# 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.

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### Sets of Fixed Points in an Ultrapower Embedding

Let $\kappa$ be a measurable cardinal and $j: V \rightarrow M$ be the elementary embedding with critical point $\kappa$. Let $X$ be a set such that for all $x\in X$, $j(x) = x$. My question is: when ...
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### $0^\sharp$ and transcendence over $L$

I've started to study the $0^\sharp$ principle from Kanamori's The Higher Infinite, and there are a few interesting, yet a bit vague, remarks scattered across the text that intrigue me and whose core ...
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### If $\kappa$ is ineffable, then so is $\{\nu \lt \kappa: \nu \text{ is weakly-compact}\}$.

So there was an exercise that I solved sometime ago which required me to prove that if $\kappa$ is ineffable, then $\{\nu \lt \kappa: \nu \text{ is weakly-compact}\}$ is stationary. At the time, I ...
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### Definability of $j:V\to M$ is an elementary embedding

I was looking at Woodin's chapter in the book Infinity: New Research Frontiers and I'm confused by some of his remarks regarding class-sized elementary embeddings. The following is taken from the book:...
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### What can we do with nonenumerable sets of formulas (e.g. formulas of Higher order Logic)?

It is well known textbook fact, that the set of (grammatically correct) sentences/formulas of higher order logic (even of the second order logic) are not enumerable. My question is - what can we do ...
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### (S)WVP - Vopenka's priciple

I would like to understand why precisely - in the nice result here on Vopenka's principle - on the 2nd page, in the last but one paragraph it is enough that there is an ordinal $\kappa$ and a ...
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### Isomorphism of $V_\lambda$ and the ultraproduct of $V_{\lambda_{\mathrm{otp}(x)}}, x \in \mathcal{P}_\kappa({\lambda})$ by a normal fine measure.

This question is actually exercise (20.5) from Set Theory by Thomas Jech. The original statement is: Let $\lambda \ge \kappa$ and let $U$ be a normal measure on $\mathcal{P}_\kappa({\lambda})$. The ...
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### Beth cardinals and inacceesible cardinals

Since my last question here about the Alephs was too imprecise and thus went over like a lead balloon, I am trying a new and simpler question which asks what I probably should have asked before. ...
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### The first uncountable ordinal and some stationary sets

We define a subset $A$ of an ordinal $\alpha$ as stationary iff we have $A$ intersecting every closed and unbounded subset of $\alpha$. Equivalently, one can define this as \forall f:\alpha\mapsto\...
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### Is being a Reinhardt cardinal first-order definable?

As is well known, Reinhardt cardinals are inconsistent with $\mathsf{ZFC}$, but many of the proofs I've seen of this rely on combinatorial or club/stationary set properties. If there's a (somewhat ...
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### Exploiting Grothendieck universes

A Grothendieck universe provides an easy-to-understand example for a model of ZFC. Because ZFC, if consistent, cannot prove the existence of any model of itself, the existence of universes needs to be ...
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### Proof of every measurable cardinal carries a normal measure

I'm reading the proof of Theorem 10.20 in Set Theory by Jech and I don't understand the last argument. The theorem says every measurable cardinal carries a normal measure. The proof goes: Let $U$ be ...
Problem 10.5 from Set Theory, Jech: A measure $U$ on $\kappa$ is normal if and only if the diagonal function $d(\alpha)=\alpha$ is the least function $f$ with the property that for all \$\gamma<\...