Combining Texts

All the ideas for 'fragments/reports', 'Set Theory and related topics (2nd ed)' and 'The Theory of Transfinite Numbers'

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4 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory

9616	A set is a collection into a whole of distinct objects of our intuition or thought [Cantor]
	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.

4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes

8920	Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz]
	Full Idea: A relation R on a non-empty set S is an equivalence relation if it is reflexive (for each member a, aRa), symmetric (if aRb, then bRa), and transitive (aRb and bRc, so aRc). It tries to classify objects that are in some way 'alike'.
	From: Seymour Lipschutz (Set Theory and related topics (2nd ed) [1998], 3.9)
	A reaction: So this is an attempt to formalise the common sense notion of seeing that two things have something in common. Presumably a 'way' of being alike is going to be a property or a part

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities

15896	Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
	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.

21. Aesthetics / C. Artistic Issues / 7. Art and Morality

468	Musical performance can reveal a range of virtues [Damon of Ath.]
	Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice.
	From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where?