Omega one of chess

\(\omega_1^\mathfrak{Ch}\) (pronounced omega one of chess) is a recursive ordinal, defined like so:


 * Consider the game of played on an infinite board.
 * Consider the set of all positions in infinite chess \(P\) and define a function \(\text{Value}: P \mapsto \omega_1\) like so:
 * Iff White has won in position \(p\), then \(\text{Value}(p) = 0\).
 * Iff White is to move in position \(p\), and if all the legal moves White can make have a minimal value of \(\alpha\), then \(\text{Value}(p) = \alpha + 1\).
 * Iff 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.