more from Richard Dedekind

### Single Idea 18841

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order]

Full Idea

It is Dedekind's categoricity result that convinces most of us that he has articulated our implicit conception of the natural numbers, since it entitles us to speak of 'the' domain (in the singular, up to isomorphism) of natural numbers.

Clarification

'Categoricity' means any two models are isomorphic (i.e. they match)

Gist of Idea

Categoricity implies that Dedekind has characterised the numbers, because it has one domain

Source

comment on Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ian Rumfitt - The Boundary Stones of Thought 9.1

Book Reference

Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.267

A Reaction

The main rival is set theory, but that has an endlessly expanding domain. He points out that Dedekind needs second-order logic to achieve categoricity. Rumfitt says one could also add to the 1st-order version that successor is an ancestral relation.