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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Many $(\kappa+\alpha)$-strong cardinals below a $(\kappa+\beta)$-strong one for $\alpha < \beta < \kappa$

Let $\lambda$ be an ordinal. A cardinal $\kappa$ is $\lambda$-strong iff there is some inner model $M$ and an elementary embedding $$ j \colon V \rightarrow M $$ s.t. $crit(j) = \kappa$ and $V_\lambda ...
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On the contradictory nature of large cardinals & choice-like axioms

Compare these two results: Theorem (Scott): $ZFC+V=L\vdash \nexists~\text{Measurable cardinal}$ Theorem (Kunen): $ZFC+AC\vdash \nexists~\text{Reinhardt cardinal}$ Now compare these two ...
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Appealing axioms incompatible with large cardinal axioms

I'm interested to know what are some 'appealing' axioms that are inconsistent with ZFC plus some large cardinal axiom. I saw the question On the contradictory nature of large cardinals & choice-...
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Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
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Let $\Phi$ denote the statement that $\mathrm{GCH}$ holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition. Definition. Call a set of axioms $\Phi$ in ...
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How to think about iterated ultrapowers?

I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, ...
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Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
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What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?

Consider the first-order theory obtained by adjoining to ZFC (or, better, Mac Lane set theory) the GrothendieckÔÇôVerdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
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Uncountable random graphs

There is a theorem from Erdos and Renyi that says that a random graph on $\aleph_0$ vertices (where each pair of vertices is connected with probability equal to $\frac{1}{2}$) will be isomorphic to ...
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Looking for extender axioms

Consider the following extender construction: Given an elementary embedding $j:V\to M$, where $M$ is transitive, with critical point $\kappa$, we can for each $a\in j(V_{\kappa})$ define a $\kappa$-...
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Is there any filter space characterization for strong cardinals?

The theory of filter spaces is introduced by Apter, Diprisco, Henle & Zwicker, in their joint paper: Arthur Apter, Carlos Di Prisco, James Henle, and William Zwicker, Filter spaces: towards a ...
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Expository Papers on Extenders

Extenders are discussed in many set theory text books. Here I am looking for some expository "papers" which are focused on this subject and its connection with forcing and large cardinals. More ...
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Proof of PFA from Supercompact

In Jech's proof (in chapter 31) that the consistency of a supercompact cardinal implies the consistency of PFA, he needs the following fact: Let $\mathbb{P}_\kappa$ be the countable support forcing ...
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Levy collapse gone bad

Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following? (1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, ...
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Weakly Normal Ultrafilters

A filter $F$ on $\lambda$ is weakly normal if and only if for all $f : \lambda \rightarrow \lambda$ such that $\{\xi < \lambda : f(\xi) < \xi\} \in F^+$, then there is some $\beta < \lambda$ ...
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Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?

The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
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Are large cardinals bi-interperable with type theory?

"Sets in Types, Types in Sets" establishes that the calculus of constructions is bi-interpretable with ZFC + infinitely many inaccessibles. Are there any more results like this higher up the large ...
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Are these strengthenings of a rank-into-rank cardinal axiom known to be inconsistent with $ZFC$?

I am just getting acquainted with "very strong" large cardinal axioms, and it seems there is a consensus that among large cardinal axioms, the rank-into-rank cardinal axioms are at the threshold of ...
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Measurable cardinals are weakly-compact?

First off, this isn't homework, but I'm doing research into large cardinal stuff so I wanna understand these theorems. I'm given this information to work with: a cardinal $\mu$ is measurable if ...
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Closeness of measures on a cardinal

Given an uncountable $\kappa$ and a $\kappa$-complete nontrivial non-normal ultrafilter on $\kappa$, and some $g:\kappa\to\kappa$ with $<_{U}$-rank $\kappa$ (where $f_0<_Uf_1$ iff $\{i<\kappa\...
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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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Large Cardinal Consequences of $\kappa$-Suslin Hypothesis

$\kappa$-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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $AC$ ...
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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 ...
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Tarski's axiom implies a proper class of inaccessible cardinals

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 ...
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Ultrapowers by extenders of potential premice

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 ...