handout03

# handout03 - x 1 x n z u → ∃ y ∀ u u ∈ y ↔ ∃ z...

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Math 5020 Handout 3 Axioms of Set Theory Extensionality A set is uniquely determined by its elements. x y [ x = y ↔ ∀ z ( z x z y ) ] Pairing The collection of a pair of sets is a set. x y z [ x z y z u z ( u = x u = y ) ] We denote the pairing set by { x, y } . Union The union of a set (of sets) is a set. x y z [ z y ↔ ∃ u ( z u u x ) ] We denote the union of x by x . Power Set The collection of all subsets of a set is a set. x y z [ z y ↔ ∀ u ( u z u x ) ] We abbreviate z ( z x z y ) as x y ( x is a subset of y ). We denote the power set of x by P ( x ). Separation A deﬁnable collection of elements of a set is a set. x x 1 . . . x n y z [ z y ( z x φ ( z, x 1 , . . . , x n )) ] for each ﬁrst-order formula φ ( v, v 1 , . . . , v n ) in the language of set theory. We denote the deﬁnable subset as { z x : φ ( z, x 1 , . . . , x n ) } . Replacement The range of a deﬁnable function with a set domain is a set. x x 1 . . . x n [ z ( z x → ∃ ! u φ (
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Unformatted text preview: x 1 , . . . , x n , z, u )) → ∃ y ∀ u ( u ∈ y ↔ ∃ z φ ( x 1 , . . . , x n , z, u )) ] for each ﬁrst-order 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.

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