174 research outputs found
Analysis in weak systems
The authors survey and comment their work on weak analysis. They describe the basic set-up of analysis in a feasible second-order theory and consider the impact of adding to it various forms of weak Konig's lemma. A brief discussion of the Baire categoricity theorem follows. It is then considered a strengthening of feasibility obtained (fundamentally) by the addition of a counting axiom and showed how it is possible to develop Riemann integration in the stronger system. The paper finishes with three questions in weak analysis.info:eu-repo/semantics/publishedVersio
Big in Reverse Mathematics: measure and category
The smooth development of large parts of mathematics hinges on the idea that
some sets are `small' or `negligible' and can therefore be ignored for a given
purpose. The perhaps most famous smallness notion, namely `measure zero',
originated with Lebesgue, while a second smallness notion, namely `meagre' or
`first category', originated with Baire around the same time. The associated
Baire category theorem is a central result governing the properties of meagre
(and related) sets, while the same holds for Tao's pigeonhole principle for
measure spaes and measure zero sets. In this paper, we study these theorems in
Kohlenbach's higher-order Reverse Mathematics, identifying a considerable
number of equivalent theorems. The latter involve most basic properties of
semi-continuous and pointwise discontinuous functions, Blumberg's theorem,
Riemann integration, and Volterra's early work circa 1881. All the
aforementioned theorems fall (far) outside of the Big Five of Reverse
Mathematics, and we investigate natural restrictions like Baire 1 and
quasi-continuity that make these theorems provable again in the Big Five (or
similar). Finally, despite the fundamental differences between measure and
category, the proofs of our equivalences turn out to be similar.Comment: 32 pages plus Technical Appendix. Same technical appendix as:
arXiv:2208.0302
Effective Choice and Boundedness Principles in Computable Analysis
In this paper we study a new approach to classify mathematical theorems
according to their computational content. Basically, we are asking the question
which theorems can be continuously or computably transferred into each other?
For this purpose theorems are considered via their realizers which are
operations with certain input and output data. The technical tool to express
continuous or computable relations between such operations is Weihrauch
reducibility and the partially ordered degree structure induced by it. We have
identified certain choice principles which are cornerstones among Weihrauch
degrees and it turns out that certain core theorems in analysis can be
classified naturally in this structure. In particular, we study theorems such
as the Intermediate Value Theorem, the Baire Category Theorem, the Banach
Inverse Mapping Theorem and others. We also explore how existing
classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into
this picture. We compare the results of our classification with existing
classifications in constructive and reverse mathematics and we claim that in a
certain sense our classification is finer and sheds some new light on the
computational content of the respective theorems. We develop a number of
separation techniques based on a new parallelization principle, on certain
invariance properties of Weihrauch reducibility, on the Low Basis Theorem of
Jockusch and Soare and based on the Baire Category Theorem. Finally, we present
a number of metatheorems that allow to derive upper bounds for the
classification of the Weihrauch degree of many theorems and we discuss the
Brouwer Fixed Point Theorem as an example
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
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