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Single Idea 9826

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite ]

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 Ref

Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.63

The 3 ideas with the same theme [what is distinctive about infinite numbers?]:

9826 A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
7556 A collection is infinite if you can remove some terms without diminishing its number [Russell]
14134 Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell]