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21
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658 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 ...
19
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0answers
737 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 ...
13
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0answers
569 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
376 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
413 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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349 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 ...
11
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0answers
237 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
votes
0answers
191 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 ...
10
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0answers
253 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 ...
9
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0answers
252 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 ...
8
votes
0answers
168 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
votes
0answers
110 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. ...
7
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157 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
424 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
199 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
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0answers
271 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
147 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
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0answers
114 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
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0answers
147 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 ...
5
votes
0answers
341 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 ...
4
votes
0answers
189 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 ...
3
votes
0answers
232 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
246 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
85 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
346 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
193 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, ...