Frege meets Dedekind: A neologicist treatment of real analysis. (English) Zbl 1014.03013

Summary: 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: \[ \text{INT}(a,b)=\text{INT}(c,d)\equiv (a+d) = (b+c).\tag{DIF} \]
\[ \begin{aligned} Q(m,n) & = Q(p,q)\equiv (n=0 \& q=0)\tag{QUOT}\\ & \vee (n\neq 0 \& q \neq 0 \& m\cdot q=n\cdot p).\end{aligned} \] 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 \(P\leq s\) 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: \[ \forall P \forall Q(C(P) =C(Q)\equiv\forall r\;(P\leq r\equiv Q\leq r)).\tag{CP} \] 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.


03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
03B15 Higher-order logic; type theory (MSC2010)
03B30 Foundations of classical theories (including reverse mathematics)
03F35 Second- and higher-order arithmetic and fragments
26A03 Foundations: limits and generalizations, elementary topology of the line
Full Text: DOI


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