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
154 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 ...
2
votes
1answer
112 views

Inaccessible cardinals

Let $M[G]$ be the full Solovay model, and let HOD be the model of hereditarily ordinal definable sets in $M[G]$. Is it possible for HOD not to have an inaccessible cardinal? Does HOD satisfy GCH?
2
votes
1answer
56 views

Normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$

I am looking at fine normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$ and I have seen that there are two (seemingly) different formulations of normality, in terms of regressive (sometimes also ...
3
votes
0answers
122 views

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 $\alpha<\kappa$, ...
5
votes
1answer
178 views

How large are measurable cardinals of higher orders?

For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows: A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure ...
1
vote
0answers
158 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 ...
6
votes
0answers
182 views

Large Cardinal Inequalities

Solovay showed that the existence of $0^\dagger$ follows from the existence of two measurable cardinals. We know existence of a measurable cardinals is weaker than existence of $0^\dagger$ so we ...
2
votes
0answers
87 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 ...
5
votes
1answer
145 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
2
votes
1answer
106 views

How many weakly compact cardinals can L have?

Q1: Assuming suitable consistency assumptions is the following consistent? $ZFC+V=L+\text{Existence of class many weakly compact cardinals}$ Q2: What is the weakest known consistency assumption for ...
1
vote
1answer
92 views

Is there some natural equivalent of a Turing machine for set theory?

My question is motivated by the fact that (at lest from my biased viewpoint), Turing machines are more related to standard math, particularly PA. But set theory, in most of its versions, seem to go ...
1
vote
1answer
77 views

the smallest inaccessible cardinal

Is there a notation for the smallest inaccessible cardinal? Is the concept even coherent? I read on Wiki that under certain assumptions even Aleph0 is strongly inaccessible. I mean the kind of ...
4
votes
1answer
73 views

Ultraproducts by countably complete ultrafilter

I've recently learned about ultraproducts, but the source I learned from almost immediately after the definitions restricted to talking about countably incomplete ultrafilters. I know that the ...
4
votes
2answers
144 views

A chess problem in Kanamori's “The Higher Infinite”?

Just after Corollary 21.17 (on p289) of Kanamori's The Higher Infinite, he outlines the direction in which he wants to take his discussion of iterated ultrapowers. However, immediately after he ...
8
votes
1answer
88 views

Sequence of Ordinals and Ordinal Definability in Levy Collapse Extensions

Let $\kappa$ be an inaccessible cardinal. Let $G$ be generic for $Col(\omega, < \kappa)$, the Levy Collapse. If $f\in \text{ }^\omega \text{Ord}^{V[G]}$, is $f \in OD_{\text{ ...
3
votes
0answers
87 views

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 $\beta < \lambda$ ...
6
votes
1answer
149 views

Large cardinal and consistency: what are the main results today?

Lately I've been studying (on Jech "Set Theory" and Kanamori "The Higher Infinite") some problems related to the extension of measures. I've never studied set theory before and I'm quite baffled from ...
1
vote
1answer
105 views

Hyper-inaccessible cardinals

in definition of $\alpha$-inaccessible cardinals on Wikipedia, we can read : For example, denote by $ψ_0(λ)$ the λth inaccessible cardinal, then the fixed points of $ψ_0$ are the 1-inaccessible ...
2
votes
1answer
80 views

How $|\mathbb R|$ is not weakly compact

$\mathbb R$, of cardinality $|\mathbb R|=2^{\aleph_0}$ is not weakly compact. So there is a function $f$ from $[{\mathbb R}]^2$ (the subsets of $\mathbb R$ of size 2) to $\{0,1\}$ such that there is ...
0
votes
1answer
88 views

Weakly Compact Cardinal is Strong Limit

A question to Lemma 9.9 in Jech's Set Theory: Every weakly compact cardinal is inaccessible. I am working on the part that for $\kappa$ weakly inaccessible, $\kappa$ is strong limit. Jech writes: ...
3
votes
2answers
163 views

A Lowenheim-Skolem-like Result for Structures of Set Theory

While reading Kanamori's proof (page 59-60 in $\textit{Higher Infinite}$) that $\kappa$ is $\Pi_1^1$-indescribable if and only if $\kappa$ is weakly compact; it appears that he uses the following ...
8
votes
1answer
178 views

If $\kappa$ is measurable, does there exist a normal measure on $\mathcal P_{\kappa}(\kappa)$?

I'm trying to do exercise $10.7$ of Jech's Set Theory: If $\kappa$ is a measurable cardinal, then there exists a normal measure on $\mathcal P_{\kappa}(\kappa)$. For a set $A$, with $|A|\geq ...
0
votes
1answer
120 views

Measurable cardinals as sets

A philosopher said that measurable cardinals are the largest possible sets. Is this true? Are those sets at all? I mean, cardinals measure size of sets and for example $2=\{\{\},\{\{\}\}\}$ but can we ...
3
votes
0answers
130 views

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 ...
1
vote
0answers
91 views

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 ...
3
votes
1answer
156 views

Is it possible to prove $|V_\kappa|=\kappa$ for strongly inaccessible $\kappa$ without AC?

