All Questions
9 questions
4
votes
0
answers
225
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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
6
votes
1
answer
419
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When does "sufficient genericity" actually suffice?
Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is $\mathbb{P}$-enforceable if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for ...
- 3,042
6
votes
2
answers
489
views
Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness
It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...
- 1,808
4
votes
1
answer
486
views
Forcing conflation for $L(\mathbb{R})$
Here's a very silly mistake I made recently: I claimed that if $\mathbb{P}\in L(\mathbb{R})$ is a forcing which adds a real, then $$(*)\quad L(\mathbb{R})^{V^\mathbb{P}}=L(\mathbb{R})^\mathbb{P}.$$ ...
CommunityBot
- 1
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: ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
9
votes
1
answer
559
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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 ...
- 32.1k
5
votes
1
answer
309
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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 ...
- 5,471
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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- 3,804