New results on unprovability and logical strength

This is a preprint for Max-Planck Institute’s proceedings. We present recent results in the area of unprovability. We find unprovable statements about a general class of functions using similtaneous diophantine approximation, unprovability results about the Riemann zeta-function using almost-periodicity and probabilistic arguments and a simple unprovability result about converging series using existence of a universal series, that is a sum of terms of the form ai · xi · yi whose re-arrangement converges to an arbitrary function of two variables. This reseach takes its origin from H.Friedman’s sine principle [2]. The statement roughly says that the sine-function, when considered discretely with rational steps, returns to smaller and smaller neighbourhoods of the same point arbitrarily often. The main idea behind the unprovability proof of the sine-principle is that it was possible to somehow formulate Ramsey-theoretic statements in this set-up without actually mentioning quantification over all possible colourings. It turns out that it is possible to extend this result to larger classes of functions in place of sine, to results of separate interest. This research also led to a general technology of using universality phenomena to imitate Ramsey-style statements in non-Ramseyan contexts while preserving logical strength and unprovability of the original Ramseyan assertions. 1 Unprovable statements obtained by simultaneous diophantine approximation Here is a quite general theorem that embraces all relevant ideas. We can think of slightly more general formulations but they wouldn’t expose new ideas. Definition 1. Let f be a function of one real argument, definable in the language of arithmetic. Af is the following statement: “for all m, there is N such that for any sequence 〈ai〉i=1 of rational numbers, there is H ⊆ N of size m such that for any two m-sequences i1 < i2 < . . . < in and i1 < k2 < . . . < kn, |f(ai1 · ai2 · . . . · ain)− f(ai1 · ak2 · . . . · akn)| < 2−i1”. Af is the statement ∀nAf . Theorem 1. Let f , g and h be three functions such that for any N ∈ N and any small ε > 0, 1. h is a periodic function with period a, continuous on its period; 2. f is such that for any b1, b2, . . . , bN , linearly independent over aQ and any c1, c2, . . . , cN ∈ [0, a), there is x ∈ Q such that for all i ∈ {1, 2, . . . , N}, |f(bi · x) mod a − ci| < ε;