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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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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Forcing Classes Into Sets

I am still studying the topics in forcing and did not yet study much about forcing with a class of conditions. I know from Jech's Set Theory that you can force that the class of ordinals in the world ...
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What are large cardinals for?

I've heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don't understand why you would bother. What's the simplest proof or whatever that ...
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“Nice” well-orderings of the reals

I have a question which I believe could be easily resolved if I happened to look at the right source - hence my asking it here as opposed to at MathOverflow. I've tried googling it, but I haven't been ...
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When does $V=L$ becomes inconsistent?

In a wonderful course I'm taking with Magidor we are finishing the proof of the Covering Theorem for $L$. The theorem, in a nutshell, says that $V$ is very close to being $L$ if and only if $0^\#$ ...
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Real-measurable cardinals that are not measurable ones

I'm reading Jech's Set Theory, and in the chapter about measurable cardinals there is a theorem that if $\kappa$ is real-measurable but not measurable then it is $\le 2^{\aleph_0}$ and so and so. ...
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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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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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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 ...
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Proving $V_{\kappa}$ is a model of ZFC for inaccessible $\kappa$

Prove that if $\kappa$ is an inaccessible cardinal, then $V_{\kappa}$ satisfies all the axioms of ZFC. How is this done for the axiom of choice and for regularity?
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What is the largest cardinal which can inject into $\mathbb{R}$ in ZF?

This question takes place in ZF. Assume some mild large cardinals; then it is consistent (in fact, it follows from AD, the consistency of which follows from mild large cardinals) that there are very ...
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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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$A^\#$ and inner models

For a set of ordinals $A$ we say that $A^\#$ exists if there exists a closed and unbounded class of indiscernibles, $I\subseteq\operatorname{Ord}$, for $L[A]$. Formally, if such class exists we define ...
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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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mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
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Equivalent ways to describe the Mitchell order

For a measurable cardinal $\kappa$, we define an ordering over $\kappa$-complete ultrafilters as follows: Suppose $W,U$ are both $\kappa$-complete free ultrafilters over $\kappa$, we say that $U\lhd ...
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Inaccessible cardinals and well-founded models of ZFC

This was a problem in an exam I took last semester, but I never got the chance to ask my professor how to solve it. Here goes: Let $\kappa$ be an inaccessible cardinal. Then $V_\kappa$ is a ...
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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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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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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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356 views

The tree property for non-weakly compact $\kappa$

In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$. One of which was that ...
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Basic constructibility question

I'm currently reading J. D. Hamkins' paper "Unfoldable cardinals and the GCH," and I've run across a comment that I think I ought to find trivial, but I don't. On page 1187, he says that ...
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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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278 views

Can forcing push the continuum above a weakly inacessible cardinal?

There is a famous quote of Paul Cohen which reads $\lt\lt$ A point of view which the author feels may eventually come to be accepted is that the continuum hypothesis is obviously false ... The ...
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Consistency of $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$

It is known that if $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$ then $0^\#$ exists. I am familiar with the Feferman-Levy model in which $\omega_1$ is singular, which has the same ...
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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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Strong cardinals and reflection

I'm new to all this large cardinal thing and I have trouble in proving the following: If $\kappa$ is a $\gamma$-strong cardinal, for some large enough $\gamma$, then $\kappa$ is ...
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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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Normal ultrafilters and Stationary sets

If $\kappa$ is a measurable cardinal, and $\mathcal{U}$ is a normal ultrafilter which is $\kappa$-complete then $\mathcal{U}$ extends the club filter (i.e. every club is a member of $\mathcal{U}$). ...
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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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168 views

Silver indiscernibles and definable injections

This is a follow up to my previous question on Silver indiscernibles. Background: Suppose that $0^\#$ exists, $\alpha<\lambda$ are limit ordinals, $i_\alpha$ is the $\alpha$th Silver ...
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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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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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Good Reference for Large Cardinals/Homotopy Theory

I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
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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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Why is Kunen inconsistency at the top of Cantor's upper attic?

Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below: So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
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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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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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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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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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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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Silver indiscernibles and constructibility

We know that if $0^\#$ exists then it's not in $L$. For an infinite ordinal $\alpha$, denote by $I_\alpha$ the initial segment of length $\alpha$ of Silver indiscernibles. Question: For which ...
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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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The real cofinality of singular cardinals in $L$ under $0^\#$

Suppose that $0^\#$ exists, is there a relatively simple way to show that for any ordinal $\lambda$, if $\lambda$ is a singular cardinal in $L$ then its real cofinality is $\omega$?
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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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Generalizing $0^\#$

Background and motivation: The following theorem is due to Silver: If there exists a Ramsey cardinal then: For every $\aleph_0 < \kappa < \lambda$, $L_{\kappa}$ is an elementary ...
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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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example of weakly inaccessible cardinal that is not a strongly inaccessible cardinal

As I study through inaccessible cardinals, I find many examples that show some cardinal being both weakly inaccessible and strongly inaccessible. So, can anyone show me the example of weakly ...
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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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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 $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ ...