All Questions
24 questions
2
votes
1
answer
161
views
Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full ...
- 3,029
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
4
votes
0
answers
166
views
Consistency of definability beyond P(Ord) in ZF
Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that ...
- 7,545
6
votes
1
answer
301
views
A variation on pinned equivalence relations
Recall (see e.g. Zapletal, Pinned equivalence relations) that a Borel equivalence relation $E$ on $\omega^\omega$ is pinned iff for every forcing $\mathbb{P}$ and every $\mathbb{P}$-name $\nu$ we have ...
- 21.2k
7
votes
0
answers
234
views
Is this equivalent to (some version of) Hechler forcing?
Let $\omega^{<\omega}$ be the set of finite strings of naturals, and let $\omega^{<\omega}_{\not=\emptyset}$ be the set of nonempty finite strings of naturals. Consider the following forcing ...
- 21.2k
3
votes
0
answers
236
views
Borel equivalence relations on Ellentuck cubes
Is there a Borel equivalence relation $E$ on $[\omega]^\omega$ such that $E \not \leq_B E_0$ and for any $a \in [\omega]^\omega$ we have that $E \upharpoonright [a]^\omega$ is Borel bireducible with $...
- 101
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
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
6
votes
1
answer
419
views
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 ...
- 21.2k
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 $$(*)\quad L(\mathbb{R})^{V^\mathbb{P}}=L(\mathbb{R})^\mathbb{P}.$$ ...
- 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
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
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 ...
- 21.2k
3
votes
1
answer
422
views
Representation of meager sets in Cohen extensions
Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $...
- 1,688
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
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
10
votes
1
answer
411
views
The least admissible above a dominating real
Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...
- 21.2k
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
8
votes
0
answers
275
views
Proving regularity properties from forcing axioms
It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...
- 391
7
votes
3
answers
518
views
When can we reach a real by forcing?
I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such ...
- 21.2k
18
votes
1
answer
871
views
Three old questions on the Sacks forcing
I came across the two following Qs in 1970.
Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
- 2,281
5
votes
3
answers
769
views
Cohen algebra (generalization)
Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets.
The Cohen algebra has a combinatorial : it is the unique atomless complete ...
- 53
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
7
votes
0
answers
378
views
Sets of reals amenable to each L[x]
If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$?
This can be proved under the Axiom of ...
- 5,471