# Questions tagged [determinacy]

The determinacy tag has no usage guidance.

63
questions

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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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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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178 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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306 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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433 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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281 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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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 ...

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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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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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163 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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207 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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137 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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192 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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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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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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497 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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196 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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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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299 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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150 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**answer

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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329 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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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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### 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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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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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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84 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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180 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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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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261 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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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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337 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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144 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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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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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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265 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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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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291 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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397 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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231 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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250 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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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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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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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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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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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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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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### 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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637 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 ...