Notre Dame Journal of Formal Logic 41 (4):335--364 (2000)

Authors
Stewart Shapiro
Ohio State University
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: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). 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≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤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.
Keywords neologicism   Frege   real numbers
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DOI 10.1305/ndjfl/1038336880
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References found in this work BETA

Logic in the Twenties: The Nature of the Quantifier.Warren D. Goldfarb - 1979 - Journal of Symbolic Logic 44 (3):351-368.
Iteration Again.George Boolos - 1989 - Philosophical Topics 17 (2):5-21.

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Citations of this work BETA

Naturalism.Davidn D. Papineau - 2007 - Stanford Encyclopedia of Philosophy.
Bad Company Tamed.Øystein Linnebo - 2009 - Synthese 170 (3):371 - 391.
Logicism, Interpretability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Review of Symbolic Logic 7 (1):84-119.
Some Criteria for Acceptable Abstraction.Øystein Linnebo - 2011 - Notre Dame Journal of Formal Logic 52 (3):331-338.

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