Notre Dame Journal of Formal Logic

Frege Meets Dedekind: A Neologicist Treatment of Real Analysis

Stewart Shapiro

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 41, Number 4 (2000), 335-364.

Dates
First available in Project Euclid: 26 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1038336880

Digital Object Identifier
doi:10.1305/ndjfl/1038336880

Mathematical Reviews number (MathSciNet)
MR1963486

Zentralblatt MATH identifier
1014.03013

Subjects
Primary: 00A25
Secondary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
logicism Frege neologicism second-order logic

Citation

Shapiro, Stewart. Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame J. Formal Logic 41 (2000), no. 4, 335--364. doi:10.1305/ndjfl/1038336880. https://projecteuclid.org/euclid.ndjfl/1038336880


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