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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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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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.
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$V_k$ being a model of ZFC whenever $k$ is strongly inaccessible

ZFC implies that the $V_k$ is a model of ZFC whenever $k$ is strongly inaccessible.. So if $k$ is weakly inaccessible, it can't be a model of ZFC? Why is it like this? And ZF implies that ...
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Does TG prove that ZFC2 has a model?

Does TG (Tarski-Grothendieck Set Theory) prove that ZFC2 (second-order ZFC) has a model? Does it at least prove the consistency of ZFC2?
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Effective Well ordering of reals

Is there an effective (constructive) well order on reals ? I know several questions were already asked on this topic, and the answers were very good to this well known problem. My question is more ...
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Ultrapowers by extenders of potential premice

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...
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Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
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Relation between inacessible cardinals and CH

I know that the two statements “There exists an inaccessible cardinal” and “Continuum Hypothesis” are both independent of ZFC. Now, are those two statements independent of each other? That older ...
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Intuition behind extenders

What's the best way to think about extenders? For instance if we have a normal $\kappa$-complete ultrafilter on $\kappa$, call it $D$, $M$ the ultrapower given by $D$, and if we look at $j: V \to M$, ...
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Kunen's result and $L(\mathbb{R})$

Recall Kunen's result: If $j:V \to M$ is a nontrivial elementary embedding then $V \ne M$. Assume $AD$ is true in $L(\mathbb{R})$. Let $j : L(\mathbb{R}) \to M$ be a nontrivial elementary embedding. ...
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$\kappa$-complete ultrafilter and bounded subsets of $\kappa$

If $U$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$ then every bounded subset of $\kappa$ has measure $0$. This is because if we had $X \in U$ with $X$ having size less than $\kappa$, ...
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How many weak/strong limit cardinals exist under different assumptions?

I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following ...
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Shelah Cardinals and Modern Set Theory

Do Shelah cardinals play an essential role in any modern set theory results or was the concept basically made obsolete by Woodin cardinals?
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Which infinite cardinals can be defined using partition relations?

Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then $\kappa$ is weakly compact if $\kappa$ satisfies ...
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An inequality in a proof of Kunen's Inconsistency

This may be a silly question, but I keep coming back to it. Let $j:V\prec M$ be a non-trivial elementary embedding with $M$ a transitive class and $\kappa$ the critical point of $j$. Define the ...
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What's the latest on Laver tables?

A couple of years ago, I was astonished and delighted to learn about Laver tables, a sequence (indexed on $n$) of Cayley-like tables for a binary operation $\star$ on numbers $i,j\leq 2^n$ that ...