Peano axioms (English Wikipedia)

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  • 24. Gray 2013, p. 133
    So Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834).

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  • 23. Fritz 1952, p. 137
    An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form of which the series of the natural numbers is one instance.

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  • 11. José Meseguer and Joseph A. Goguen (Dec 1986). "Initiality, induction, and contputability". In Maurice Nivat and John C. Reynolds (ed.). Algebraic Methods in Semantics (PDF). Cambridge: Cambridge University Press. pp. 459–541. ISBN 9780521267939. Here: sect.2.3, p.464 and sect.4.1, p.471

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