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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Universe cardinals and models for ZFC

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal ...
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A question regarding the status of CH in the Gitik model

Consider models of ZF+"Every uncountable cardinal is singular" (eg. Moti Gitik: "All uncountable cardinals can be singular", Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be ...
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Large cardinals and $V$

I am confused by something: $\mu$ is a large cardinal if $\lambda<\mu\Rightarrow 2^{\lambda}<\mu$ and any union of less than $\mu$ sets of size less than $\mu$ is less than $\mu$. On Wikipedia ...
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Question about rank into rank cardinals

I do not understand why λ is smaller than $\kappa_0$ if λ is the supremum of a growing sequence that starts at $\kappa_0$ (see definition below). From cantor's attic ...
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Are the objects associated with large cardinals still sets?

When set theorists speak of large cardinals, are they still referring to the cardinality of some collection? If so, is this object a (hypothetical) set, a proper class, or something else?
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Ineffable Cardinals and Critical Point of Elementary Embeddings

A cardinal $\kappa$ is a ineffable if and only if for all sequences $\langle A_\alpha : \alpha < \kappa\rangle$ such that $A_\alpha \subseteq \alpha$ for all $\alpha < \kappa$, then there exists ...
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Supercompact cardinals and being witnessed by a structure of limited rank

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal ...
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Nonconstructible Subsets of Singular Cardinals

I am trying to understand the proof of Corollary 18.34 in Jech's Set Theory: If $0^\sharp$ does not exist, then if $\kappa$ is a singular cardinal and if there exists a nonconstructible subset of ...
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Removing sets from models of set theory

I have a naive and open-ended question: How can one remove a set from a model of set theory in such a way that the result is again a model of set theory? Directly related: what kinds of sets can ...
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about the smallest $k$ that $V_k$ is a model of ZFC

Let $k$ to be the smallest ordinal that $V_k$ is a model of ZFC. I know that $k$ need not to be inaccessible cardinal,and $k$ has confinality $\omega$. Then how big is $k$? How to write down $k$ in ...
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Inaccessible cardinals

Let $M[G]$ be the full Solovay model, and let HOD be the model of hereditarily ordinal definable sets in $M[G]$. Is it possible for HOD not to have an inaccessible cardinal? Does HOD satisfy GCH?
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Normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$

I am looking at fine normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$ and I have seen that there are two (seemingly) different formulations of normality, in terms of regressive (sometimes also ...
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Levy collapse gone bad

Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following? (1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, ...
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Zero sharp and Kunen inconsistence.

There is some (deep/shallow/interesting/trivial) relation between: (1) $\exists 0^\sharp$ iff $\exists j:L\longrightarrow L$ nontrivial elementary embedding. (2) The Kunen inconsistency $\not\exists ...
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Follow up question for an exam question

Some time ago I got an answer to this old exam question. Now I found the following question, which strikes me as very similar: Let $\kappa$ measurable cardinal, and $\preceq$ a partial ordering on ...
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Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
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109 views

How many weakly compact cardinals can L have?

Q1: Assuming suitable consistency assumptions is the following consistent? $ZFC+V=L+\text{Existence of class many weakly compact cardinals}$ Q2: What is the weakest known consistency assumption for ...
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Is there some natural equivalent of a Turing machine for set theory?

My question is motivated by the fact that (at lest from my biased viewpoint), Turing machines are more related to standard math, particularly PA. But set theory, in most of its versions, seem to go ...
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the smallest inaccessible cardinal

Is there a notation for the smallest inaccessible cardinal? Is the concept even coherent? I read on Wiki that under certain assumptions even Aleph0 is strongly inaccessible. I mean the kind of ...
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Ultraproducts by countably complete ultrafilter

I've recently learned about ultraproducts, but the source I learned from almost immediately after the definitions restricted to talking about countably incomplete ultrafilters. I know that the ...
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A chess problem in Kanamori's “The Higher Infinite”?

Just after Corollary 21.17 (on p289) of Kanamori's The Higher Infinite, he outlines the direction in which he wants to take his discussion of iterated ultrapowers. However, immediately after he ...
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Sequence of Ordinals and Ordinal Definability in Levy Collapse Extensions

Let $\kappa$ be an inaccessible cardinal. Let $G$ be generic for $Col(\omega, < \kappa)$, the Levy Collapse. If $f\in \text{ }^\omega \text{Ord}^{V[G]}$, is $f \in OD_{\text{ ...
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Weakly Normal Ultrafilters

A filter $F$ on $\lambda$ is weakly normal if and only if for all $f : \lambda \rightarrow \lambda$ such that $\{\xi < \lambda : f(\xi) < \xi\} \in F^+$, then there is some $\beta < \lambda$ ...
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Large cardinal and consistency: what are the main results today?

