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10
votes
1answer
150 views
n odd: $\bf\Delta^1_n$ wadge degrees are $< \bf\delta^1_{n+1}$
My adviser is out of town and there is a comment in the Van Wesep paper "wadge degrees and descriptive set theory" that I can't figure out.
Work in ZF+AD throughout.
As stated in the title, the ...
8
votes
1answer
209 views
$\Sigma^0_1\wedge\Pi^0_1$-Determinacy holds in the second admissible above the game
Let $T$ be a game tree and $T\in N\in M$, where $N,M$ are the two least admissibles containing $T$. Let $A$ be a boolean combination of two lightface open sets in $[T]$, or alternatively, a boolean ...
9
votes
1answer
250 views
Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set
Does anyone know of a reference for the fact that if $\lambda$ is a limit of Woodin cardinals, then the pointclass $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ is $\omega$-parameterized? By ...
7
votes
1answer
204 views
Determinacy from $\omega_1\rightarrow(\omega_1)^{\omega_1}$
Assuming the Axiom of Determinacy (abbreviated AD), Martin showed how to derive a rather strong partition on $\omega_1$, namely that $\omega_1\rightarrow(\omega_1)^{\omega_1}$. In "Infinitary ...
10
votes
1answer
212 views
Consistency strength of projective determinacy (PD)
Let PD stand for projective determinacy, and consider the two claims:
(1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals)
(2) Con(ZFC+PD) implies Con(ZFC + there are ...
8
votes
1answer
307 views
Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game
As a preface to this question, this is my first time asking on Math overflow, and this seemed like the sort of question that would be acceptable here. However, I apologize if it is not.
A method for ...
13
votes
3answers
749 views
Counterintuitive consequences of the Axiom of Determinacy?
I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
10
votes
1answer
209 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 ...
3
votes
1answer
207 views
sigma-algebra generated by OD sets
Assume $V=L(\mathbb{R})$ and the Axiom of Determinacy. Is every set of reals generated by ordinal-definable sets of reals under the operations of countable union and intersection?
The class of sets ...
6
votes
2answers
292 views
Weakly homogeneous trees under AD
If AD$_\mathbb{R}$ holds and $\kappa < \Theta$ then every tree $T$ on $\kappa$ is weakly homogeneous (Martin–Woodin, "Weakly homogeneous trees.") I recall hearing that the hypothesis can be ...
13
votes
0answers
358 views
How to prove projective determinacy (PD) from I0?
Martin and Steel (in 1987?) showed that if there are infinite many Woodin cardinals then every projective set of reals is determined (PD).
However, it is mentioned in many texts that in 1983/1984 ...
10
votes
1answer
343 views
Consistency strengths related to the perfect set property
I want a model of $\mathrm{MA}_{\sigma\mathrm{-centered}}+\neg\mathrm{CH}$ in which every set of reals in $L(\mathbb{R})$ has the perfect set property. In terms of consistency strength, it is known ...
6
votes
2answers
652 views
How additive is Lebesgue measure in ZF+AD ?
What is known about the additivity of Lebesgue measure under the Axiom of Determinacy?
That is, for what cardinals $\kappa$ do we have
with $|I| = \kappa$, for all functions $f : I \to ...
5
votes
1answer
269 views
value of Theta in ZF+AD
Since I found out about it, I've always been interested in the Axiom of Determinacy rather than the Axiom of Choice. Along these lines, I've kept flipping back to ...
9
votes
2answers
703 views
Martin's cone theorem and recursion theory
Martin's remarkable cone theorem in the theory of determinacy says the following:
Suppose $A\subseteq \omega^\omega$ is Turing invariant and determined. If $\forall x\exists y(x\le_T y\& y\in ...
0
votes
1answer
179 views
3
votes
1answer
231 views
Related Open Game in Analytic Determinacy
For this question, please refer to Chapter 33 page 638, Set Theory Millennium Edition, by Thomas Jech.
The proof of analytic games $G_A$ is converted into an open game $G^\ast$ on some suitable ...
