Set Theory Axioms

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Mathematics for Computer Science Zermelo-Frankel Set Theory MIT 6.042J/18.062J Axioms of Zermelo-Frankel Set Theory: with the Choice axiom (ZFC) define the standard ZFC Theory of Sets Albert R Meyer, March 4, 2015 ZF.1 Albert R Meyer, March 4, 2015 ZF.2 Some Axioms of Set Theory Some Axioms of Set Theory Extensionality Extensionality x and y have the same elements ∀x[x ∈ y IFF x ∈ z] iff x and y are members of the same sets Albert R Meyer, March 4, 2015 ZF.3 Albert R Meyer, March 4, 2015 ZF.4 1 Some Axioms of Set Theory Some Axioms of Set Theory Extensionality Power set ∀x[x ∈ y IFF x ∈ z] Every set has a power set iff ∀ ∃ ∀ ⊆ ∈ x p s.s x IFF s p ∀x[y ∈ x IFF z ∈ x] Albert R Meyer, March 4, 2015 ZF.5 Albert R Meyer, March 4, 2015 ZF.6 Some Axioms of Set Theory Sets are Well Founded Comprehension According to ZF, the elements If S is a set, and P(x) is a of a set have to be “simpler” predicate of set theory, than the set itself. In then particular, {x ∈ s| P(x)} no set is a member of itself, is a set or a member of a member… Albert R Meyer, March 4, 2015 ZF.7 Albert R Meyer, March 4, 2015 ZF.8 2 Sets are Well Founded Sets are Well Founded Def: x is ∈-minimal in y Def: x is ∈-minimal in y ∈ x is in y, but no element x y AND of x is in y [∀z.z ∈ x IMPLIES z ∉ y] Albert R Meyer, March 4, 2015 ZF.9 Albert R Meyer, March 4, 2015 ZF.10 Some Axioms of Set Theory Some Axioms of Set Theory Foundation Foundation Every nonempty set has Every nonempty set has an ∈-minimal element an ∈-minimal element ∀x.[x ≠∅ IMPLIES ∃y. y is ∈ -minimal in x] Albert R Meyer, March 4, 2015 ZF.11 Albert R Meyer, March 4, 2015 ZF.12 3 S ∉ S S ∉ S Let R::= {S}. Let R::= {S}. If S ∈ S, then R has no ∈-minimal element. If it exists, it must be S, but S ∈ R and S ∈ S, so S is not ∈-minimal in R. Albert R Meyer, March 4, 2015 ZF.13 Albert R Meyer, March 4, 2015 ZF.14 Zermelo-Frankel Set Theory Zermelo-Frankel Set Theory S ∉ S implies that S ∉ S implies that (1) the collection of all sets is not (1) the collection of all sets is not a set, and a set, and (2) W = {s ∈Sets|s ∉s} (2) W = { s ∈ Sets|s ∉ s } is the collection of all sets -- which is why it’s not a set. Albert R Meyer, March 4, 2015 ZF.15 Albert R Meyer, March 4, 2015 ZF.16 4 MIT OpenCourseWare https://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2015 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms..

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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection).
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Equivalents to the Axiom of Choice and Their Uses A
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Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
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