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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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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Similarity of the Universe $V$ and its $\kappa$-Fragments

Intuitively, if a cardinal $\kappa$ is "large" then the $\kappa$-fragment of the universe $V_{\kappa}$ is so "similar" to the entire universe $V$. For example if $\kappa$ is supercompact then ...
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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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Large Cardinals with Elementary Extension Characterization

Question: Which large cardinals $\kappa$ have a characterization in the following form: $\kappa$ is large if and only if for all cardinals $\lambda>\kappa$, $\langle W_{\kappa},\in\rangle\prec ...
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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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37 views

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 ...
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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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How large are measurable cardinals of higher orders?

For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows: A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure ...
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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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Large Cardinal Inequalities

Solovay showed that the existence of $0^\dagger$ follows from the existence of two measurable cardinals. We know existence of a measurable cardinals is weaker than existence of $0^\dagger$ so we ...
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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 ...