The foundations of mathematics in the theory of sets
type: Item
instance of: scholarly article
zbMATH Open document ID: 0972.03001
publication date: 2001-03-06T00:00:00Z
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 00A30
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03E30
Mathematics Subject Classification ID: 03E65
zbMATH DE Number: 1572542
zbMATH Keywords: finiteness
zbMATH Keywords: set theory
zbMATH Keywords: foundations of mathematics
zbMATH Keywords: finite set theory
author: John P. Mayberry
MaRDI profile type: Publication
title: The foundations of mathematics in the theory of sets
published in: Encyclopedia of Mathematics and Its Applications
review text: The author's core problem is to give a set-theoretic foundation for the whole of (modern) mathematics. To reach this goal he first reflects about the problem of foundations, and argues that such a foundation cannot be given via an elementary theory formalized within first-order logic. This is, because neither (the metatheory of) first-order logic as a tool to axiomatically describe mathematical structures, nor even the syntax of the formalized language of (first-order) logic can be understood without a previous understanding of the notion of set. Therefore he comes back to the ``traditional'' understanding of the notion of an axiom as some statement which is obvious by itself. NEWLINENEWLINENEWLINEThe axioms he chooses for his purpose form a weak version of the standard ZF axioms (without infinity). The crucial point is that the author does not allow unrestricted, ``global'' quantifications over the universe of sets. More formally: \(\Sigma_0\)-formulas are allowed for comprehension, but \(\Sigma_1\)- formulas are not. That means, restricted quantifications are considered as unproblematic, unrestricted ones, however, as problematic. NEWLINENEWLINENEWLINEMoreover, he argues that the usual laws of classical logic hold true only for restricted formulas, and fail (to some extent) for ``global'' formulas. NEWLINENEWLINENEWLINEAt all, the author adheres to a static understanding of mathematics. The notion of an effective procedure, i.e.\ the notion of a construction, is not an essentially mathematical notion for him (cf.~pp. 15/16). NEWLINENEWLINENEWLINEA very basic observation is that the author reads the Greek notion of number, that means the Greek notion of ``arithmoi'', which means the ``multitudes composed of units'', essentially as finite sets. And argues then that the existence of such numbers is more or less obvious from our experience. Furthermore, number words like ``five'' from his point of view get a meaning like words as ``horse''. And these ``arithmoi'' are …
reviewed by: Siegfried J. Gottwald
description: scientific article