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6
votes
1answer
225 views

Solovay's Theorem on Partitions of Stationary Sets and Weak Choice Principles

There is a weak choice principle called $DC_\lambda$ which holds in $L(V_{\lambda+1})$ under the assumption of a non-trivial elementary embedding $$j:L(V_{\lambda+1})\prec L(V_{\lambda+1})$$ and it is ...
12
votes
1answer
209 views

extending elementary embeddings from initial segments of V to all of V

Suppose $j:V_\lambda \rightarrow V_\eta$ is (elementary and) cofinal. Can $j$ be extended to all of $V$? (Subsidiary question: What conditions are there on an ultrafilter/extender/whatever so that ...
3
votes
4answers
262 views

What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$. It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...
5
votes
1answer
229 views

Adding large sets not containing countable ground model sets

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension). Before I state the question, let me add ...
2
votes
1answer
214 views

Are measures of a measurable cardinal measurable? (Edited and Updated Version)

Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers. Definition: Let $\kappa$ be a measurable cardinal. Define: ...
11
votes
1answer
328 views

Is the inclusion version of Kunen inconsistency theorem true?

The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the ...
5
votes
1answer
175 views

what kind of ordinal is the degree of strongness of a partially strong cardinal (Edited and revised)

For an infinite cardinal $\kappa$ and an ordinal $\lambda>\kappa,$ $\kappa$ is called $\lambda-$strong, if there is a non-trivial elementary embedding $j: V \rightarrow M$ with $crit(j)=\kappa$ ...
5
votes
1answer
179 views

What are the Possible Large Cardinals of $L[X]$?

I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the ...
3
votes
1answer
178 views

Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)

We know that if ‎$‎‎\kappa‎‎$ is a‎ ‎measurable ‎cardinal ‎and ‎‎$‎‎\mu$ be a‎ ‎two-valued ‎non-trivial‎‎$‎‎‎\kappa‎$-additive ‎measure ‎on ‎it ‎the‎n the corresponding inner model produced by ...
2
votes
1answer
197 views

Can we flex the rigid models by enough power?

Definition (1): ‎An ‎‎$‎‎‎\mathcal{L}‎$ -‎ ‎structure ‎‎$‎‎‎\mathcal{M}‎$ ‎called ‎"‎‎rigid" ‎iff ‎‎there ‎is ‎no ‎non-trivial automorphism on ‎$‎\mathcal{M}‎$.‎ ‎‎ Definition (2): ‎An ...
10
votes
4answers
573 views

Proving ZFC results using large cardinals

There are many $ZFC$ results that their proof uses forcing: The idea is that we force the statement to be true, and then using absoluteness (or other reasons) we conclude that the result is true in ...
14
votes
3answers
538 views

Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...
7
votes
1answer
203 views

Character of normal ultrafilters

The character of an ultrafilter $U$, denoted $\chi(U)$, is the minimal size of an $A \subseteq U$ such that $(\forall x \in U ) (\exists y \in A) y \subseteq x$. This cardinal characteristic has been ...
5
votes
1answer
147 views

Is there a truth approximation on a‎ cumulative hierarchy‎‎‎‎?

‎‎‎Note ‎to ‎the ‎following ‎well known theorem:‎ Theorem (1): ‎If ‎‎$‎‎\kappa‎‎$ ‎be a ‎‎"measurable" ‎cardinal ‎and ‎‎$‎‎‎\mathcal{F}‎$ be a‎ ‎"non-principal ‎$‎‎‎\kappa‎$-complete normal‎" ...
6
votes
1answer
266 views

Are larger large cardinals less expressible?

First note to the following well known theorems:‎‎ Theorem (1): ‎The ‎notion ‎of ‎"‎$‎‎x$ ‎is a strongly inaccessible cardinal‎" ‎is ‎first ‎order ‎expressible ‎and ‎‎$‎‎\Pi_{1}$‎. Theorem (2):‎‎ ...
7
votes
1answer
348 views

How strong are large cardinal properties of Ord?

Ordinal numbers are generalizations of natural numbers. In this sense the "proper class" of all ordinals ($Ord$) is very similar to "infinite" set of all natural numbers ($\omega$). In the other ...
6
votes
1answer
662 views

Is there a monster behind the trees?

