8 research outputs found
On the mathematical and foundational significance of the uncountable
We study the logical and computational properties of basic theorems of
uncountable mathematics, including the Cousin and Lindel\"of lemma published in
1895 and 1903. Historically, these lemmas were among the first formulations of
open-cover compactness and the Lindel\"of property, respectively. These notions
are of great conceptual importance: the former is commonly viewed as a way of
treating uncountable sets like e.g. as 'almost finite', while the
latter allows one to treat uncountable sets like e.g. as 'almost
countable'. This reduction of the uncountable to the finite/countable turns out
to have a considerable logical and computational cost: we show that the
aforementioned lemmas, and many related theorems, are extremely hard to prove,
while the associated sub-covers are extremely hard to compute. Indeed, in terms
of the standard scale (based on comprehension axioms), a proof of these lemmas
requires at least the full extent of second-order arithmetic, a system
originating from Hilbert-Bernays' Grundlagen der Mathematik. This observation
has far-reaching implications for the Grundlagen's spiritual successor, the
program of Reverse Mathematics, and the associated G\"odel hierachy. We also
show that the Cousin lemma is essential for the development of the gauge
integral, a generalisation of the Lebesgue and improper Riemann integrals that
also uniquely provides a direct formalisation of Feynman's path integral.Comment: 35 pages with one figure. The content of this version extends the
published version in that Sections 3.3.4 and 3.4 below are new. Small
corrections/additions have also been made to reflect new development
Pincherle's theorem in Reverse Mathematics and computability theory
We study the logical and computational properties of basic theorems of
uncountable mathematics, in particular Pincherle's theorem, published in 1882.
This theorem states that a locally bounded function is bounded on certain
domains, i.e. one of the first 'local-to-global' principles. It is well-known
that such principles in analysis are intimately connected to (open-cover)
compactness, but we nonetheless exhibit fundamental differences between
compactness and Pincherle's theorem. For instance, the main question of Reverse
Mathematics, namely which set existence axioms are necessary to prove
Pincherle's theorem, does not have an unique or unambiguous answer, in contrast
to compactness. We establish similar differences for the computational
properties of compactness and Pincherle's theorem. We establish the same
differences for other local-to-global principles, even going back to
Weierstrass. We also greatly sharpen the known computational power of
compactness, for the most shared with Pincherle's theorem however. Finally,
countable choice plays an important role in the previous, we therefore study
this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi
The Gandy-Hyland functional and a computational aspect of Nonstandard Analysis
In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving nonstandard numbers inside Nelson's internal set theory. From this classical and ineffective proof in Nonstandard Analysis, a term from Godel's system T can be extracted which computes the Gandy-Hyland functional in terms of a modulus-of-continuity functional and the special fan functional. We obtain several similar relative computability results not involving Nonstandard Analysis from their associated nonstandard theorems. By way of reversal, we show that certain relative computability results, called Herbrandisations, also imply the nonstandard theorem from whence they were obtained. Thus, we establish a direct two-way connection between the field Computability (in particular theoretical computer science) and the field Nonstandard Analysis