Full Idea
Two sets are the same size if they can be placed in one-to-one correspondence. But even numbers have one-to-one correspondence with the natural numbers. So a set is infinite if it has one-one correspondence with a proper subset.
Clarification
For 'proper subset', see Idea 8665
Gist of Idea
Infinite sets correspond one-to-one with a subset
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
Book Reference
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.15
A Reaction
Dedekind's definition. We can match 1 with 2, 2 with 4, 3 with 6, 4 with 8, etc. Logicians seem happy to give as a definition anything which fixes the target uniquely, even if it doesn't give the essence. See Frege on 0 and 1, Ideas 8653/4.
Related Ideas
Idea 8653 Nought is the number belonging to the concept 'not identical with itself' [Frege]
Idea 8654 One is the Number which belongs to the concept "identical with 0" [Frege]