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21
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694 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 ...
21
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0answers
807 views

Supercompact and Reinhardt cardinals without choice

A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it: Definition. A cardinal $\kappa$ is supercompact if for all ordinals ...
15
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0answers
356 views

What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ ...
15
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0answers
613 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 ...
13
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0answers
394 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 ...
13
votes
0answers
391 views

How to prove projective determinacy (PD) from I0?

Martin and Steel (in 1987?) showed that if there are infinite many Woodin cardinals then every projective set of reals is determined (PD). However, it is mentioned in many texts that in 1983/1984 ...
12
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0answers
433 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 ...
11
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0answers
192 views

Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
11
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0answers
251 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 > ...
10
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0answers
220 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 ...
9
votes
0answers
162 views

Consistency strength of $\aleph_2$-Souslin hypothesis

Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis? Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and ...
9
votes
0answers
167 views

Preserving Jonsson cardinals

I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ...
9
votes
0answers
269 views

Reinhardt cardinals and iterability

Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M_\alpha, \alpha\in ON,$ with $M_0=V,$ and elementary embeddings ...
7
votes
0answers
184 views

Countable choice in $L(\mathbb{R}^*_G)$

Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} ...
7
votes
0answers
211 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 ...
7
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0answers
370 views

$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars

Hi, I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem: Definition: Given some class of ...
6
votes
0answers
170 views

Where does this strengthening of I1 stand?

Let's call a cardinal $\delta$ an $\text{I1}$-tower cardinal if for each $A\subseteq V_{\delta}$, there exists a $\kappa<\delta$ such that whenever $\kappa<\alpha<\delta$ there is some ...
6
votes
0answers
172 views

Singular Jonsson cardinals

Is the consistency of the following well-known: $(*)$: There exists a singular cardinal $\kappa$ such that : (1) $\kappa$ is a Jonsson cardinal, (2) $\kappa$ is not a fixed point of the ...
6
votes
0answers
180 views

Core model for supercompact cardinals and iteration trees

I have a few somehow related questions: Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...
6
votes
0answers
222 views

$\delta$-strong compactness and generalized strong tree properties

Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties? Recall the definitions as per Joan ...
6
votes
0answers
210 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," ...
6
votes
0answers
429 views

A proposed axiom of Laver (updated)

A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom: (L) Some elementary embedding ...
6
votes
0answers
290 views

PFA and the compactness/incompactness of $\omega_2$

There are many consequences of the Proper Forcing Axiom (PFA) which essentially say that $\omega_2$ has a lot of compactness. For instance, Matteo Viale and Christoph Weiss have a few papers in ...
5
votes
0answers
102 views

Does the critical sequence for subalgebras of elementary embeddings with finitely many generators have order type $\omega$?

Suppose that $\lambda$ is a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j,k\in\mathcal{E}_{\lambda}$, then define ...
5
votes
0answers
178 views

Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?

Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$. Clearly, $\kappa$ is a cardinal. Question: Is it consistent that $\kappa = ...
5
votes
0answers
261 views

Cardinal characteristics without choice

(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...
5
votes
0answers
162 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 ...
5
votes
0answers
233 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 ...
5
votes
0answers
124 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
0answers
150 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 ...
3
votes
0answers
168 views

name for an intermediate notion between huge and 2-huge

I am employing a large cardinal notion that has been used explicitly before, and I am wondering if someone has given it a good succinct name. A cardinal $\kappa$ is huge if there is an elementary $j ...
3
votes
0answers
246 views

Internal large cardinal embeddings. How deep can we go?

Suppose, in V, that $\kappa$ is a large cardinal, say $\kappa$ is supercompact, and $j:V \to M$ is an ultrapower embedding witnessing the large cardinal properties of $\kappa$. From here, we can take ...
3
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0answers
262 views

On Successive Regular Cardinals With No Ladders

Definition: Let $\kappa$ be an $\aleph$ cardinal, we say that $\langle f_\alpha\colon\alpha\to\kappa\mid\alpha<\kappa^+\rangle$ is a ladder if every $f_\alpha$ is injective. Equivalently this is ...
2
votes
0answers
92 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 ...
2
votes
0answers
358 views

large cardinal tree properties as properties of sheaves

As follows from this talk Large Properties for Small Cardinals, p.7,p.4 http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf, the definitions of weakly compact ...
2
votes
0answers
195 views

a partial order not dense iff a measurable exists

For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order: (i) elements(objects) of $Ht_\kappa$ are classes X of sets of size $\kappa$ with the property that, ...
1
vote
0answers
119 views

A question regarding extendible cardinals and a result of M. Magidor

The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971): "Definition: Logic is called ...
0
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0answers
146 views

If C is a closed, unbounded subset of Ord, are the sets generated by the new class of ordinals Ord restricted to C, a Universe?

Given any universe, and some closed unbounded subset of Ord, is it possible toy generate a universe in the following way: Restrict Ord to a target club. Then generate all look the sets necessary to ...