Foundational and Mathematical Uses of Higher Types

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 (Φ n -WKL + , Ψ n -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 1 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 ω plus quantifierfree 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 Ψ 1 -WKL + is sufficient. Our conservation results for (Φ n -WKL + , Ψ n -WKL + ) are proved via a new elimination result for a strong non-standard principle of uniform Σ 0 1 -boundedness which we introduced in 1996 and which implies the WKL-extensions studied in this paper.

Light monotone Dialectica methods for proof mining

“…Remark 4.6 (F nc vs. F) As proved by Kohlenbach in [18], in a fully extensional context, the weaker axiom F − actually implies F, hence the purely syntactic elimination of F is possible via the simulation in terms of F − , followed by, e.g., Theorem 3.5 (since F − is a Delta axiom). On the other hand, the model-interpretation treatment of F in a weakly extensional setting (via Theorem 4.3) appears technically impossible, due to its unsuitable logical shape.…”

A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

“…hyperbolic spaces can be proved. ∃-UB X extends the principle Σ 0 1 -UB (introduced in [12] and further studied in [14,5]) for the Cantor space C by including abstract bounded metric spaces X in addition to C. Σ 0 1 -UB has proved to be useful in the cause of proof mining in the context of compact Polish spaces (see [13] and [1]) as it allows one to give very short and coding free proofs of many of the usual applications of weak König's lemma WKL. In addition Σ 0 1 -UB proves various classically false theorems such as the uniform continuity (with modulus of continuity) of all extensional functionals Φ : 2 IN → IN which makes it possible to treat continuous functions without explicitly having to refer to moduli of continuity.…”

Decidable fan theorem and uniform continuity theorem with continuous moduli

“…In the context of classical (second or higher‐order) reverse mathematics, is equivalent to the weak Kőnig lemma , the classical equivalent of the decidable fan theorem [19, Theorem IV.2.3][15, § 4]. In the constructive context [5, 13], however, the relation between and the fan theorem becomes somewhat subtle.…”

“…Beeson [1, Chapter VI, § 8, Exercise 8]), which in turn is equivalent to the existence of a code in the sense of reverse mathematics (cf. Kohlenbach [15, Proposition 4.4]). Since Loeb's code can be considered as an associate of a real‐valued function it induces, one would expect that the characterisation for integer‐valued functions extends to the context of functions from [0, 1] to .…”