State-strategies for games in Fσδ ∩ Gδσ

Abstract

The Cantor-Bendixson theorem says: For every closed set , ∣∣ ≤ ω0 ∨ ⊇ perfect (and therefore is countable or has the power of the continuum). W. H. Young proved the alternative for Gδ-sets, and Hausdorff extended it to Souslin-sets. Cantor used his ordinals to prove the theorem (I like to think he invented them for this purpose). In Hausdorff's Mengenlehre ordinals do not officially enter in either of the three proofs. You should try to put them back. It is very important never to hide away ω1 when it actually is there, and it is there when the continuum problem is the subject.In Davis [6] you find that Cantor's alternative is equivalent to another alternative: (I wins ) ∨ (J wins ). Here means a lopsided game in which one of the players makes long moves on the game tree. So, the classical theorems can be restated as determinacy results for lopsided games, and now I make this observation: The original CB-proof “actually presents” a winning strategy. The HD-proofs do no such thing; all you know at the end is existence of a winning strategy (and Davis' remark is needed to gain this knowledge).You understand now why I like CB-proofs, particularly for determinacy of games. They do exist also for games in which both players make short moves. Many years ago I tried to publish such a proof for determinacy of Fσ-games. The referee said it was not worth the trouble. I say you should remake this proof (it is a special case of the one presented in this paper), because it is the ideal of a determinacy proof.