10
votes
0answers
218 views

Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem: Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
7
votes
1answer
231 views

Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...
2
votes
1answer
145 views

The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$. One important example of left distributive algebras arises ...
7
votes
1answer
116 views

Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding. Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...
10
votes
1answer
271 views

What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...
6
votes
0answers
192 views

Does Sageev's result need an inaccessible?

In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...
10
votes
1answer
421 views

Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined. In "Turing ...
14
votes
6answers
2k views

What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$). Their non-existence is consistent with axioms of usual mathematics. It is provable that some of ...
10
votes
2answers
580 views

Questions about Prikry forcing and Cohen forcing

I have some questions. The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...
9
votes
1answer
282 views

Inner model in which every uncountable cardinal is large

The following is known: $(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$. My question is that: Are there any large cardinal ...
4
votes
0answers
173 views

$\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...
10
votes
0answers
245 views

cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension? To focus on the ...
13
votes
0answers
562 views

Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
5
votes
1answer
239 views

Woodin Cardinals and Inner Models

I have a few questions I have been thinking about that I could definitely use some insights on: Question 1. Since a Woodin cardinal is a "local" notion, defined with respect to some rank-initial ...
2
votes
0answers
102 views

Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse. Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...
15
votes
1answer
873 views

Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
22
votes
4answers
964 views

What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom. My question is that which one of these definitions are more popular or standard amongst set theorists? Any ...
5
votes
3answers
615 views

PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$. Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...
7
votes
1answer
240 views

Are superstrong stronger than strongly compact cardinals? (or vice versa)

In the last part of Kanamori's excellent "The Higher Infinite" there is a small diagram about the strength and consistency strength of some major large cardinal axioms. Below supercompact cardinals ...
25
votes
3answers
1k views

Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004: ...
10
votes
1answer
345 views

V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...
2
votes
1answer
216 views

$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem

Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define: $Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in ...
1
vote
1answer
255 views

$(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization: Definition 1: Let $M$ be a ...
8
votes
2answers
424 views

Is the tree of large cardinals linear?

Kanamori in the introduction of his book "The Higher Infinite" says: "The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a ...
14
votes
4answers
1k views

A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$. What happens after exponentiation? We have the following equation: $2^{N_n}=N_{2^{n}}$. (Which says: For all finite cardinal $n$ ...
3
votes
1answer
170 views

Failure of GCH at indescribable cardinals

Can $\Pi^m_n$ indescribable cardinal be the first one where $\text{GCH}$ fails? Hauser showed in Hauser,K.: Indescribable cardinals and elementary embeddings. J. Symb. Logic 56, 439457 (1991) that ...
3
votes
4answers
245 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 $HOD$. ...
2
votes
1answer
196 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
320 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 ...
6
votes
1answer
251 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):‎‎ ...
6
votes
1answer
219 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
260 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
359 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 ...
10
votes
2answers
526 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 ...
6
votes
1answer
272 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
4answers
680 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 ...
4
votes
1answer
256 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 ...
5
votes
1answer
212 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 ...
5
votes
1answer
156 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 ...
10
votes
2answers
543 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 ...
7
votes
2answers
398 views

What can the degrees of constructibility be?

If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the ...
9
votes
1answer
294 views

Homogeneous Namba-like forcing

Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly ...
8
votes
1answer
262 views

Elementary Embeddings and Relative Constructibility

Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding ...
7
votes
2answers
273 views

Relation between $\neg \square(\kappa)$ and the tree property at $\kappa$.

If $\kappa$ is an inaccessible cardinal then the tree property at $\kappa$ is equivalent to weak compactness of $\kappa$, which implies that $\square(\kappa)$ fails---that is, that every coherent ...
13
votes
2answers
624 views

Can measures be added by forcing?

The Lévy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be ...
8
votes
1answer
360 views

Can $\omega_1$ be supercompact?

Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"? In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where ...
12
votes
1answer
845 views

What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...
5
votes
2answers
270 views

Measures that are not OD

Is anything known about the consistency strength of the statement: "There is a normal measure (on a cardinal) that is not ordinal-definable"? In particular, is it consistent relative to the ...
20
votes
2answers
805 views

Can there be an embedding j:V to L, from the set-theoretic universe V to the constructible universe L, when V not= L?

Main Question. Can there be an embedding $j:V\to L$ of the set-theoretic universe $V$ to the constructible universe $L$, if $V\neq L$? By embedding here, I mean merely a proper class isomorphism from ...
4
votes
2answers
683 views

Large cardinals without the ambient set theory?

In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan Talk about cardinals without the (ambient) ...