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ZF ⊦ Σ40 determinateness

  • J. B. Paris (a1)


In this paper we show that in Zermelo-Fraenkel set theory (ZF) sets of reals are determinate.

Before proceeding to the proof it will be helpful to consider some previous work in this area. The first major result was obtained by Gale and Stewart [3] who showed that in ZF open games are determinate. This was then successively improved by Wolfe [4] to (and so of course ) and then by Morton Davis [1] to . The results of Morton Davis further showed that countable unions of sufficiently ‘simple’ determinate sets are also determinate. At this time, however, sets did not appear sufficiently simple for this method to be applied in order to get determinacy.

The next major advance was made by D. A. Martin who showed, using indiscernibles, that with large cardinal assumptions games are equivalent to certain open games (i.e., player I (II) has a winning strategy for the game iff I (II) has a winning strategy for the open game). Thus, by the Gale–Stewart result, games are determinate. Martin's result also showed that under these large cardinal assumptions sets are sufficiently simple for the Morton Davis method to be applied to them.



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[1]Davis, M., Infinite games with perfect Information, Advances in game theory, Annals of Mathematical Studies, No. 52, Princeton University Press, Princeton, N.J., 1964, pp. 85101.
[2]Friedman, H., Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2 (1971).
[3]Gale, D. and Stewart, F. M., Infinite games with information, Contributions to the theory of games, Annals of Mathematical Studies, No. 28, Princeton University Press, Princeton, N.J., 1953, pp. 245266.
[4]Wolfe, P., The strict determinateness of certain infinite games, Pacific Journal of Mathematics, vol. 5 (1955), pp. 841847.

ZF ⊦ Σ40 determinateness

  • J. B. Paris (a1)


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