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Single Idea 7556

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

Full Idea

A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together.

Gist of Idea

A collection is infinite if you can remove some terms without diminishing its number

Source

Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86)

Book Ref

Russell,Bertrand: 'Mysticism and Logic' [Unwin 1989], p.86


A Reaction

He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics.


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

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