First, let me note that my definition of "strongly inaccessible" is that a nonzero ordinal $\kappa$ is strongly inaccessible if ${\rm cf}(\kappa)=\kappa$ and $\forall\alpha<\kappa, {\cal ...
3
votes
2answers
134 views

Weakly inaccessible cardinals and Discovering Modern Set Theory

So I've been trying to teach myself some set theory and I've come across some exercises in Just and Weese's Discovering Modern Set Theory. To whit: Pg. 180 Definition 20: A cardinal $\kappa$ is ...
3
votes
1answer
82 views

Measurable $\rightarrow$ Mahlo

On page 135 of Jech there is a step that I do not understand in the proof of "measurable implies Mahlo". We have already proved that "measurable implies strong limit". Jech writes "As $\kappa$ is ...
3
votes
2answers
125 views

Is there still any hope that the GCH could be equivalent to some large cardinal axiom?

Is there still any hope that the GCH could be equivalent to some large cardinal axiom? Even a simple yes or not answer will be fine. Thanks!!
3
votes
0answers
114 views

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 ...
3
votes
1answer
110 views

Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set

Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
17
votes
1answer
462 views

On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
3
votes
0answers
83 views

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 ...
2
votes
1answer
164 views

Is the expressive power of infinitary logic language $L(\infty,\infty)$ larger, the same or smaller than that of ZFC+large cardinal axioms?

In a previous question I learned that the power of statements of the form $\Pi_m^n$ or $\Sigma_m^n$ for arbitrary positive $m$ and $n$, is smaller than that of ZFC. For instance, the GCH cannot be ...
4
votes
0answers
79 views

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 Grothendieck–Verdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
4
votes
2answers
84 views

A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals

I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$. Apparently ...
1
vote
2answers
197 views

Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
6
votes
1answer
207 views

Weakly Compact Cardinals and the Extension Property

From $\textit{The Higher Infinite}$ by Kanamori, in his proof (page 39 - 41) that $\kappa$ is weakly compact if and only if $\kappa$ satisfies the Extension Property, the proof does something that I ...
3
votes
4answers
208 views

Are all large cardinal axioms expressible in terms of elementary embeddings?

An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ ...
2
votes
1answer
199 views

Did large cardinals exist before 1963?

I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
2
votes
1answer
107 views

Strongly inaccessible cardinals and certain kinds of universes

Write $\mathrm{tc}(*)$ for transitive closure. For all cardinals $\kappa$, let $H(\kappa)$ denote the collection of all sets $X$ such that $|X|<\kappa$, and For all $x \in \mathrm{tc}(X)$, it ...
7
votes
1answer
137 views

Does asserting the existence of large cardinals allow one to prove the existence of new subsets of $\mathbb{R}$?

ZFC proves the existence of certain subsets of $\mathbb{R}$. Suppose ZFC is consistent, and we adjoin a large cardinal axiom to ZFC, obtaining ZFC'. Assume ZFC' is also consistent. It it possible to ...
3
votes
1answer
152 views

How is Mahlo cardinal used?

I would like to know how Mahlo cardinals are used - as such examples may help me understand why they were created and so on.
2
votes
1answer
142 views

$V_k$ being a model of ZFC whenever $k$ is strongly inaccessible

ZFC implies that the $V_k$ is a model of ZFC whenever $k$ is strongly inaccessible.. So if $k$ is weakly inaccessible, it can't be a model of ZFC? Why is it like this? And ZF implies that ...
2
votes
1answer
87 views

Does TG prove that ZFC2 has a model?

Does TG (Tarski-Grothendieck Set Theory) prove that ZFC2 (second-order ZFC) has a model? Does it at least prove the consistency of ZFC2?
4
votes
1answer
147 views

Effective Well ordering of reals

Is there an effective (constructive) well order on reals ? I know several questions were already asked on this topic, and the answers were very good to this well known problem. My question is more ...
1
vote
0answers
77 views

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 ...
4
votes
2answers
277 views

Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
4
votes
2answers
103 views

Relation between inacessible cardinals and CH

I know that the two statements “There exists an inaccessible cardinal” and “Continuum Hypothesis” are both independent of ZFC. Now, are those two statements independent of each other? That older ...
6
votes
1answer
134 views

Intuition behind extenders

What's the best way to think about extenders? For instance if we have a normal $\kappa$-complete ultrafilter on $\kappa$, call it $D$, $M$ the ultrapower given by $D$, and if we look at $j: V \to M$, ...