All Questions
36 questions with no upvoted or accepted answers
17
votes
0
answers
908
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 ...
- 32.1k
14
votes
0
answers
353
views
The failure of GCH al $\aleph_\omega$ by nice forcing
There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:
1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.
2) Woodin's ...
- 32.1k
12
votes
0
answers
213
views
Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$
The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...
- 490
12
votes
0
answers
458
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?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...
- 18.6k
11
votes
0
answers
490
views
$\Sigma^2_1$ and the Continuum Hypothesis
This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...
- 7,125
11
votes
0
answers
299
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. ...
- 2,111
11
votes
0
answers
747
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 well-...
- 32.1k
10
votes
0
answers
561
views
stationary reflection in $[\kappa]^\omega$
It is well-known that the following reflection principle is consistent relative to a supercompact:
For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
- 18.6k
9
votes
0
answers
177
views
Inner model of "CH + large cardinals" that satisfies MM?
I've been told that if $V$ has CH and large cardinals, then there is an inner model in which MM holds. The sketch is as follows:
Any $\Sigma^2_1$ sentence that can be forced is already true in such $V$...
- 335
9
votes
0
answers
258
views
Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$
I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
- 877
9
votes
0
answers
240
views
Homogeneity of a variant of Prikry forcing
Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...
- 32.1k
8
votes
0
answers
389
views
PCF conjecture and fixed points of the $\aleph$-function
Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and ...
- 32.1k
7
votes
0
answers
260
views
A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice ...
- 7,545
7
votes
0
answers
197
views
Query about iterated collapse forcing
I have heard it said that it is possible to use Woodin's iterated collapse forcing to prove that the theory $\textsf{ZF}$+"there exists a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow ...
- 2,125
7
votes
0
answers
443
views
Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC
I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...
- 3,038
7
votes
0
answers
294
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 ...
- 32.1k
7
votes
0
answers
213
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 ...
user42090
6
votes
0
answers
125
views
From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
- 7,545
6
votes
0
answers
318
views
Temporary destruction of measures in intermediate models
It is a well-known theorem that if $\kappa$ is measurable, then there is a generic extension in which $\kappa$ is no longer weakly compact, but we can force its weak compactness back and recover the ...
- 39.7k
6
votes
0
answers
171
views
Preserving supercompactness in intermediate forcing extensions
Let $\kappa$ be supercompact in $V$. Let $\mathbb{P}$ be one of the standard forcing notions (or an iteration of such), and for simplicity assume that $\mathbb{P}$ is ${<}\kappa$-directed closed (e....
- 1,281
6
votes
0
answers
206
views
Collapse successor of singular while preseving supercompactness
Suppose $\kappa$ is a supercompact cardinal. Is it possible to find a forcing which collapses $\kappa^{+\omega+1}$ to $\kappa^{+\omega}$ (all those $\kappa^{+n}$'s are preserved) while the ...
- 490
6
votes
0
answers
344
views
Inner models with all sets generic
Question: Under large cardinal axioms, what is the intersection of all inner models $M$ of ZFC such that every set in $V$ is set-generic over $M$?
Every set belongs to a generic extension of HOD, and ...
- 7,545
6
votes
0
answers
406
views
Mildly destroying all worldly cardinals below an inaccessible by forcing
We call a cardinal $\theta$ worldly if $V_\theta \models \mathsf{ZFC}$. If $\kappa$ is an inaccessible cardinal, then we have an unbounded set of worldly cardinals below $\kappa$, in fact even a club ...
- 325
6
votes
0
answers
472
views
A question about the HOD Conjecture
Is it known what other equivalence the HOD Conjecture has only under the assumption of a supercompact cardinal?
That is, under the assumption of an extendible cardinal $\delta$ (Or even HOD-...
6
votes
0
answers
242
views
Adding minimal subsets to $\aleph_\omega$
Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...
- 32.1k
6
votes
0
answers
406
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 ...
- 3,982
5
votes
0
answers
150
views
Consistency upper bounds for $\neg\square_{\aleph_\omega}$
In the introduction of Cummings and Friedman's $\square$ on the singular cardinals the following is written:
Failure of $\square_\lambda$ for $\lambda$ singular is stronger and rather more ...
- 2,286
5
votes
0
answers
276
views
Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$
Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property?
Because conditional $Σ^2_2$ absoluteness under $◊$ ...
- 7,545
5
votes
0
answers
304
views
Symmetry between V and HOD
Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$?
Note that $Σ_2^V$ is the best ...
- 7,545
5
votes
0
answers
143
views
Normal ideals on $[\lambda]^{<\kappa}$ concentrating on maximal cardinality
Is the following consistent? $2^\omega > \omega_2$, and there is a normal precipitous ideal $I$ on $[\omega_2]^\omega$ such that every $X \subset [\omega_2]^\omega$ of size $< 2^\omega$ is in $...
- 18.6k
4
votes
0
answers
225
views
Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same ...
- 7,545
3
votes
0
answers
200
views
Weak extender models for supercompactness without choice
Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N$...
- 7,545
3
votes
0
answers
249
views
Independence through forcing vs generic collapses
Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
- 7,545
3
votes
0
answers
90
views
What does the algebras of elementary embeddings look like in this forcing extension?
In this paper, Laver claims (on page 14 of the Arxiv version) that if there exists a rank-into-rank embedding, then in some upward Easton forcing extension, there are elementary embeddings $j,k:V_{\...
3
votes
0
answers
173
views
Two questions about the behavior of the continuum function
The first question asks about the global behavior of the power function in the case of finite gaps.
Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\...
- 32.1k
2
votes
0
answers
44
views
Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
- 51