All Questions
Tagged with determinacy infinite-games
10 questions
6
votes
0
answers
185
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Complexity of transfinite 5-in-a-row and other games
Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
- 7,545
7
votes
0
answers
198
views
Strengthening Determinacy in constructive set theory?
Recall how games work. Let $X$ be a set (the "game space") and $\alpha$ an ordinal (the "game clock"). Alice and Bob take turns naming elements of $X$. We write them down in order ...
- 63.9k
2
votes
0
answers
267
views
Is determinacy of (some) very long open games consistent?
For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play ...
- 21.1k
7
votes
0
answers
258
views
Is this determinacy principle consistent?
Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"):
If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\...
- 21.1k
24
votes
2
answers
1k
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What is the complexity of the winning condition in infinite Hex? In particular, is infinite Hex a Borel game?
Consider the game of infinite Hex, where two players Red and Blue alternately place their stones on the infinite hex grid, each aiming to create a winning configuration.
Red wins after infinite play, ...
- 236k
2
votes
0
answers
140
views
Weakening of open determinacy for uncountably long games
For a cardinal $\kappa$ I'll use the phrase "$(\kappa,\kappa)$-game" to mean "two-player, perfect-information, deterministic game on $\kappa$ of length $\kappa$."
Say that a ...
- 21.1k
6
votes
1
answer
260
views
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 ...
- 2,286
2
votes
0
answers
196
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: ...
- 59
11
votes
1
answer
686
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 ...
- 5,031
12
votes
0
answers
357
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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 ...
- 21.1k