Unformatted text preview: x 1 , . . . , x n , z, u )) → ∃ y ∀ u ( u ∈ y ↔ ∃ z φ ( x 1 , . . . , x n , z, u )) ] for each ﬁrstorder formula φ ( v 1 , . . . , v n , v, w ) in the language of set theory. Here ∃ ! xφ ( x ) is an abbreviation of ∃ x [ φ ( x ) & ∀ y ( φ ( y ) → y = x )] (intuitively, there is a unique x such that φ ( x )). Inﬁnity There exists an inﬁnite set. ∃ x [ ∃ y ( y ∈ x & ∀ z ¬ z ∈ y ) & ∀ z ( z ∈ x → ∃ u ( u ∈ x & z ∈ u & ∀ w ( w ∈ u ↔ w = z )))] Regularity Every nonempty set contains an ∈minimal element (every set is regular). ∀ x [ ∃ y ( y ∈ x ) → ∃ y ( y ∈ x & ∀ z ( z ∈ x → ¬ z ∈ y )) ] Axiom of Choice (AC) For every set (of sets) there exists a choice function....
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This note was uploaded on 04/11/2011 for the course MATH 5020 taught by Professor Staff during the Spring '11 term at North Texas.
 Spring '11
 staff
 Set Theory, Sets

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