The dichotomy in the determinacy of certain two-person infinite games with moves from {lcub}0,1{rcub}.

摘要

We investigate certain well-known games from the field of set theory; namely, certain two-person games of perfect information with small complexity and with small infinite length. We consider games with moves from the natural numbers and games with moves from {lcub}0,1{rcub}. We show that the determinacy of open games with length o·n and with moves from {lcub}0,1{rcub} is true regardless of the existence of large cardinals for n ≥ 2. We show that this is not true, however, for some more complex games: For k ≥ 3 and n ≥ 2, the determinacy of P0k games with length o·n and with moves from {lcub}0,1{rcub} is equivalent to the determinacy of P0k games with length o·n and with moves from o, which in turn requires the existence of large cardinals. We also examine the question of whether for classes Gamma properly between S01 and P03 , large cardinals are required for the determinacy of Gamma games with length o·n and with moves from {lcub}0,1{rcub} for n ≥ 2.