A machine can play Lucas's game
A machine programmed to do transfinite counting could play Lucas's game as well as Lucas can. Lucas relies on the fact that transfinite counting hasn't been formalised. But the fact it hasn't doesn't mean humans are better at it than machines are.
Irving J. Good, (1967).

Transfinite numbers: Numbers that go beyond the magnitude of any finite set.

Transfinite counting: A form of arithmetic that works with transfinite numbers instead of just with finite numbers.

Notes: Also, see the "Can improved machines beat the Lucas argument?" arguments on this map, and "The Gödelization procedure can be algorithmically specified" Box 52.
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