Non-principal Ultrafilters, Program Extraction and Higher Order Reverse Mathematics

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a non-principal ultrafilter on N exists and let ACA0 be the higher order extension of ACA0. We show that ACA0 + (U) is Π2-conservative over ACA0 and thus that ACA0 +(U) is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly Π2 statement ∀f ∃g Aqf(f, g) in ACA0 + (U) a realizing term in Gödel’s system T can be extracted. This means that one can extract a term t ∈ T , such that ∀f Aqf(f, t(f)). In this paper we will investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. We will classify the consequences of this statement in the spirit of reverse mathematics. Furthermore, we will provide a program extraction method. Let (U) be the statement that a non-principal ultrafilter on N exists. Let RCA0 , ACA0 be the extensions of RCA0 resp. ACA0 to higher order arithmetic as introduced by Kohlenbach in [16]. In RCA0 or ACA0 the statement (U) can be formalized using an object of type NN −→ N. Further, let Feferman’s μ be a functional of type NN −→ N satisfying f(μ(f)) = 0 if ∃x f(x) = 0 and let (μ) be the statement that such a functional exists. It is clear that (μ) implies arithmetical comprehension. However, μ is not definable in ACA0 . We will show that • over RCA0 the statement (U) implies (μ) and therefore is strictly stronger than ACA0 , and that • ACA0 + (μ) + (U) is Π2-conservative over ACA0 and therefore also conservative over PA. Moreover, we will show that from a proof of ∀f ∃g Aqf(f, g) in ACA0 +(μ)+(U), where Aqf is quantifier free, one can extract a realizing term t in Gödel’s system T , i.e. a term such that ∀f Aqf(f, t(f)). The system ACA0 + (μ) + (U) is strong in the sense that one can carry out many common ultralimit and non-standard arguments. For instance one can carry out in this theory the construction of Banach limits and many Loeb measure constructions. Date: February 17, 2012 14:23. 2010 Mathematics Subject Classification. 03B30, 03F35, 03F60.