A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

Abstract

We extend the principle Σ 0 1 -UB of uniform Σ 0 1 -boundedness introduced earlier by the author to a uniform boundedness principle ∃-UB X for abstract bounded metric and hyperbolic spaces which are not assumed to be compact. Despite the fact that this principle implies numerous results which in general are true only for compact spaces (and continuous functions) we can prove that for a large class K of such consequences A the conclusion A is true in arbitrary bounded spaces even when ∃-UB X is used to facilitate the proof of A. For a somewhat more restricted class of sentences A even effective uniform bounds can be extracted from such proofs.

Gödel's Functional Interpretation and its Use in Current Mathematics*

“…The shift of emphasis towards applications in mathematics deviates from this original motivation and replaces the issue of foundational reductions by concrete mathematical applications. However, there is one aspect of the original preoccupation with consistency proofs that has shown up again in the course of this applied reorientation: monotone functional interpretation can be used to prove a useful elimination result for a classically false strong uniform boundedness principle ∃‐UB X (Kohlenbach 2006) over e.g. 𝒜 ω [ X , d , W ] (i.e.…”

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…In recent years, general logical metatheorems based on functional interpretations have been proved which for large classes of proofs and theorems guarantee the extractability of effective and strongly uniform bounds: [28,56,64,66,87]. This shows that the concrete applications are not 'ad hoc' and so meet the critique expressed in [21] of early stages of the unwinding program (see also [90] for a discussion of 'unwinding' in general).…”

“…This principle allows one (among many other things) to prove (over A ω [X, d, W ]) that every nonexpansive mapping f : X → X has a fixed point which is known to be false already for bounded closed convex subsets of Banach spaces such as c 0 . Nevertheless, for a large class of sentences A provable using ∃-UB X (including so-called asymptotic regularity statements) one can show that they are classically correct (see [66] and [63,70] for concrete instances of this). Since in metric fixed point theory many proofs of asymptotic regularity exist which use as an assumption that f has fixed points this can (and has been) applied for…”

“…The shift of emphasis towards applications in mathematics deviates from this original motivation and replaces the issue of foundational reductions by concrete mathematical applications. However, there is one aspect of the original preoccupation with consistency proofs that has shown up again in the course of this applied reorientation: monotone functional interpretation can be used to prove a useful elimination result for a classically false strong uniform boundedness principle ∃-UB X ( [66]) over e.g. (X, d)).…”

On Tao's “finitary” infinite pigeonhole principle

“…Σ 0 1 -UB was first introduced in [4] and is studied in detail in [8] (for a systematic prooftheoretic treatment of even more general forms of uniform boundedness by a specially designed so-called bounded functional interpretation see [2]). Recently in [7,8], Σ 0 1 -UB was generalized to a principle ∃-UB X dealing with uniformities in the absence of compactness for abstract bounded metric and hyperbolic spaces. Again, while not valid in the intended model, the principle satisfies strong conservation theorems and so can be used safely for proofs of large classes of statements.…”