Foundational and Mathematical Uses of Higher Types

In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are proof-theoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on non-collapsing hierarchies (Φn-WKL+, Ψn-WKL+) of principles which generalize (and for n = 0 coincide with) the so-called ‘weak’ König’s lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X → Y between Polish spaces X,Y , the more flexible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN → IN can be seen (in our higher order context) ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.