more from Peter Smith

Single Idea 10068

[catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic]

Full Idea

The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence.

Gist of Idea

Natural numbers have zero, unique successors, unending, no circling back, and no strays


Peter Smith (Intro to Gödel's Theorems [2007], 01.1)

Book Reference

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.1

A Reaction

These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems.