103 research outputs found
The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems
We investigate wether three statements in analysis, that can be proved
classically, are realizable in the realizability model of extensional
continuous functionals induced by Kleene's second model . We prove that a
formulation of the Riemann Permutation Theorem as well as the statement that
all partially Cauchy sequences are Cauchy cannot be realized in this model,
while the statement that the product of two anti-Specker spaces is anti-Specker
can be realized
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
Banach Spaces as Data Types
We introduce the operators "modified limit" and "accumulation" on a Banach
space, and we use this to define what we mean by being internally computable
over the space. We prove that any externally computable function from a
computable metric space to a computable Banach space is internally computable.
We motivate the need for internal concepts of computability by observing that
the complexity of the set of finite sets of closed balls with a nonempty
intersection is not uniformly hyperarithmetical, and thus that approximating an
externally computable function is highly complex.Comment: 20 page
A rich hierarchy of functionals of finite types
We are considering typed hierarchies of total, continuous functionals using
complete, separable metric spaces at the base types. We pay special attention
to the so called Urysohn space constructed by P. Urysohn. One of the properties
of the Urysohn space is that every other separable metric space can be
isometrically embedded into it. We discuss why the Urysohn space may be
considered as the universal model of possibly infinitary outputs of algorithms.
The main result is that all our typed hierarchies may be topologically
embedded, type by type, into the corresponding hierarchy over the Urysohn
space. As a preparation for this, we prove an effective density theorem that is
also of independent interest.Comment: 21 page
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