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.