Questions tagged [determinacy]

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Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game

Consider the following infinite perfect information game with two players (the name I gave in the title of the post it totally made up): at each round $i \in \omega$, player $\mathrm{I}$ picks a ...
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  • 927
7 votes
1 answer
278 views

Uniformization under AD

Can the following uniformization statement be proved by $ZF+AD+DC$? For any binary relation $R\subseteq \mathbb{R}^2$ with the property that $\forall x (\{y\mid R(x,y)\}\mbox{ is at most countable ...
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  • 3,771
3 votes
1 answer
109 views

Analytic sets and Turing determinacy

I wonder whether the following question have a positive answer within $ZFC$. Question If $\{A_n\}_{n\in \omega}$ is a sequence of analytic sets so that $\bigcup_n A_n=2^{\omega}$, then there must be ...
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  • 3,771
3 votes
1 answer
733 views

I can't believe it's not Replacement!

(I feel like I might have to apologise in advance for this question, but oh well..) I just rediscovered a comment from Asaf K here on MO that states that full Replacement is not needed for Borel ...
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7 votes
0 answers
194 views

Determinacy of symmetric games

Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all ...
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16 votes
1 answer
586 views

What sets can be unraveled?

A set $X\subseteq\omega^\omega$ is unravelable iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $...
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12 votes
0 answers
458 views

Can a generic $\mathbb{R}$ have a new cardinality?

This question was asked and bountied at MSE, without success. My main question is whether, starting with a model of determinacy, a "generic $\mathbb{R}$" could be different in cardinality ...
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12 votes
1 answer
330 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 ...
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14 votes
1 answer
720 views

Is there a minimal inner model for determinacy?

Assume $\sf ZF+AD$. Is there some inner model $M$ containing all the ordinals such that $M\models\sf ZF+AD$ as well? What if we require $\omega_1$ and/or $\omega_2$ to be computed correctly? Can we ...
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3 votes
1 answer
251 views

Pointwise definable models of determinacy

Suppose that the Axiom of Determinacy (AD) holds in $L(ℝ)$, and for some statement φ, $α$ is minimal such that $L_α(ℝ)⊨φ$. Do definable (in $L_α(ℝ)$) elements of $L_α(ℝ)$ form an elementary ...
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7 votes
1 answer
178 views

Aronszajn Trees when AC fails

This question may be easy and indicative of my ignorance about the failure of the axiom of choice. If so, I apologize. Below assume $\mathsf{DC}$ but not $\mathsf{AC}$. Suppose we have a partial order ...
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2 votes
0 answers
169 views

An analytic game and Ramsey's theorem

Let $$P=\{(l_n, m_n)_{n=1}^t: l_n, m_n,t\in\mathbb{N}, l_1<\ldots <l_t, m_1<\ldots <m_t\}$$ and suppose that $T$ is some subset of $P$. Suppose that $T$ also has the following properties: ...
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5 votes
0 answers
221 views

Forcing absoluteness in the setting of second-order arithmetic

There are some results about (set-size) forcing absoluteness for first-order properties in $L(\mathbb{R})$ and for $\mathbf{\Pi}^1_{\infty}$ properties when one works over $\mathsf{ZFC}$. My question ...
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0 votes
0 answers
140 views

Strength of $Δ^1_{2n}$ determinacy

According to Lightface mice with finitely many Woodin cardinals from optimal determinacy hypotheses by Yizheng Zhu, theorem 1.1, over $\mathbf{Σ^1_{2n+1}}$ determinacy, $Δ^1_{2n+2}$ determinacy is ...
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4 votes
1 answer
204 views

Strong partition property + DC + existence of non-principal ultrafilter on $\omega$

It was mentioned after Theorem 30.27 in Kanamori's Higher Infinite that Woodin constructed a model of $DC$ + there exists unboundedly many many $\kappa<\Theta$ such that $\kappa \to (\kappa)^\...
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1 vote
0 answers
1k views

The Rise and Fall of Dictators & How it Depends on Our Choice

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively. Shelah, Saharon, On the Arrow property. Adv. in Appl. ...
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2 votes
1 answer
273 views

Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$

Let $Z_2$ and $Z_3$ be second and third order arithmetics respectively. In $Z_2$'s language, $\text{AD}$ (the axiom of determinacy) and $\text{PD}$ (projective determinacy) are stated the same way (...
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11 votes
1 answer
548 views

The Axiom of Determinacy and the Banach-Mazur game

The Wikipedia article on the Axiom of Determinacy (AD) claims: Equivalent to the axiom of determinacy is the statement that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is ...
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9 votes
0 answers
241 views

A bi-modal logic related to determinacy

The short version of my question is as follows. There is a natural (I hope!) way to associate a bimodal theory to a game (two-player, perfect-information, length-$\omega$, on $\omega$); are there "...
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9 votes
1 answer
616 views

Can the Turing degrees be linearly ordered?

Assuming the axiom of choice, every set can be linearly (indeed, well-) ordered. However, without choice this can fail, as witnessed most drastically by the consistency of amorphous sets. More ...
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3 votes
2 answers
320 views

Strong limit cardinals in AD

In this question, I will be working in ZF. Let $h(\kappa)$ for a cardinal $\kappa$ (not necessarily an ordinal) be the smallest ordinal $\alpha$ such that there is no surjection from a set of size $\...
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2 votes
1 answer
158 views

Determinacy and polynomial time degrees

Is there a function $f:2^{<ℕ}→\{0,1\}$ such that for all $X∈{2^ℕ}$ with $X_{2i+1}=f(X_0,...,X_i)$ all hyperarithmetical properties of the polynomial time degree of $X$ are independent of $X$? The ...
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1 vote
1 answer
286 views

Under ZF + DC + AD, is it known what the properties are of the Hartogs number for $\mathcal{P}(\kappa)$ for some $\kappa>\aleph_0$?

