Combining Texts

Ideas for 'fragments/reports', 'The Myth of the Aesthetic Attitude' and 'A Tour through Mathematical Logic'

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2 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV

13529	Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
	Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3)
	A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension

13526	Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
	Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2)
	A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot.