Foundational and Mathematical Uses of Higher Types

  • Ulrich Kohlenbach

Abstract

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 (Phi_n-WKL+, Psi_n-WKL+) of principles
which generalize (and for n = 0 coincide with) the so-called `weak' K¨onig'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 : N^N -> N can be seen (in our higher order context) to be equivalent
to the existence of a continuous modulus of pointwise continuity.
For the direct representation of F the existence of such a modulus is
independent even of full arithmetic in all finite types E-PA^omega plus quantifier-free
choice, as we show using a priority construction due to L. Harrington.
The usual WKL-based proofs of properties of F given in reverse mathematics
make use of the enrichment provided by codes of F, and WKL does not seem
to be sufficient to obtain similar results for the direct representation of F in
our setting. However, it turns out that   Psi_1-WKL+ is sufficient.
Our conservation results for (Phi_n-WKL+,  Psi_n-WKL+) are proved via a new
elimination result for a strong non-standard principle of uniform Sigma^0_1-
boundedness
which we introduced in 1996 and which implies the WKL-extensions studied
in this paper.
Published
1999-12-01
How to Cite
Kohlenbach, U. (1999). Foundational and Mathematical Uses of Higher Types. BRICS Report Series, 6(31). https://doi.org/10.7146/brics.v6i31.20100