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$$\omega_1^\mathfrak{Ch}$$ (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 $$\omega_1^{\mathfrak{Ch}'}$$.
• 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, $$\omega_1^\mathfrak{Ch} = \omega_1^{\mathfrak{Ch}_3} = \omega_1^\text{CK}$$. Note it has been proven that $$\omega_1^{\mathfrak{Ch}'} = \omega_1$$.

Video Edit

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.