‎First Fix the following notation:‎ ‎ $‎‎\forall ‎\kappa‎\in Card~~~Tp(‎\kappa‎):="‎\kappa‎~has~tree~property"$‎ ‎‎‎‎ The large cardinals as "monsters of heaven" live everywhere in the land of ...
21
votes
0answers
667 views

categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result? Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...
11
votes
0answers
244 views

Absoluteness of “$\kappa$-homogeneously Suslin” for sets of reals

What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals? For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > ...
6
votes
1answer
244 views

Vopenka's Principle for non-first-order logics

(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.) Vopenka's Principle ($VP$) states that, given any ...
10
votes
2answers
273 views

Singular successors without large cardinals

Assuming the axiom of choice we have that successor cardinals are regular. However as one of the first examples of uses of forcing show, it is consistent relative to $\sf ZF$ that $\omega_1$ is ...
13
votes
3answers
394 views

Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...
15
votes
2answers
774 views

Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following: Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$ In the paper "The ...
3
votes
1answer
238 views

Large cardinals and mild extensions

It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are: Question ...
12
votes
0answers
419 views

Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...
2
votes
2answers
177 views

Preservation of measurable cardinals in mild extensions

I would like some help with the proof of the preservation of measurable cardinals in mild extensions, as I am a bit new with forcing. By mild extensions, I mean the generic extension produced from a ...
10
votes
2answers
584 views

What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals? Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...
3
votes
1answer
201 views

ultrafilter characterisation of huge cardinals

A cardinal $\kappa$ is huge iff there is $\lambda>\kappa$ and a $\kappa$-complete normal ultrafilter on $P_{\leq \kappa}(\lambda)$, or, equivalently, on the set of families of subsets of $\lambda$ ...
10
votes
2answers
548 views

Failure of diamond at large cardinals

What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact. Remark. The problem of forcing ...
5
votes
2answers
523 views

Good Books about Large Cardinals

I am very new to set theory and have only learned the basics up to cardinal and ordinal aritmetic. I would like to learn about large cardinals and I am reading Thomas Jech's Set Theory. I have read ...
6
votes
1answer
671 views

Do Measurable Cardinals Exist? (assuming ZFC)

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes: It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable ...
6
votes
1answer
280 views

Indescribability of cardinals and categoricity of $V_\kappa$

If $\kappa$ is an inaccessible cardinal then $V_\kappa$ is a model of $\sf ZFC_2$ ($\sf ZFC$ with a second-order replacement axiom). If there are many inaccessible cardinals then there are many ...
9
votes
2answers
424 views

Every weakly compact cardinal is Mahlo

This is a reference question. Does anyone know any book or paper that has the proof that every weakly compact cardinal is Mahlo, using only combinatorics? I know the definition of weak compactness ...
10
votes
1answer
287 views

Consistency strength of projective determinacy (PD)

Let PD stand for projective determinacy, and consider the two claims: (1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals) (2) Con(ZFC+PD) implies Con(ZFC + there are ...
10
votes
1answer
485 views

Why does the generalised Galvin-Prikry Theorem only hold at Ramsey cardinals?

The Galvin-Prikry theorem says that Borel sets are Ramsey. This means that for every Borel set $S\subseteq[\omega]^\omega$, there is an $A\in\[\omega]^\omega$ such that either $[A]^\omega \subseteq S$ ...
9
votes
4answers
758 views

On the large cardinals foundations of categories

(This question was posted on math.SE over two weeks ago, but received no answer. I am therefore posting it here as well.) It is well-known that there are difficulties in developing basic category ...
10
votes
1answer
437 views

Forcing mildly over a worldly cardinal.

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a ...
29
votes
2answers
895 views

Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...
5
votes
2answers
299 views

The Kunen inconsistency and definable classes

There is a tension between (1) interpreting proper class talk in set theory as talk about first-order formulas and satisfaction; and (2) taking it to be an interesting and non-trivial result that ...
6
votes
2answers
376 views

Partition relation, almost a Ramsey cardinal?

Is it consistent with ZFC to have a cardinal $\kappa$ which is not Ramsey and $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds for some $n\in \omega$? The partition relation $\kappa ...
7
votes
1answer
392 views

More on Kunen's inconsistency result

I would like to suggest another argument for Kunen's inconsistency result, and I wonder to know if the argument is correct. I am also interested to see, if the proof is correct, which part of the ...
2
votes
1answer
500 views

Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject. Part 1 is about foundations. Much of the ...
4
votes
1answer
263 views

Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal

Is anything known about the consistency strength of the following statement? $\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap ...
7
votes
1answer
278 views

Applications of higher-order reflection principles

Let $L_3$ be the third-order language of set theory with identity on the first sort. Variables $x$ are first-order, $y$ are second-order, and $z$ are third-order. In the style of Lévy and Bernays, we ...
5
votes
1answer
231 views

Consistency of many Erdos cardinals

Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$? Or of various weakenings about how far such cardinals extend into ...
4
votes
1answer
331 views

Models of Determinacy

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...
4
votes
1answer
154 views

$\omega$-small and properly small premice.

Let $\mathcal M$ be a premouse. $\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M$, then $\mathcal ...
5
votes
1answer
162 views

Versions of large cardinals with target model in a generic extension

(I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see http://www.jstor.org/stable/4093051 or http://arxiv.org/abs/0711.0680.) A measurable cardinal is a cardinal ...
5
votes
3answers
317 views

A question about Mitchell/Steel Fine Structure and Iteration Trees

In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the ...
9
votes
2answers
572 views

Kunen's inconsistency result

A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know ...