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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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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Inaccessibility of $\omega^V_1$ in $L$ under determinacy

I'm looking for the proof of the following fact: Assume axiom of determinacy. Then $\omega^V_1$ is a strongly inaccessible cardinal in $L$. I have seen this result mentioned in few places (e.g. ...
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1answer
88 views

$M$-amenable ultrafilters on $\kappa$ are $\kappa$-powerset preserving

Let $M$ be a transitive model of $\operatorname{ZFC-}$ and let $$ j \colon M \rightarrow N $$ be elementary with $\operatorname{crit}(j) = \kappa \in \operatorname{wfp}(N)$. Let $U_j$ be defined by ...
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139 views

“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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1answer
69 views

Proving that the existence of strongly inaccessible cardinals is independent from ZFC?

ZFC can't prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options: ZFC proves there are no ...
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66 views

Does the following define a Mahlo cardinal?

Let M be a cardinal with the following properties: - M is regular - $\kappa < M \implies 2^\kappa < M$ - $\kappa < M \implies s(\kappa) < M$ where $s(\kappa)$ is the smallest strongly ...
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40 views

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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1answer
85 views

Members of $\Sigma^1_4$ sets

I'm pretty sure this is an easy descriptive set theory question that I'm just blanking on. Is it consistent with large cardinals - say, with a measurable - that every (nonempty) $\Sigma^1_4$ class ...
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Defining the critical point of an elementary embedding when there exist incomparable cardinals.

Assume one has. for $V$ and for some transitive class $M$, an elementary embedding $j$: $V$$\rightarrow$$M$ and that $j$$\neq$$id$, where $id$ is the identity. If $V$ and $M$ satisfy $ZFC$ then the ...
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2answers
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Elementary embeddings,V,set theory,L,cardinals

I would like to make clear some properties of $j,M,L$ and $V$ in ZFC. let $j:V \to M$ denote a (nonidentity) elementary emedding and $M$ a transitive $\in$-model of ZFC.Is there an example of ...
4
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2answers
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$L$ models set theory and so does $V_\kappa$ for $\kappa$ inaccessible

Background: I have been reading the 1980 edition of Kunen. Theorem VI.2.1 states it is provable from ZF that $\mathbf L$ (Kunen writes classes in bold) is a model of ZF. Also, it is a well-known ...
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A function on an LCH space that is sequentially continuous but nowhere continuous

This question is an extension of Looking for example of topological spaces where sequential continuity does not imply continuity. In my answer to that question, I gave an example of a topological ...
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1answer
108 views

Gauging the “size” of measurable cardinals and inaccessible cardinals

I have been told that measurable cardinals are much "larger" than ordinary strongly inaccessible cardinals. I have even heard that the former make the latter look "tiny" in comparison. This is ...
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3answers
176 views

Is the critical point of an embedding of a model of set theory inaccessible in it?

Can we find an elementary embedding $j:M\to N$ with $M,N$ transitive $ZFC$-models, $\kappa$ being the critical point, so that $\kappa$ is not inaccessible in $M$ ? ($\kappa$ is regular in $M$.) I ...
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53 views

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 ...
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76 views

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 ...
3
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1answer
76 views

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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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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1answer
95 views

Ramsey Combinatorics and linear order

Prove that the following are equivalent for an infinite cardinal $\kappa$. (1) $\kappa \to (\kappa)^2_2$ (2) In any linearly ordered set of cardinality $\kappa$ there is either a well-ordered or a ...
2
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1answer
51 views

ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$

My understanding is that Solovay (1970)'s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence ...
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1answer
137 views

What is wrong with this “proof” that there is no $\omega$th inaccessible cardinal?

"Theorem": There is no $\omega$th inaccessible cardinal. "Proof": Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, ...
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118 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 ...
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1answer
114 views

Breaking the AC barrier using Kelley-Morse set theory

In her blog post Variants of Kelley-Morse set theory, Prof. Gitman proves that every model of the the common version of Kelley-Morse set theory ($\mathsf{KM}$) is a model of the Wikipedia version of ...
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1answer
120 views

Why does a nontrivial $V \to V$ have a critical point?

