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Questions tagged [large-cardinals]

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.

Large cardinals are such cardinals whose existence cannot be proved within $\sf ZFC$, and requires stronger axioms to be added to $\sf ZFC$, 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 talking about small categories, and perhaps at one or…
Asaf Karagila
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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 satisfies $p\star 1\stackrel{\text{def}}{=}p+1\bmod…
Steven Stadnicki
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Can forcing push the continuum above a weakly inacessible cardinal?

There is a famous quote of Paul Cohen which reads “A point of view which the author feels may eventually come to be accepted is that the continuum hypothesis is obviously false ... The continuum $\mathfrak c$ is greater than…
Ewan Delanoy
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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 will be countable in the generic extension, which…
Asaf Karagila
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In what sense are inaccessible cardinals inaccessible?

Another title for this question could be: where do inaccessible cardinals live? It may be that this question does not make any sense. So I will try to explain what I mean. I think of the ZFC axioms as a recipe that allows us to structure the…
User0112358
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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 requires the use of large cardinals? Is there some branch…
Seamus
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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 consistency strength as $ZF$. I have three…
Asaf Karagila
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Why is $0^\sharp$ not definable in $ZFC$?

I have a little question about $0^\sharp$. I'm sure has a nice and easy answer, but I'm just not seeing it and I think it'll help my understanding of $L$ quite a bit if I can piece the answer together. Given Godel's constructible universe $L$, we…
Neil Barton
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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 able to find a satisfactory answer. Question: is…
user13568
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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. (Corollary 10.10) How can a cardinal number be…
Asaf Karagila
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Collapse $\aleph_{\omega_1}$ to $\aleph_\omega$.

Is there a forcing notion that collapses $\aleph_{\omega_1}$ to $\aleph_\omega$ while preserving every cardinal below $\aleph_\omega$?
user123
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What are the motivations of large cardinal research?

Why do set theorists research large cardinals? Is it about the consistency results, or is there a type of mathematical beauty to large cardinals? If so, are there any examples of beauty in large cardinals someone with a knowledge of some…
littleman
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A tame(ish) fragment of second-order logic

This question is about a tame(?) fragment of second-order logic with the standard semantics $\mathsf{SOL}$, motivated by the Tarski-Vaught test. The general setup is as follows. Given structures $\mathfrak{A},\mathfrak{B}$ and a logic $\mathcal{L}$,…
Noah Schweber
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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 $0^\#$ does not exist. It is consistent that no large…
Asaf Karagila
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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?
Everett Piper
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