Questions tagged [determinacy]
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96 questions
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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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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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Determinacy interchanging the roles of both players
Let me refer to Jech's "Set Theory" Chap. 33 Determinacy:
"With each subset A of $\omega^\omega$ we associate the following game $G_A$, played by two players I and II. First I chooses a natural ...
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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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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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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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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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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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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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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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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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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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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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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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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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$\operatorname{AD}$ 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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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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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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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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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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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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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 $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} \mathbb{R}^{...
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Which forcing types preserve the axiom of determinacy?
Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...
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What axioms (other than choice) have a taming effect on the ordering of cardinalities?
Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...
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$\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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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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$\Sigma^0_1\wedge\Pi^0_1$-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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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 ...
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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 $f : I \to 2^\mathbb{R}$,...
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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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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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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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The projection of a weakly homogeneous tree is determined
Where can I read a proof of this?
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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 http://en.wikipedia.org/wiki/%CE%98_%...
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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 space....
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