It is a well-known result by Woodin that the Hartogs number $h(\mathbb{R})$ (more commonly known as $\Theta$) is a Woodin cardinal (in HOD) assuming ZF + AD + DC. This is equivalent to $h(\mathcal{P}(\...
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12 votes
0 answers
335 views

Undetermined copy/diagonalize games without CH

This question was the motivation behind an earlier question of mine; having thought about it some more, that question seems nontrivial and the connection is actually pretty tenuous anyways, so I've ...
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3 votes
1 answer
238 views

Does determinacy in $L(\mathbb{R})$ implies projective determinacy (in $V$)?

Does $AD^{L(\mathbb{R})}$ directly implies projective determinacy? At least it certainely implies $PD$'s consistency.
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7 votes
1 answer
438 views

Limitations of determinacy hypotheses in ZFC

When considering (set-theoretic) games, we have three parameters we can adjust: Definability of the payoff set The set of legal moves The length of the game When working in $\textsf{ZFC}$, what are ...
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10 votes
1 answer
399 views

Is there a natural inner model of AD$_\mathbb{R}$?

The question is as in the title, but let me explain a bit. Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm ...
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11 votes
1 answer
476 views

Is determinacy on an infinite Dedekind finite set consistent?

Consider $\mathrm{AD}_X$, determinacy for games where players pick moves from $X$. We know that it is consistent for $X = \omega$ or $\mathbb{R}$ (under large cardinal assumptions), but inconsistent ...
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6 votes
1 answer
342 views

How much real determinacy can live in $L(\mathbb{R})$?

It's well-known that AD$_\mathbb{R}$ fails in $L(\mathbb{R})$, provably in ZFC. This is because: AD$^{L(\mathbb{R})}$ implies DC$^{L(\mathbb{R})}$. Over ZF+DC, AD + "Every set of reals has a scale" ...
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2 votes
0 answers
95 views

Forgetful Determinacy and Gale-Stewart theorem

I have been studying a 1969 article by Rabin that proofs his apparently very influential Tree Theorem (that says the monadic second-order theory of 2 successors, S2S, is decidable). To give a bit of ...
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  • 171
5 votes
0 answers
183 views

The club filter in definable preorders

So this is an embarrassing question. Call a preorder $\mathbb{P}$ good if it has the following properties: Every countable chain in $\mathbb{P}$ has a least upper bound. $\mathbb{P}$ is directed (any ...
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1 vote
1 answer
334 views

Defining cones and Turing cones

In Set Theory Jech defines a cone to be a subset of the Baire Space $\mathcal{N}$ of the form $$\operatorname{cone}(x_0)= \{x : x_0 \in L[x]\}$$ where $x_0 \in \mathcal{N}$. Jech then defines the ...
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  • 180
3 votes
1 answer
278 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: ...
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6 votes
1 answer
355 views

$\operatorname{AD}$ and the measurability of $\omega_1$

Are there proofs of the measurability of $\omega_1$ (under $\operatorname{AD}$) that do not use Turing degrees nor the $\Sigma_1^1$ boundedness lemma? I've been struggling to find an "elementary" ...
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4 votes
0 answers
353 views

Why is this transfinite game not determined?

This question originates from the paper On the Axiom of Determinateness by Jan Mycielski, section 7. Given a set $X$ and an ordinal $\alpha$, the author defines a transfinite game of length $\alpha$ ...
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6 votes
0 answers
153 views

Breaking determinacy with forcing, and then fixing it

While forcing is usually presented over models of ZFC, it works equally well over models of ZF (or even less). However, the general theory of forcing becomes much stranger (much like the general ...
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5 votes
1 answer
220 views

Spreading sets - especially without choice

For what follows, I work in ZF+AD+DC. However, the questions below are not obviously trivial in ZFC, so I'm also interested in results in that system. Suppose I have a set $X\subseteq \mathbb{R}$. ...
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5 votes
1 answer
439 views

Comparing the sizes of uncountable sets of reals under AD

Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\...
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6 votes
1 answer
271 views

Ordinal-definable witnesses to the perfect set property?

This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate. Throughout we work in ZF+AD. My question is: If $A$ is an uncountable OD set of reals, ...
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6 votes
1 answer
259 views

Proof of a soft version of Moschovakis's lemma

The following fact, which I've heard being called "soft version of Moschovakis's lemma" (see top answer here) is the following: Under AD, if there is a surjection $\Bbb R\rightarrow\alpha$, then ...
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11 votes
0 answers
298 views

The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...
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10 votes
1 answer
405 views

Assuming AD, is every infinite cardinal closed under power set in a choice model?

Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$. Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...
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7 votes
0 answers
232 views

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} \mathbb{R}^{...
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9 votes
0 answers
258 views

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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5 votes
1 answer
452 views

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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  • 1,529
5 votes
1 answer
545 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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  • 51
6 votes
1 answer
240 views

Forcing over models of determinacy

Consider a ctm $\mathfrak{M}$ of $ZF+AD^+$. Is it possible to force over $\mathfrak{M}$ to get a model of ZFC which satisfies further the following: Every projectively definable family of sets of ...
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  • 1,406
9 votes
1 answer
213 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 ...
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9 votes
1 answer
285 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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9 votes
1 answer
394 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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