Non-principal ultrafilters, program extraction and higher-order reverse mathematics

Journal of Mathematical Logic 12 (1):1250002- (2012)
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Abstract

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher-order arithmetic. Let [Formula: see text] be the statement that a non-principal ultrafilter on ℕ exists and let [Formula: see text] be the higher-order extension of ACA0. We show that [Formula: see text] is [Formula: see text]-conservative over [Formula: see text] and thus that [Formula: see text] is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly [Formula: see text] statement ∀ f ∃ g A qf in [Formula: see text] a realizing term in Gödel's system T can be extracted. This means that one can extract a term t ∈ T, such that ∀ f A qf).

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