# Questions tagged [large-cardinals]

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### Thinning chains of elementary extensions

I'm bumping this question, since I'm still curious regarding the answer but this question seems to have gone unnoticed. This is a follow-up to this question, regarding a stronger variant of ...
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### What's the consistency status/strength of this limitation principle?

$\DeclareMathOperator\iCard{iCard}$In a prior posting If we limit matters what ZFC can prove, would that be consistent? to MO, I tried to capture the informal principle of whatever ZFC proves, it is, ...
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### If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$?

Here we choose the definition of "is a cardinal" as there is no surjective map from a smaller ordinal to it. It's easy to prove that, if $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...
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### Consistency strength about Ramsey M-rank and Mahlo-Ramsey cardinal

In the website "Cantor's attic", there are a long list of large cardinal axioms arranged by consistency strength. In the list, "α-Mahlo Ramsey" is placed higher than "Ramsey M-...
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### How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST?

Fix a language $\mathcal{L}$ of first-order set theory. For this question, we can assume that $\mathcal{L}$ is the language described in Chapter 1 of “An introduction to set theory” [William A. R. ...
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### What is known about the least cardinal where $\kappa$ fails to be supercompact?

Assume $\kappa$ is $\lambda$-supercompact for some $\lambda$ but not fully supercompact. Are there any known restrictions (or provably non-restrictions) on the least $\delta$ such that $\kappa$ is not ...
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### An inconsistency about Magidor models

Recall that a model $M\prec V_\theta$ is $\kappa$-Magidor if it has transitive intersection with $\kappa$ and its transitive collapse is equal to $V_\alpha$ for some $\alpha<\kappa$. The ...
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### Can HCD accommodate all known large cardinal axioms?

HOD has been found to be useful in that it is an inner model that can accommodate essentially all known large cardinals. However, there is a definable well ordering over HOD, so it cannot satisfy ...
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### Dehornoy's proof that the application of two elementary embeddings is an elementary embedding

What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book Braids and Self-Distributivity? That lemma states "Assume that $j_1$ and $j_2$ are elementary ...
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### Motivation for Laver's use of large cardinals to show finite combinatorial properties of Laver tables

Laver showed in 1995 that the period of the first row of certain Laver tables is unbounded, assuming that a rank-into-rank cardinal exists. The most accessible proof of his result that I was able to ...
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### A restricted form of the inner model hypothesis

Previously asked and bountied at MSE, with slight difference. To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal ...