# Small game

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In CGT, a small game is an abstract game smaller than every positive number (even infinitesimal numbers) and greater than every negative number: i.e. it is a game G " such that " -x < G < x " for every number " x > 0 . Examples of small games are uarr (UP), darr (DOWN) and ast (STAR). All small games are infinitesimal, and the only number which is a small game is, of course, zero. Small games can still be positive, negative or fuzzy, like uarr, darr " and " ast  respectively.

A game is called all small if all its positions are small games. Theorem 57 of ONAG (on page 101) states that this is equivalent to all its stopping positions (positions which are numbers) being zero. Equivalently, if all its numerical positions are small (i.e. 0), then so are all others.

Not all infinitesimal games are small, as seen in ONAG in the list under The Gamut revealed[1], e.g. the largest infinitesimal games alpha | bbb "R"^+" || "bbb "R"^+ = (1/ oo )_alpha  for large ordinals alpha.

## Notes and references

[1] ONAG, second edition (2001), page 214.

Small game last edited by Dieter on October 23, 2023 - 09:44