Chess Wiki

ω1Ch (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 Value:P -> ω1 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.
  • ω1Ch 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:

Evans and Hamkins proved that ω1Ch and ω1Ch3 are at most the Chruch-Kleene ordinal, and ω1Ch' = ω1. Although it has not been proven, it is believed that some of these ordinals are as large as possible, that is, ω1Ch = ω1Ch3 = ω1Ck. Note it has been proven that ω1Ch' = ω1.



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