The tag has no wiki summary.

learn more… | top users | synonyms

6
votes
1answer
127 views

higher-order reflection

In the first-order context, "reflection" of a formula $\varphi(x)$ below $\kappa$ refers to the the following situation: There are many ordinals $\alpha<\kappa$ such that for all $a \in ...
6
votes
8answers
2k views

ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE. $\aleph_0, \aleph_1,\aleph_2\dots$ the lists ...
5
votes
0answers
135 views

A variant of Chang's model with choice

Let $M_n$, $n < \omega$, be a models of $ZFC$ with the same ordinals, closed under countable sequences. Let $\alpha_n$ be an ordinal which is a regular cardinal in $M_n$. Question 1: Is it ...
10
votes
2answers
472 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 ...
16
votes
4answers
1k views

What are the known implications of “There exists a Reinhardt cardinal” in the theory “ZF + j”?

This is, alas, in large part a series of questions on unpublished work of Hugh Woodin; it's also quite frivolous if Reinhardt cardinals turn out inconsistent. Definitions: Call $\kappa$ an ...
10
votes
1answer
320 views

Erdős cardinals and ineffable cardinals

In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have ...
10
votes
2answers
588 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}, ...
10
votes
0answers
275 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
237 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
150 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
2answers
227 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
1answer
117 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 ...
6
votes
1answer
198 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 ...
5
votes
1answer
137 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
1answer
151 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 ...
11
votes
1answer
278 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 ...
3
votes
2answers
393 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 ...
13
votes
0answers
563 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 ...
6
votes
0answers
195 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," ...
12
votes
1answer
345 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
1answer
424 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
1answer
150 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
183 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 ...
6
votes
2answers
426 views

Characterizing elementary embeddings of $L$ and $L_\alpha$ under 0#

Suppose 0# exists. It is clear that every order preserving map from the indiscernibles to the indiscernibles gives an elementary embedding from $L$ to $L$. Furthermore, following lemmas 18.7 and 18.8 ...
20
votes
0answers
631 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 ...
10
votes
2answers
417 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 ...
9
votes
1answer
297 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 ...
10
votes
0answers
172 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
172 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 ...
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 ...
2
votes
0answers
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 ...
9
votes
1answer
286 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
181 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
246 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
206 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
208 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 ...
5
votes
0answers
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
2answers
177 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
142 views
-2
votes
1answer
136 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
241 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 ...
15
votes
1answer
881 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 ...
2
votes
0answers
104 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 $\text{Coll}(\omega,\lt\kappa)$ ...
4
votes
2answers
186 views

Relation between indiscernibles for $L$ and for $L[A]$

It is known that $L\models 2^\kappa=\kappa^+$, and that for a set of ordinals $A$ we know that $L[A]\models \exists\lambda\forall\kappa>\lambda(2^\kappa=\kappa^+)$. In this sense, there is some ...
5
votes
1answer
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
3answers
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
2answers
287 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
240 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
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
0answers
221 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 ...