All Questions
Tagged with descriptive-set-theory forcing
11 questions
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
14
votes
1
answer
1k
views
Reverse-engineer forcing: am I reinventing the wheel?
In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but
I have a sinking feeling I’m reinventing the wheel; does ...
- 21.2k
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
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
9
votes
3
answers
530
views
What is the descriptive complexity of a set added by Cohen forcing?
I want to think of ZFC as not fully determining the powerset of the naturals, because you can add subsets with forcing and otherwise have a lot of control over the cardinality of the powerset of the ...
- 247
9
votes
1
answer
559
views
Just a little absoluteness might be cheaper?
Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...
- 21.2k
6
votes
1
answer
378
views
Iteration of random reals
Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\...
- 1,688
6
votes
1
answer
417
views
$\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$
Let me first recall some pretty standard notations:
$\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$;
$\mathfrak{b}$ is the bounding ...
- 1,151
5
votes
1
answer
309
views
Universally Baire Tree Representation of Projective Sets
In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
- 1,750
4
votes
2
answers
501
views
"Potentially club" filters on $\omega_2$
Short version: what can we say about subsets of $\omega_2$ which - in a generic extension where $\omega_2$ is the new $\omega_1$ - contain a club?
We could of course generalize beyond $\omega_2$, but ...
- 21.2k
4
votes
1
answer
746
views
Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?
My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...
- 135