Lately I've been studying (on Jech "Set Theory" and Kanamori "The Higher Infinite") some problems related to the extension of measures. I've never studied set theory before and I'm quite baffled from ...
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Hyper-inaccessible cardinals

in definition of $\alpha$-inaccessible cardinals on Wikipedia, we can read : For example, denote by $ψ_0(λ)$ the λth inaccessible cardinal, then the fixed points of $ψ_0$ are the 1-inaccessible ...
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How $|\mathbb R|$ is not weakly compact

$\mathbb R$, of cardinality $|\mathbb R|=2^{\aleph_0}$ is not weakly compact. So there is a function $f$ from $[{\mathbb R}]^2$ (the subsets of $\mathbb R$ of size 2) to $\{0,1\}$ such that there is ...
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Weakly Compact Cardinal is Strong Limit

A question to Lemma 9.9 in Jech's Set Theory: Every weakly compact cardinal is inaccessible. I am working on the part that for $\kappa$ weakly inaccessible, $\kappa$ is strong limit. Jech writes: ...
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A Lowenheim-Skolem-like Result for Structures of Set Theory

While reading Kanamori's proof (page 59-60 in $\textit{Higher Infinite}$) that $\kappa$ is $\Pi_1^1$-indescribable if and only if $\kappa$ is weakly compact; it appears that he uses the following ...
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If $\kappa$ is measurable, does there exist a normal measure on $\mathcal P_{\kappa}(\kappa)$?

I'm trying to do exercise $10.7$ of Jech's Set Theory: If $\kappa$ is a measurable cardinal, then there exists a normal measure on $\mathcal P_{\kappa}(\kappa)$. For a set $A$, with $|A|\geq ...
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Measurable cardinals as sets

A philosopher said that measurable cardinals are the largest possible sets. Is this true? Are those sets at all? I mean, cardinals measure size of sets and for example $2=\{\{\},\{\{\}\}\}$ but can we ...
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Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
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Tarski's axiom implies a proper class of inaccessible cardinals

I'm trying to prove this theorem, but is it even true? A Tarski's class is a set $T$ such that: For each $y\in T$, ${\cal P}(y)\subseteq T$ and ${\cal P}(y)\in T$, and for each $A\subseteq T$ such ...
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Is it possible to prove $|V_\kappa|=\kappa$ for strongly inaccessible $\kappa$ without AC?

First, let me note that my definition of "strongly inaccessible" is that a nonzero ordinal $\kappa$ is strongly inaccessible if ${\rm cf}(\kappa)=\kappa$ and $\forall\alpha<\kappa, {\cal ...
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Weakly inaccessible cardinals and Discovering Modern Set Theory

So I've been trying to teach myself some set theory and I've come across some exercises in Just and Weese's Discovering Modern Set Theory. To whit: Pg. 180 Definition 20: A cardinal $\kappa$ is ...
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Measurable $\rightarrow$ Mahlo

On page 135 of Jech there is a step that I do not understand in the proof of "measurable implies Mahlo". We have already proved that "measurable implies strong limit". Jech writes "As $\kappa$ is ...
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Is there still any hope that the GCH could be equivalent to some large cardinal axiom?

Is there still any hope that the GCH could be equivalent to some large cardinal axiom? Even a simple yes or not answer will be fine. Thanks!!
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Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?

The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
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Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set

Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
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On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
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Are large cardinals bi-interperable with type theory?

"Sets in Types, Types in Sets" establishes that the calculus of constructions is bi-interpretable with ZFC + infinitely many inaccessibles. Are there any more results like this higher up the large ...
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Is the expressive power of infinitary logic language $L(\infty,\infty)$ larger, the same or smaller than that of ZFC+large cardinal axioms?

In a previous question I learned that the power of statements of the form $\Pi_m^n$ or $\Sigma_m^n$ for arbitrary positive $m$ and $n$, is smaller than that of ZFC. For instance, the GCH cannot be ...
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What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?

Consider the first-order theory obtained by adjoining to ZFC (or, better, Mac Lane set theory) the Grothendieck–Verdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
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A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals

I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$. Apparently ...
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Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
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Weakly Compact Cardinals and the Extension Property

From $\textit{The Higher Infinite}$ by Kanamori, in his proof (page 39 - 41) that $\kappa$ is weakly compact if and only if $\kappa$ satisfies the Extension Property, the proof does something that I ...
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Are all large cardinal axioms expressible in terms of elementary embeddings?

An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ ...
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Did large cardinals exist before 1963?

I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
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Strongly inaccessible cardinals and certain kinds of universes

Write $\mathrm{tc}(*)$ for transitive closure. For all cardinals $\kappa$, let $H(\kappa)$ denote the collection of all sets $X$ such that $|X|<\kappa$, and For all $x \in \mathrm{tc}(X)$, it ...
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Does asserting the existence of large cardinals allow one to prove the existence of new subsets of $\mathbb{R}$?

ZFC proves the existence of certain subsets of $\mathbb{R}$. Suppose ZFC is consistent, and we adjoin a large cardinal axiom to ZFC, obtaining ZFC'. Assume ZFC' is also consistent. It it possible to ...
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How is Mahlo cardinal used?

I would like to know how Mahlo cardinals are used - as such examples may help me understand why they were created and so on.