Forcing the truth of a weak form of Schanuel’s conjecture

Schanuel’s conjecture states that the transcendence degree over Q of the 2n-tuple (λ1, . . . , λn, eλ1 , . . . , eλn ) is at least n for all λ1, . . . , λn ∈ C which are linearly independent over Q; if true it would settle a great number of elementary open problems in number theory, among which the transcendence of e over π. Wilkie [11], and Kirby [4, Theorem 1.2] have proved that there exists a smallest countable algebraically and exponentially closed subfield K of C such that Schanuel’s conjecture holds relative to K (i.e. modulo the trivial counterexamples, Q can be replaced by K in the statement of Schanuel’s conjecture). We prove a slightly weaker result (i.e. that there exists such a countable field K without specifying that there is a smallest such) using the forcing method and Shoenfield’s absoluteness theorem. This result suggests that forcing can be a useful tool to prove theorems (rather than independence results) and to tackle problems in domains which are apparently quite far apart from set theory. A brief introduction We want to give an example of how we might use forcing to study a variety of expansions of the complex (or real) numbers enriched by arbitrary Borel predicates, still maintaining certain “tameness” properties of the theory of these expansions. We clarify what we intend by “tameness” as follows: in contrast with what happens for example with o-minimality in the case of real closed fields, we do not have to bother much with the complexity of the predicate P we wish to add to the real numbers (we can allow P to be an arbitrary Borel predicate), but we pay a price reducing significantly the variety of elementary superstructures (M,PM ) for which we are able to lift P to PM so that (R, P ) ≺ (M,PM ) and for which we are able to use the forcing method to say something significant on the first order theory of (M,PM ). Nonetheless the family of superstructures M for which this is possible is still a large class, as we can combine (Woodin and) Shoenfield’s absoluteness for the theory of projective sets of reals with a duality theorem relating certain spaces of functions to forcing constructions, to obtain the following1: Theorem 1 (V. and Vaccaro [10]). — Let X be an extremally disconnected (i.e. such that the closure of open sets is open) compact Hausdorff space. Let C+(X) be the space of continuous functions f : X → S2 = C ∪ {∞} such that the preimage of ∞ is nowhere dense (S2 is the one point compactification of C). For any p ∈ X, let C+(X)/p be the ring of germs in p of functions in C+(X). Given any Borel predicate R on C, define a predicate RX/p ⊆ (C+(X)/p)n by the Math. classification: 03E57, 03C60, 11U99.