All Questions
Tagged with forcing determinacy
13 questions
12
votes
1
answer
448
views
Comparing generic versions of $\mathbb{R}$
This question was previously asked and bountied at MSE, unsuccessfully.
I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and ...
- 21.2k
12
votes
0
answers
506
views
Can a generic $\mathbb{R}$ have a new cardinality?
This question was asked and bountied at MSE, without success.
My main question is whether, starting with a model of determinacy, a "generic $\mathbb{R}$" could be different in cardinality ...
- 21.2k
11
votes
2
answers
709
views
Which forcings preserve (some) determinacy?
The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
- 21.2k
9
votes
0
answers
271
views
Which forcing types preserve the axiom of determinacy?
Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...
- 39.7k
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
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
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 ...
- 21.2k
5
votes
1
answer
258
views
Forcing over models of determinacy
Consider a ctm $\mathfrak{M}$ of $ZF+AD^+$. Is it possible to force over $\mathfrak{M}$ to get a model of ZFC which satisfies further the following:
Every projectively definable family of sets of ...
- 1,416
5
votes
1
answer
651
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.
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- 51
5
votes
0
answers
246
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Forcing absoluteness in the setting of second-order arithmetic
There are some results about (set-size) forcing absoluteness for first-order properties in $L(\mathbb{R})$ and for $\mathbf{\Pi}^1_{\infty}$ properties when one works over $\mathsf{ZFC}$. My question ...
- 5,821
4
votes
0
answers
107
views
Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of ...
- 7,545
3
votes
1
answer
334
views
Getting measures (especially on $\omega_2$) from potential clubs
This is a spinoff of this earlier question of mine.
Short version:
What measures in $L(\mathbb{R})$ can be gotten from "potentially club" filters, under appropriate hypotheses?
Long version: ...
- 21.2k
2
votes
0
answers
62
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545