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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“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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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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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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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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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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$V_k$ 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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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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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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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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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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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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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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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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Weakly-compact cardinals

I am reading Jech's Set Theory, in particular the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in ...