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### Countable choice in $L(\mathbb{R}^*_G)$

Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} ...

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### 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 ...

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### What axioms (other than choice) have a taming effect on the ordering of cardinalities?

Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...

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240 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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### 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 ...

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219 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 ...

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273 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 ...

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### 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 ...

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280 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 ...

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381 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 ...

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### 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 ...

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### 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 ...

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217 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 ...

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313 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 ...

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380 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 ...

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369 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 ...

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### 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 ...

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297 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 ...

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### 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 ...

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### 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 ...