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..