# Tagged Questions

**10**

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**0**answers

224 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

**1**answer

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

**1**answer

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 ...

**8**

votes

**2**answers

220 views

### A Weakening of the Tree Property

If $f$ and $g$ are two functions, define $f \sim g$ if they differ only finitely often on their common domain.
The following property of a large cardinal arose from a problem in model theory. I am ...

**7**

votes

**1**answer

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 ...

**5**

votes

**1**answer

135 views

### A realcompact analogue of the Baire category theorem

Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...

**5**

votes

**1**answer

150 views

### Covering properties of strongly compact embedding

Let $\kappa$ be a $\mu$-strongly compact cardinal, which means that there is an elementary embedding $j:V\rightarrow M$, with critical point $\kappa$ such that $M$ is well founded (even closed under ...

**10**

votes

**1**answer

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**

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**0**answers

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

**1**answer

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 ...

**3**

votes

**2**answers

387 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

**1**answer

149 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

**1**answer

177 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**

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**0**answers

167 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

**1**answer

169 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**

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**0**answers

82 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

**6**answers

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

**2**answers

581 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

**1**answer

283 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

**0**answers

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**

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**0**answers

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 ...

**5**

votes

**1**answer

200 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

**1**answer

204 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 ...

**13**

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**0**answers

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 ...

**10**

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**2**answers

414 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**

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**0**answers

113 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

**2**answers

176 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

**1**answer

138 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?

**5**

votes

**1**answer

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

**0**answers

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 $\text{Coll}(\omega,\lt\kappa)$ ...

**5**

votes

**1**answer

82 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

**3**answers

293 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 ...

**6**

votes

**2**answers

286 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

**2**answers

239 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

**1**answer

482 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. ...

**1**

vote

**0**answers

219 views

### If there is a Reinhardt cardinal, then there is one universe? [closed]

If there is a nontrivial elementary embedding $j:V \to V$, then there is a universe which contains all the large cardinals.
Is there such a universe? Does this imply there is one universe from ...

**4**

votes

**1**answer

160 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 ...

**15**

votes

**1**answer

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 ...

**5**

votes

**1**answer

245 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

**1**answer

251 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

**1**answer

169 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

**4**answers

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

**3**answers

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 ...

**6**

votes

**3**answers

528 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

**1**answer

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 ...

**8**

votes

**1**answer

282 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

**1**answer

198 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 ...

**25**

votes

**3**answers

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

**1**answer

377 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

**1**answer

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 ...