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for Computer Science Zermelo-Frankel Theory MIT 6.042J/18.062J of Zermelo-Frankel : with the Choice (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 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 ∀x[x ∈ y IFF x ∈ z] Every set has a power set iff ∀x∃p∀.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

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