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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.
10059 | In mathematic we are ignorant of both subject-matter and truth [Russell] |
7554 | Self-evidence is often a mere will-o'-the-wisp [Russell] |
7556 | A collection is infinite if you can remove some terms without diminishing its number [Russell] |
7557 | To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts [Russell] |