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2000 Frege Meets Dedekind: A Neologicist Treatment of Real Analysis
Stewart Shapiro
Notre Dame J. Formal Logic 41(4): 335-364 (2000). DOI: 10.1305/ndjfl/1038336880

Abstract

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume's Principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers:

$(\mathrm{DIF})\qquad \mathrm{INT}(a,b) = \mathrm{INT}(c,d)\equiv(a+d)=(b+c).$

$(\mathrm{QUOT}) \qquad \begin{eqnarray}\mathrm{Q}(m,n)=\mathrm{Q}(p,q)\equiv(n=0 \: \& \: q=0) \\ \qquad \vee\: (n\neq 0 \: \& \: q\neq 0 \: \& \: m \cdot q=n \cdot p).\end{eqnarray}$

The development of the real numbers is an adaption of the Dedekind program involving "cuts" of rational numbers. Let $P$ be a property (of rational numbers) and $r$ a rational number. Say that $r$ is an upper bound of $P$, written $P\leq r$, if for any rational number $s$, if $Ps$ then either $s<r$ or $s=r$. In other words, $P\leq r$ if $r$ is greater than or equal to any rational number that $P$ applies to. Consider the Cut Abstraction Principle:

$(\mathrm{CP}) \qquad \forall P \forall Q(C(P)=C(Q) \equiv \forall r(P\leq r \equiv Q\leq r)).$

In other words, the cut of $P$ is identical to the cut of $Q$ if and only if $P$ and $Q$ share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano Arithmetic can be derived from Hume's Principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies.

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Stewart Shapiro. "Frege Meets Dedekind: A Neologicist Treatment of Real Analysis." Notre Dame J. Formal Logic 41 (4) 335 - 364, 2000. https://doi.org/10.1305/ndjfl/1038336880

Information

Published: 2000
First available in Project Euclid: 26 November 2002

zbMATH: 1014.03013
MathSciNet: MR1963486
Digital Object Identifier: 10.1305/ndjfl/1038336880

Subjects:
Primary: 00A25
Secondary: 03A05

Keywords: Frege , logicism , neologicism , second-order logic

Rights: Copyright © 2000 University of Notre Dame

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