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Chess

Opening position of a game of chess on a finite board.

\(\omega_1^\mathfrak{Ch}\) (pronounced omega one of chess) is a large countable ordinal, defined like so:[1][2]

  • Consider the game of chess played on a (countably) infinite board. Only a finite number of pieces are allowed.
  • Consider the set of all positions in infinite chess \(P\) and define a function \(\text{Value}: P \mapsto \omega_1\) like so:
    • If White has won in position \(p\), then \(\text{Value}(p) = 0\).
    • If White is to move in position \(p\), and if all the legal moves White can make have a minimal value of \(\alpha\), then \(\text{Value}(p) = \alpha + 1\).
    • If Black is to move in position \(p\), and if all the legal moves Black can make have a supremum of \(\alpha\), then \(\text{Value}(p) = \alpha\).
  • \(\omega_1^\mathfrak{Ch}\) is the supremum of the values of all the positions from which White can force a win.

There are a few variants of this ordinal:

  • If an infinite number of pieces are allowed, the supremum is called \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\).
  • With 3D chess, the supremum is called \(\omega_1^{\mathfrak{Ch}_3}\).
  • With 3D chess with an infinite number of pieces, the supremum is called \(\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}\). This ordinal has been proven to equal the first uncountable ordinal.

Evans and Hamkins proved that \(\omega_1^\mathfrak{Ch}\) and \(\omega_1^{\mathfrak{Ch}_3}\) are at most the Church-Kleene ordinal, and \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}} \leq \omega_1\). Although it has not been proven, it is believed that all these ordinals are as large as possible — that is, \(\omega_1^\mathfrak{Ch} = \omega_1^{\mathfrak{Ch}_3} = \omega_1^\text{CK}\) and \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}} = \omega_1\).

Sources Edit

  1. Transfinite Game Values in Infinite Chess
  2. Omega one of chess at Cantor's Attic

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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