicate]

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
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how big is the smallest inaccessible cardinal?

I know a cardinal is inaccessible if it is uncountable,regular,strong limit.And we cannot prove its existence in ZFC.but by axiom of choice,every infinite cardinal is an aleph.so can you write the ...
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How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
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Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
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Is there a largest large cardinal?

In ZFC, a cardinal is an isomorphism class of sets. However ZFC doesn't explicitly have classes; NBG, which is a conservative extension of ZFC does. There is no largest cardinal by Cantors Theorem ...
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Let $\Phi$ denote the statement that $\mathrm{GCH}$ holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition. Definition. Call a set of axioms $\Phi$ in ...
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How to think about iterated ultrapowers?

I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, ...
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Model for replacement

If $R_{k} $ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...