53 research outputs found
Approaches to analysis with infinitesimals following Robinson, Nelson, and others
This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins. Keywords: axiomatisations, infinitesimal, nonstandard analysis, ultraproducts, superstructure, set-theoretic foundations, multiverse, naive integers, intuitionism, soritical properties, ideal elements, protozoa
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/
On effective compactness and sigma-compactness
Using the Gandy -- Harrington topology and other methods of effective
descriptive set theory, we prove several theorems on compact and sigma-compact
pointsets. In particular we show that any set of the Baire
space either is covered by a countable union of compact
sets, or contains a subset closed in and homeomorphic to (and
then is not covered by a sigma-compact set, of course)
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Cauchy's sum theorem is a prototype of what is today a basic result on the
convergence of a series of functions in undergraduate analysis. We seek to
interpret Cauchy's proof, and discuss the related epistemological questions
involved in comparing distinct interpretive paradigms. Cauchy's proof is often
interpreted in the modern framework of a Weierstrassian paradigm. We analyze
Cauchy's proof closely and show that it finds closer proxies in a different
modern framework.
Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation;
uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc
Hypernatural Numbers as Ultrafilters
In this paper we present a use of nonstandard methods in the theory of
ultrafilters and in related applications to combinatorics of numbers
Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics
We examine some of Connes' criticisms of Robinson's infinitesimals starting
in 1995. Connes sought to exploit the Solovay model S as ammunition against
non-standard analysis, but the model tends to boomerang, undercutting Connes'
own earlier work in functional analysis. Connes described the hyperreals as
both a "virtual theory" and a "chimera", yet acknowledged that his argument
relies on the transfer principle. We analyze Connes' "dart-throwing" thought
experiment, but reach an opposite conclusion. In S, all definable sets of reals
are Lebesgue measurable, suggesting that Connes views a theory as being
"virtual" if it is not definable in a suitable model of ZFC. If so, Connes'
claim that a theory of the hyperreals is "virtual" is refuted by the existence
of a definable model of the hyperreal field due to Kanovei and Shelah. Free
ultrafilters aren't definable, yet Connes exploited such ultrafilters both in
his own earlier work on the classification of factors in the 1970s and 80s, and
in his Noncommutative Geometry, raising the question whether the latter may not
be vulnerable to Connes' criticism of virtuality. We analyze the philosophical
underpinnings of Connes' argument based on Goedel's incompleteness theorem, and
detect an apparent circularity in Connes' logic. We document the reliance on
non-constructive foundational material, and specifically on the Dixmier trace
(featured on the front cover of Connes' magnum opus) and the Hahn-Banach
theorem, in Connes' own framework. We also note an inaccuracy in Machover's
critique of infinitesimal-based pedagogy.Comment: 52 pages, 1 figur
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