# Tagged Questions

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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### Independence of existence of inaccessible cardinals

Let $I$ be the formula which states that there exists strongly inaccessible cardinals. My question is regarding the proof of $ZFC\nvdash I$ appearing in Jech (part of theorem 12.12). He starts by ...
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### Model for replacement

If $R_{k}$ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...
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### to prove some subset of a club set is club

Let $\kappa$ be a weak inaccessible cardinal and $C$ closed unbounded in $\kappa$. Then why is $C' = \{\alpha \in C:|\alpha \cap C|= \alpha\}$ also closed unbounded in $\kappa$?
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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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### If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U = j_D$, is $U=D$?

Here, $j_U$, $j_D$ are the canonical elementary embeddings induced by the measures $U,D$ on $\kappa$. I actually wish to ask three related questions. Let Ult$_U(V)$, Ult$_D(V)$ be the transitive ...
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### “Clubbiness” of $\Pi^1_n$ sets

I'm sure this is just my google-fu failing me, but: what are sufficient large cardinal axioms to guarantee "Every (boldface) $\Pi^1_n$ set of countable ordinals contains or is disjoint from a club ...
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### Do large cardinal properties tend to be semiabsolute?

I don't know much about large cardinals; so, I want to get a feeling of the landscape. Hence this question. Definition. Whenever $C$ is a unary predicate in the language of ZFC, let us call $C$ ...
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### Inaccessible Cardinals and Grothendieck Universes

I'm trying to prove that the following statements are equivalent: 1.$\forall\alpha\in\mathbb{ON}\ \exists\ \kappa>\alpha$ is an inaccessible cardinal. 2.$\forall x\ \exists\ U\ x\in U$ and $U$ is ...
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### Is the following notion equivalent to subtle cardinals?

Let $\kappa$ be a regular, uncountable cardinal. We call $\kappa$ $\dagger$ if for every sequence $(A_\alpha \colon \alpha < \kappa)$, $A_\alpha \subseteq \alpha$ and every $\xi_0 < \kappa$ ...
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### Is there any inconsistent large cardinal axiom which its inconsistency proof is essentially different from proof of Kunen inconsisteny theorem?

There is a long list of large cardinal axioms. Most of them deemed to be consistent with ZFC but there are also some axioms like existence of Reinhardt or $\omega$-huge cardinals which are natural ...
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### Kanamori's proof that Vopenka's principle implies the existence of extendibles.

In The Higher Infinite, Kanamori gives a proof that Vopenka's principle implies the existence of (many) extendibles. The relevant theorem is Proposition 24.15 on p.337. Working in a $V_\kappa$ which ...
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$\kappa$-Suslin Hypothesis ($\kappa$-SH) for the infinite regular cardinal $\kappa$ says that every tree of height $\kappa$ either has a branch of length $\kappa$ or an antichain of cardinality $\... 0answers 82 views ### Is consistency strength order dense? Definition: Let$\sigma$,$\theta$be two statement in the language of set theory. We say$\sigma <_{c} \theta$($\sigma$is strictly lower than$\theta$in consistency strength order) if within ... 0answers 179 views ### How large or small can the gap in Shelah's main gap theorem be, up to consistency? Shelah's main gap theorem in model theory says: For each first order complete theory$T$in a countable language if$I(T,\kappa)$denotes the number of its models of size$\kappa$then one of ... 0answers 95 views ### Follow up question for an exam question Some time ago I got an answer to this old exam question. Now I found the following question, which strikes me as very similar: Let$\kappa$measurable cardinal, and$\preceq$a partial ordering on ... 0answers 67 views ### Minimal model of ZF with$0\sharp$We know that the constructible universe$L$is an absolute and minimal model of ZF (every standard model of ZF contains "an"$L$, and it is actually the same$L$for all of them). It is also my ... 0answers 131 views ### How can nontrivial elementary embeddings of the universe to some inner model be surjective? Consider$\kappa$to the least measurable cardinal, or equivalently$\kappa$is the critical point for an elementary embedding$j:V \rightarrow M$from the universe$V$to an inner model$M$(critical ... 0answers 165 views ### Choice-less Set Theory for Dummies In almost every graduate set theory text there are some parts about equivalences of$AC$, its consequences, some axioms like$AD$which imply$\neg AC$, some well-known axiomatic systems which$AC$... 0answers 221 views ### Zero sharp and Kunen inconsistence. There is some (deep/shallow/interesting/trivial) relation between: (1)$\exists 0^\sharp$iff$\exists j:L\longrightarrow L$nontrivial elementary embedding. (2) The Kunen inconsistency$\not\exists ...
I'm trying to prove this theorem, but is it even true? A Tarski's class is a set $T$ such that: For each $y\in T$, ${\cal P}(y)\subseteq T$ and ${\cal P}(y)\in T$, and for each $A\subseteq T$ such ...
I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...