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By G. Milne

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Clearly C0 ⊇ C1 ⊇ . . and each Ck = ∅ (by the hypothesis of the lemma). Hence, again by ℵ1 -saturation, k∈N Ck = ∅, and any β from the intersection satisﬁes condition (2). We derive a simple corollary about witnessing of quantiﬁers for particular families F. 2 Family F is closed under deﬁnition by cases if for any α0 , α1 ∈ F and any B ∈ A there is β ∈ F such that β(ω) = α0 (ω) α1 (ω) if ω ∈ B otherwise. 1 is satisﬁed for families closed under deﬁnitions by cases. 3 Assume that F is deﬁnable in M and that it is closed under deﬁnitions by cases.

E. (recursively enumerable) subsets of (deﬁned using a code for the Boolean combination and a universal 10 -formula) and measure ν giving to a string w the weight 2n−1−2|w| . However, we have no application for these more general constructions here and we restrict to the counting measure on M-ﬁnite . e. the formula α = β is K(F)-valid) if they differ for an inﬁnitesimal fraction of samples ω ∈ {0, 1}n . e. less than n−k , for any k ∈ N. Whenever necessary (not in this book, however) this can be remedied analogously to complexity theory: build the model from the same random variables but computed on a tuple of independent samples.

Assume Ck , k ∈ N, are deﬁnable sets such that F∩ C = ∅,