infinite chess

The study of ∞ chess is taken seriously in some circles. I found out by way of Cantor’s attic where one of the results by a certain Prof. Hamkins sits near the smallest of the infinities around ω1 i.e. the first uncountable ordinal.  Hamkins’ website has interesting diagrams from his paper on “Transfinite Game Values in Infinite Chess”.

Transfinite refers to those ω symbols (the first of which is the “number” denoted by 1,2,…ω).  ω corresponds to the cardinality of the integers or aleph-0, and ω1 to the first uncountable cardinality (which is not necessarily the cardinality of the reals, as the continuum hypothesis remains open or is unprovable) but unlike with the cardinalities you can do arithmetic with ordinals, and that messes with your mind. There is also an extension of mathematical induction to transfinite induction which works on ordinal numbers: assume P(n) is true for all n<m; then P(m) implies P is true everywhere.

Game values in this context are the “mate in N” values of the positions. Specific infinite game values come from artificial positions with mechanical win sequences, such as a king being pursued down
a corridor of blocked pawns that can’t attack it. Hamkins gave specific constructions for ω3 and then for ω4 here. The papers give whimsical names in describing the constructions due to the lack of standard terms. That also lends a cozy recreational air.  New fields in mathematics are like gold rushes that open a new realm for exploration, where personal glory is possible at first by working a few plots by hand before heavy tools and custom machinery overrun the field.

these games can often have a somber absurd character of play

 

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