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3
votes
2answers
450 views

A question about “local” versus “global” large cardinal axioms

The terms "local" and "global" when applied to large cardinal axioms seem to have a well understood intuitive meaning, although a formalized definition of them in (a meta-language for)ZFC might be ...
3
votes
1answer
161 views

Are normal ultrafilters generated by conditional closure systems?

Suppose that $\kappa$ is a cardinal, $X$ is a set with $|X|>\kappa$, and $\mathcal{U}\subseteq P(P_{\kappa}(X))$ is a normal ultrafilter. We say that a collection $C\subseteq P_{\kappa}(X)$ is a ...
5
votes
1answer
210 views

Is there a large-cardinal completeness theorem for $L$?

I cannot currently find the original, but if memory serves, Goedel once speculated that there might be a "large-cardinal completeness theorem for $V.$" This theorem would state: *Theorem. For every ...
10
votes
0answers
210 views

Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident. Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...
4
votes
1answer
195 views

Can a Measureable Cardinal Become the Least Weakly Compact Cardinal in a Forcing Extension?

I am trying to establish whether it is consistent that some property holds at the least weakly compact cardinal. I know that the property holds at measureables. Hence (hoping everything else goes ...
2
votes
0answers
87 views

Do there exist projective realcompact covers?

In 1958 Gleason [1] constructed projective covers in the category of compact Hausdorff spaces. These may be characterized in many ways. One description that is most interesting to me: $p:EX\to X$ is a ...
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
622 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
309 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
202 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 ...
11
votes
1answer
296 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 ...
5
votes
1answer
250 views

$\omega$ universally Baire sets, tree representations

I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me. ...
6
votes
1answer
267 views

regularity of ultrafilters

An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap_{\alpha \in y} X_\alpha ...
15
votes
0answers
588 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 ...
10
votes
2answers
459 views

Does there exist a supercompactness theorem?

Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of ...
5
votes
0answers
120 views

A result of Steel on characterizing lightface pointclasses

In the article Projectively wellordered inner models, Steel proves the following theorem (4.12): Theorem: Let $n < \omega$ and suppose $\mathcal{M}_n^{\sharp}$ exists. Let ...
5
votes
2answers
214 views

Can we have a $\kappa$-Suslin tree where $\kappa$ is above a measurable cardinal?

Question: Can we have a set theory in which there exists a $\kappa$-Suslin tree with $\kappa$ larger than the least measurable cardinal? A $\kappa$-Suslin tree is a tree with levels indexed by ...
-2
votes
1answer
182 views

is the existence of an inaccessible cardinal stronger than just CON(ZFC)? [closed]

is it even stronger than that ZFC has a transtitive model?
-2
votes
1answer
137 views

what's the limit of cardinals can be proved to exist in ZFC

what is the smallest cardinal k can not be proved to exist in ZFC?And what is the smallest cardinal k ,that the existence of k can imply CON(ZFC)?
5
votes
1answer
255 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
108 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 $\text{Coll}(\omega,\lt\kappa)$ ...
5
votes
1answer
91 views

Property of $L$ Relating to Reflection

The idea of the question is whether it is ever possible that $L$ is so nice in the sense that $\{L_\alpha\}$ does not incorrectly "guess" a bigger inaccessible than $L$ really has, as long as ...
1
vote
3answers
304 views

Can the structure of an ultrafilter determine the structure of its ultrapower?

Usually we work with ultrafilters as pure sets without any structure. Q1. Is there any important notion of structure on an ultrafilter? Q2. Is there any non-trivial notion of structure on ...
7
votes
2answers
297 views

What is the shape of large cardinal tree in implication strength order?

