Full Idea
A system S is said to be infinite when it is similar to a proper part of itself.
Clarification
We would now call a 'system' a 'set'
Gist of Idea
A system S is said to be infinite when it is similar to a proper part of itself
Source
Richard Dedekind (Nature and Meaning of Numbers [1888], V.64)
Book Reference
Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.63
A Reaction
This definition of infinity is widely accepted. It evidently says that a collection is infinite if removing part of it makes no inroads on its size. (But surely you could remove all the natural numbers except the first three? But what do I know?).
Related Idea
Idea 24301 If we subtract a part from a multitude, will that part not itself be a multitude? [Plato]