Omega one of chess

\(\omega_1^\mathfrak{Ch}\) (pronounced omega one of chess) is a large countable ordinal, defined like so:Transfinite Game Values in Infinite ChessOmega one of chess at Cantor's Attic
* Consider the game of 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 \(\omega_1^{\mathfrak{Ch}'}\).
* 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 \(\omega_1^{\mathfrak{Ch}_3'}\). 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 \(\omega_1^{\mathfrak{Ch}'}=\omega_1\). Although it has not been proven, it is believed that some of these ordinals are as large as possible — that is, \(\omega_1^\mathfrak{Ch} = \omega_1^{\mathfrak{Ch}_3} = \). Note it has been proven that \(\omega_1^{\mathfrak{Ch}'} = \omega_1\).
== Video ==
Here is a video which explains Infinite Chess in a less formal way that should give you an idea of how chess ordinals are calculated.
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== Sources ==
== See also ==
Category:Transfinite ordinals
Category:Unsolved problems
Category:Chess