There are two natural orders on large cardinal axioms. (a) Consistency strength order $\sigma \leq_C \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta)\longrightarrow Con(ZFC+\sigma)$ (b) ...
6
votes
2answers
270 views

Weakly Compact Cardinal and Iterability

In $\textit{Set Theory}$ by Jech 1978 edition, in the proof of Lemma 32.5 which you can hopefully see at the Google book link. In the course of the proof using the tree property, he produces from any ...
4
votes
1answer
533 views

Origin of “Woodin cardinal”

Sorry if this is a completely stupid question (I'm a not a set-theorist, though I've been doing some reading in the subject), but I was wondering, specifically, about the exact provenance of the name. ...
4
votes
1answer
167 views

What does the set of cardinals admitting a k-additive measure look like?

Consider an infinite cardinal $\kappa$. Is it the case that the existence of a $\kappa$-additive measure on some infinite set implies the existence of such a measure on every infinite set of size ...
16
votes
1answer
1k 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 ...
6
votes
1answer
311 views

Which large cardinals are upward reflecting?

Let the first order formulas $p(x)$ and $wi(x)$ assert "$x$ is a large cardinal of type $p$" and "$x$ is weakly inaccessible" respectively. The large cardinal type $p$ is upward reflecting if ...
9
votes
1answer
324 views

Suslin trees in ccc ideals

A countably complete ideal $I$ on a set $Z$ ideal is c.c.c. when there is no uncountable family of pairwise disjoint $I$-positive subsets of $Z$. If such an ideal exists, then there exists a weakly ...
1
vote
1answer
172 views

Different Methods for Forcing Total Failure of Generalized Continuum Hypothesis

It seems there are few papers on total failure of Generalized Continuum Hypothesis. As an example Merimovich in 2007 constructed a model of ZFC such that GCH fails everywhere assuming consistency of ...
22
votes
4answers
1k 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 ...
7
votes
3answers
692 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
3answers
589 views

Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?
7
votes
1answer
261 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 ...
8
votes
1answer
315 views

What is known about equiconsistency of PFA and existence of supercompact cardinals?

Question: What is the last status of known partial results and new approaches on the problem of equiconsistency of PFA and existence of supercompact cardinals?
4
votes
1answer
201 views

Increasing and Descending Chains of Inner Models for Measurable Cardinals

Notation: For each measurable cardinal $\kappa$ and a non-trivial $\kappa$-additive two-valued measure $\mu$ on it let $M_{\kappa,\mu}$ be the corresponding inner model. Question: Assuming suitable ...
27
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: ...
12
votes
1answer
408 views

How strong is the iterated consistency of ZFC?

Let $T_0$ be $\mathsf{ZFC}$ and, for $n\in\omega$, set $T_{n +1}=T_{n}+\mathrm{Con}(T_{n})$. Question 1: Is there a natural number $n$ such that $T_{n}$ is equiconsistent with $\mathsf{ZFC}+$ ...
10
votes
1answer
375 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
220 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
262 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 ...
12
votes
1answer
358 views

capturing small sets in small factors

Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known: ...
10
votes
2answers
198 views

What is the strength of chains of 1-extendibles?

Let $X$ be a collection of cardinals such that if $\kappa,\lambda\in X$ and $\kappa<\lambda$, then there is a non-trivial elementary embedding $j:V_{\kappa+1} \to V_{\lambda+1}$ with $crit(j) = ...
8
votes
2answers
470 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 $n$ ...
5
votes
0answers
149 views

Impact of Supercompacts on Measurables

It is consistent that the least measurable cardinal can carry exactly one normal measure but in almost all models for this theory there is no supercompact cardinal. It seems existence of a ...
8
votes
0answers
183 views

Critical Points of Rank-into-Rank Embeddings

A rank-into-rank embedding is a non-trivial elmentary embedding from a rank initial segment of $V$ into itself: $j:V_\delta\prec V_\delta$. Define the critical sequence of such an embedding by setting ...
6
votes
2answers
385 views

The First Failure of GCH in Large Cardinals Smaller than Measurables

A well known theorem by Scott says: If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then ...
3
votes
1answer
177 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 ...