Ideas from 'works' by Giuseppe Peano [1890], by Theme Structure

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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers

3338	Numbers have been defined in terms of 'successors' to the concept of 'zero'

			Full Idea: Dedekind and Peano define the number series as the series of successors to the number zero, according to five postulates.
			From: report of Giuseppe Peano (works [1890]) by Simon Blackburn - Oxford Dictionary of Philosophy p.279

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

5897	0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n)

			Full Idea: 1) 0 is a number; 2) The successor of any number is a number; 3) No two numbers have the same successor; 4) 0 is not the successor of any number; 5) If P is true of 0, and if P is true of any number n and of its successor, P is true of every number.
			From: report of Giuseppe Peano (works [1890]) by Antony Flew - Pan Dictionary of Philosophy 'Peano'
			A reaction: Devised by Dedekind and proposed by Peano, these postulates were intended to avoid references to intuition in specifying the natural numbers. I wonder if they could define 'successor' without reference to 'number'.