Questions tagged [determinacy]

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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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0answers
351 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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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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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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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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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 ...
14
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1answer
676 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 ...
3
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237 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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1answer
171 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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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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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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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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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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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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1answer
242 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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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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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 "...
9
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1answer
558 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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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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1answer
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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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1answer
270 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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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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1answer
215 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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1answer
412 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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359 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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1answer
458 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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1answer
310 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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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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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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1answer
311 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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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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1answer
337 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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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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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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1answer
214 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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1answer
418 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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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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1answer
254 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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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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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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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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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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1answer
424 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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1answer
494 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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1answer
234 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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1answer
208 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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1answer
281 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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1answer
389 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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1answer
454 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 ...
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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 ...