All Questions
16 questions
6
votes
0
answers
179
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the ...
- 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
8
votes
1
answer
413
views
Precipitous ideal and inner model
Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$.
If we don't have the inner model condition, then we only know that $\...
- 1,490
17
votes
3
answers
1k
views
Minimum transitive models and V=L
Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?
You may assume that ZFC has transitive models. ...
- 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
5
votes
1
answer
265
views
Consistency strength of lifting through a lot of collapsing
What is the consistency strength of the following situation?
$j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$.
$\mathbb P$ is a forcing that ...
- 18.6k
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
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
6
votes
0
answers
242
views
Does $0^{\#}$ imply that $L$'s set generic multiverse has irreconcilable theories?
Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P}...
- 1,080
18
votes
1
answer
2k
views
What is the modal logic of outer multiverse?
The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as possible worlds connected to each other via a certain accessibility relation.
The modal logic associated ...
14
votes
3
answers
934
views
Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model theory
I have two unrelated question.
First question. To motivate the question, let me explain an example. The natural way to force the failure of singular cardinals hypothesis ($SCH$), is to start with a ...
- 32.1k
6
votes
1
answer
385
views
Absoluteness for the Chang model
Often, large cardinals imply that many definable inner models and transitive sets have theories which are absolute under forcing. For example, assuming a proper class of Woodin cardinals, the theory ...
- 20.9k
6
votes
0
answers
176
views
Breaking determinacy with forcing, and then fixing it
While forcing is usually presented over models of ZFC, it works equally well over models of ZF (or even less). However, the general theory of forcing becomes much stranger (much like the general ...
- 20.9k
8
votes
2
answers
478
views
Consistency strength of being strong cardinal and indestructible under collapses
What is the consistency strength of the following statement:
$\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...
- 32.1k
3
votes
4
answers
439
views
What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?
ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$.
It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...
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