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.

learn more… | top users | synonyms

4
votes
2answers
54 views

Is it provable in $ZFC$ that if $V_\kappa\vDash ZFC$, then $\kappa$ is strongly inacessible?

The other direction of this implication is pretty obvious, but I'm having a hard time seeing why this direction might be true. I suspect that it isn't, but part of my suspicion comes from my inability ...
2
votes
1answer
27 views

Is there an agreed upon convention for naming ZFC+Large Cardinal Axioms?

Is there an agreed upon convention in general for what to name ZFC+[Large Cardinal Axiom]? Or would one have to state explicitly which axiom was being added? To explain what I mean, note that anyone ...
0
votes
1answer
41 views

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 $\kappa$?
2
votes
2answers
75 views

What are some applications of large cardinals?

Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger ...
1
vote
0answers
46 views

Supercompact cardinals and $H_{\theta^+}$ [on hold]

Given $\kappa\leq \theta$ two cardinal numbers we say that $\kappa$ is $\theta-$supercompact if there exists an elementary embedding $j: V\rightarrow M$ such that $cp(j)=\kappa$, $j(\kappa)>\theta$ ...
2
votes
1answer
138 views

Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
1
vote
1answer
28 views

Does the relationship between Jonsson Cardinals and Jonsson Algebras require the axiom of choice?

Using Skolem functions, one can see in ZFC that a cardinal $\kappa$ is Jonsson iff there are no Jonsson algebras on $\kappa$. (I.e. every algebra of size $\kappa$ has a proper subalgebra of size ...
4
votes
1answer
107 views

What is the *exact* consistency strength of $MK$?

It's well known that the existence of an inaccessible cardinal proves $Con(MK)$. Joel Hamkins has a nice blog post (http://jdh.hamkins.org/km-implies-conzfc/) that explains what you get out of $MK$, ...
1
vote
1answer
64 views

Proving ZFC is consistent

I've heard from a friend that we can actually prove the consistency of ZFC if we assume at least one inaccessible cardinal exists. How is this carried out, precisely? Googling doesn't help and my ...
3
votes
1answer
61 views

Weakly Compact Cardinals are Mahlo Proof

I have a question about a corollary in Jech's set theory text which states: Corollary 17.19. Every Weakly Compact cardinal $ \kappa $ is a Mahlo cardinal, and the set of Mahlo cardinals below $ ...
6
votes
0answers
70 views

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 ...
1
vote
1answer
48 views

Elementary embeddings and measurable cardinals

Given a measurable cardinal $\kappa$ we can consider its associated embedding $j:V\longrightarrow M\cong Ult_U(V)$ where $U$ is a $\kappa-$complete non principal normal ultrafilter on $\kappa$. In ...
4
votes
1answer
62 views

When can “$j: V \rightarrow M$ is an elementary embedding” be defined in ZF?

This regards elementary embeddings of inner models of set theory. It seems that it is in general "stated" via an axiom schema each member of which states that the class function is elementary with ...
3
votes
1answer
69 views

Erdös cardinals in $L$

I've readed in Jech's book that the existence of the $\omega-$Erdös cardinal $\kappa(\omega)$ (that is the minimumm cardinal $\kappa$ for which $\kappa\rightarrow (\omega)^{<\omega}$) it is ...
3
votes
1answer
83 views

Ordering of Large Cardinal Axioms

One of the unexplained phenomena about large cardinal axioms is that they tend to be in a linear order (in terms of consistency strength). So it brings to mind other natural questions about this ...
1
vote
2answers
57 views

Standard Complete Model of ZFC and Reflection

I am reading Levy's Axiom Schemata of Strong Infinity in Axiomatic Set Theory. In theorem 6, which says that this Reflection principle $N_0$ is equivalent to Replacement + Infinity in $ZF$ (in $S$ ...
3
votes
1answer
60 views

Diamond in Subtle Cardinals

In Jensen's manuscript on combinatorial principles, he defines the notion of a subtle cardinal: Definition. A regular cardinal $\kappa$ is subtle if for all $C$ club, $(A_\alpha)_{\alpha\in C}$ a ...
1
vote
0answers
66 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 ...
7
votes
1answer
80 views

Fixed points in the enumeration of inaccessible cardinals

Let inaccessible cardinal mean uncountable regular strong limit cardinal. Consider $\mathsf{ZFC}$ with an additional axiom: For every set $x$ there is an inaccessible cardinal $\kappa$ such that ...
1
vote
0answers
38 views

Is it possible to define a group structure on arbitrary set? [duplicate]

Is it possible to define a group structure on arbitrary set? It is obvious for finite sets and also sets with cardinality |Q| and |R| and also we don't know that is there other cardinality betwen them ...
7
votes
2answers
379 views

mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
15
votes
1answer
288 views

What's the latest on Laver tables?

A couple of years ago, I was astonished and delighted to learn about Laver tables, a sequence (indexed on $n$) of Cayley-like tables for a binary operation $\star$ on numbers $i,j\leq 2^n$ that ...
2
votes
0answers
73 views

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 ...
4
votes
2answers
257 views

about the smallest $k$ that $V_k$ is a model of ZFC

Let $k$ to be the smallest ordinal that $V_k$ is a model of ZFC. I know that $k$ need not to be inaccessible cardinal,and $k$ has confinality $\omega$. Then how big is $k$? How to write down $k$ in ...
3
votes
1answer
47 views

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. ...
6
votes
1answer
96 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 ...
5
votes
0answers
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 ...
4
votes
1answer
85 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 ...
3
votes
3answers
76 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 ...
3
votes
0answers
43 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 ...
5
votes
1answer
86 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 ...
-1
votes
1answer
91 views

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 ...
2
votes
2answers
97 views

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
votes
2answers
119 views

$L$ 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 ...
6
votes
2answers
93 views

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 ...
5
votes
1answer
120 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 ...
3
votes
3answers
190 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 ...
3
votes
0answers
57 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 ...
2
votes
0answers
78 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
votes
1answer
79 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 ...
5
votes
0answers
105 views

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 ...
5
votes
1answer
96 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
votes
1answer
54 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 ...
4
votes
1answer
157 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 $\kappa$, ...
3
votes
2answers
534 views

On the definition of weakly compact cardinals

I am reading in Jech's Set Theory the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in the book. I ...
0
votes
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 ...
1
vote
0answers
127 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 ...
2
votes
1answer
125 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 ...
4
votes
1answer
158 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, ...
4
votes
1answer
72 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 ...