# 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
1 vote
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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 ...
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 ...