Ideas from 'The Theory of Transfinite Numbers' by George Cantor [1897], by Theme Structure

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
A set is a collection into a whole of distinct objects of our intuition or thought
                        Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought.
                        From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2
                        A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
Cantor needed Power Set for the reals, but then couldn't count the new collections
                        Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections.
                        From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I
                        A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue.