Mathematical Logic and Foundations of Set Theory, by Yehoshua Bar-Hillel (Eds.)

By Yehoshua Bar-Hillel (Eds.)

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Extra info for Mathematical Logic and Foundations of Set Theory, Proceedings of an International Colloquium Held Under the Auspices of The Israel Academy of Sciences and Humanities

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T,)}. ,B) E W 5 , A ) . Let n:w x R x R + R be the canonical recursive homeomorphism. Each ( < K must fall under one of the cases below. 5 = 0. Put Case 2 . 5 = 5 +1. Case 1. Case 3. 4 ( ( ) = (n(l,a,a): ~ E R } . Put 4(5) = {71(2,a,cc): cc~4(4')). 4 is a singular limit ordinal. Put c [ < 5 and some cojinal m a p f : + 5 , cc E +(() = {n(3,~,/3): f o r some = C o d ( g * ; &), $(c) and G,(St) where g* is a choice subfunction of the function f *(q) = 4 ( f ( q ) ) } . Here G(St;) is the canonical universal set for the class of E:(Sr) subsets of R x R , where Z i ( 5 ) is associated with each relation 5 via (3-4) and (3-5).

In [ R x RIA) with length [. The assumed closure properties of A imply 5 is realized in A & < t * 5 is realized in A , [ is realized in A a [ [>0 &c + 1 is realized in A , is realized in A 5 5 is the length of some prewellordering of R in A . It is also clear that the same ordinals are realized if we allow prewellorderings on subsets of any product space X . Put 0th) = supremum {[: 5 is realized in A}. g. A = Ai,A:,R2. The basic theory in which we work is Zermelo-Fraenkel without choice but with the countable axiom of choice for sets of reals.

Our chief result was that (with A D and DC) each uk is the order type of a projective prewellordering of R and that larger ordinals, like n,, are order types of hyperanalytic prewellorderings. e. games whose definition guarantees that player I cannot win; such games were first in the codes”. utilized by Solovay in his proof that (with AD) every subset of uI is Friedman’s early contribution was a technique of utilizing games where it is not clear which player wins, by proving the desired result by cases; we call games used in such proofs Friedman games.

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