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2 ideas
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. |
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. |