92 research outputs found
From Nonstandard Analysis to various flavours of Computability Theory
As suggested by the title, it has recently become clear that theorems of
Nonstandard Analysis (NSA) give rise to theorems in computability theory (no
longer involving NSA). Now, the aforementioned discipline divides into
classical and higher-order computability theory, where the former (resp. the
latter) sub-discipline deals with objects of type zero and one (resp. of all
types). The aforementioned results regarding NSA deal exclusively with the
higher-order case; we show in this paper that theorems of NSA also give rise to
theorems in classical computability theory by considering so-called textbook
proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/
Grilliot's trick in Nonstandard Analysis
The technique known as Grilliot's trick constitutes a template for explicitly
defining the Turing jump functional in terms of a given
effectively discontinuous type two functional. In this paper, we discuss the
standard extensionality trick: a technique similar to Grilliot's trick in
Nonstandard Analysis. This nonstandard trick proceeds by deriving from the
existence of certain nonstandard discontinuous functionals, the Transfer
principle from Nonstandard analysis limited to -formulas; from this
(generally ineffective) implication, we obtain an effective implication
expressing the Turing jump functional in terms of a discontinuous functional
(and no longer involving Nonstandard Analysis). The advantage of our
nonstandard approach is that one obtains effective content without paying
attention to effective content. We also discuss a new class of functionals
which all seem to fall outside the established categories. These functionals
directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page
An algorithmic approach to the existence of ideal objects in commutative algebra
The existence of ideal objects, such as maximal ideals in nonzero rings,
plays a crucial role in commutative algebra. These are typically justified
using Zorn's lemma, and thus pose a challenge from a computational point of
view. Giving a constructive meaning to ideal objects is a problem which dates
back to Hilbert's program, and today is still a central theme in the area of
dynamical algebra, which focuses on the elimination of ideal objects via
syntactic methods. In this paper, we take an alternative approach based on
Kreisel's no counterexample interpretation and sequential algorithms. We first
give a computational interpretation to an abstract maximality principle in the
countable setting via an intuitive, state based algorithm. We then carry out a
concrete case study, in which we give an algorithmic account of the result that
in any commutative ring, the intersection of all prime ideals is contained in
its nilradical
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
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