Let $V$ denote the von Neumann universe, and let $j: V \to V$ be a nontrivial (non-identity) elementary embedding. The critical point is the smallest $\kappa$ such that $j(\kappa) > \kappa$. The ...
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1answer
143 views

Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, ...
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1answer
61 views

Replacing an ordinal with its cardinality in a partition relation

In The Higher Infinite, Kanamori claims that if $\alpha$ is a cardinal, and $\beta \to (\alpha)^\gamma_\delta$ for some $\beta$, then the least such $\beta$ is a cardinal. I can't seem to think of a ...
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147 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 $AC$ ...
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1answer
103 views

Trouble understanding elementary embedding proofs

Here are two pretty standard results about elementary embeddings that I don't understand. (1) Let $\pi:V \to M$ be a non-trivial elementary embedding with $M$ a transitive class. Let ...
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2answers
145 views

How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam's original argument about measure theory and measurable cardinals. Here is the result I am looking for: ...
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125 views

A question regarding a paper of M. Magidor [closed]

I am interested in the following paper of M. Magidor: "On the role of supercompact and extendible cardinals in logic", Israel Journal of Mathematics, 05/1971; 10(2): 147-157. The abstract (which I ...
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150 views

2 Questions regarding Relative Consistency Proofs

First Question: Let IC be the statement "There is an inaccessible cardinal." I have read that one cannot prove (in ZFC) the relative consistency of ZFC + IC w.r.t. ZFC. i.e. $ Con(ZFC) \rightarrow ...
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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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137 views

Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?

A bit of philosophy: under the usual definition of the aleph numbers, ZFC proves the sentence "$\aleph_1$ is an ordinal." However, in some sense $\aleph_1$ isn't really an ordinal (in my opinion), ...
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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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135 views

How large is an uncountable regular cardinal which is closed under arbitrary fast operators?

Let $Card$ be the proper class of all cardinals, define an infinite set of operators like $\otimes_{n}:(Card\setminus \omega)\times (Card\setminus\{0\})\longrightarrow Card$ which are defined for each ...
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$\kappa$ ineffable $\Rightarrow$ $\kappa$ tree-property

Let $\kappa$ be an uncountable, regular cardinal. We call $\kappa$ ineffable iff for every sequence $(A_\xi \colon \xi < \kappa)$ of subsets $A_\xi \subseteq \xi$ there is a stationary subset $S ...
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52 views

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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181 views

Consistency strength of the “club ultrafilter”

What are the consistency strengths of $$ZF+``\text{The club filter on $\omega_1$ is an ultrafilter}"$$ and $$ZF + DC + ``\text{The club filter on $\omega_1$ is an ultrafilter}"?$$ I know that the ...
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100 views

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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ineffable, weakly compact, and ? cardinal

In the online book, page 312, http://projecteuclid.org/download/pdf_1/euclid.pl/1235419485 what cardinal notion do we get, by requireing that X in the bottom of the page, is not only stationary ...
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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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179 views

Consistency strength of 0-1 valued Borel measures

The following is an overly fancy way of asking a question suggested in Borel Measures: Atoms vs. Point Masses Let $\phi$ be a property that topological spaces can have (such as "compact", "$T_1$", ...
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59 views

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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100 views

$0^\sharp$ and the regularity of $\aleph_\omega$

I'm sure I'm missing something trivial, and the most likely of it is that I'm simply wrong on my understanding of the constructible universe $L$, or maybe one of the Wikipedia entries I'm about to ...
4
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1answer
123 views

Are there any large cardinal properties of the critical point of a $j: L \longrightarrow L$?

I've recently been thinking a bit about $L$ and $0 \sharp$. As is well known, the existence of $0 \sharp$ is equivalent to the existence of a non-trivial elementary embedding $j: L \longrightarrow ...
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Is this extension of ZFC known to be outright inconsistent?

Is the following first-order theory known to be outright inconsistent? Adjoin to ZFC a unary function $U$ together with the following axioms. If $\alpha$ is an ordinal, then there exists an ...
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Proof of “If $ZFC$ proves there is an inaccessible cardinal, then $ZFC$ is inconsistent”.

Let $I$ be the statement "there is an inaccessible cardinal". I'm aware of two proofs of "If $ZFC\vdash I$ then $ZFC$ is inconsistent". One proof uses the Second Incompleteness Theorem, which I ...