Reals by Abstraction

Philosophia Mathematica 8 (2):100--123 (2000)
On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities.
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DOI 10.1093/philmat/8.2.100
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Logicism, Interpretability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Review of Symbolic Logic 7 (1):84-119.
Abstraction and Identity.Roy T. Cook & Philip A. Ebert - 2005 - Dialectica 59 (2):121–139.
What is Neologicism?Bernard Linsky & Edward N. Zalta - 2006 - Bulletin of Symbolic Logic 12 (1):60-99.

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