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Single Idea 13949

[from 'Principles of Arithmetic, by a new method' by Giuseppe Peano, in 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic ]

Full Idea

Peano's axioms are categorical (any two models are isomorphic). Some conclude that the concept of natural number is adequately represented by them, but we cannot identify natural numbers with one rather than another of the isomorphic models.

Gist of Idea

All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers

Source

comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11

Book Reference

Cartwright,Richard: 'Philosophical Essays' [MIT 1987], p.48


A Reaction

This is a striking anticipation of Benacerraf's famous point about different set theory accounts of numbers, where all models seem to work equally well. Cartwright is saying that others have pointed this out.