**ω _{1}^{Ch}** (pronounced

**omega one of chess**) is a large countable ordinal, defined like so:

^{[1]}

^{[2]}

- Consider the game of chess https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/15px-Wikipedia-logo-v2.svg.png 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 Value:P -> ω
_{1}like so:- If White has won in position p, then 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 α, then Value(p) = α + 1.
- If Black is to move in position p, and if all the legal moves Black can make have a supremum of α, then Value(p) = α.

- ω
_{1}^{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 ω
_{1}^{Ch}'.This ordinal has been proven to equal the first uncountable ordinal, Ω https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/15px-Wikipedia-logo-v2.svg.png. - With 3D chess, the supremum is called ω
_{1}^{Ch3}. - With 3D chess with an infinite number of pieces, the supremum is called ω
_{1}^{Ch3}'. This ordinal has been proven to equal the first uncountable ordinal, Ω https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/15px-Wikipedia-logo-v2.svg.png.

Evans and Hamkins proved that ω_{1}^{Ch} and ω_{1}^{Ch3} are at most the Chruch-Kleene ordinal https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/15px-Wikipedia-logo-v2.svg.png, and ω_{1}^{Ch}' = ω_{1}. Although it has not been proven, it is believed that some of these ordinals are as large as possible, that is, ω_{1}^{Ch} = ω_{1}^{Ch3} = ω_{1}^{Ck}. Note *it has* been proven that ω_{1}^{Ch}' = ω_{1}.

## Gallery[]

## Video[]

Here is a video which explains how Infinite chess ordinals are calculated: