Truth, Deduction, ComputationLecture ISet Theory, ZFC axioms

Vlad PatryshevSCU2013

Sets, Formally (ZFC)

Ernst Friedrich Ferdinand

Zermelo (pronounced [tsermelo])

Abraham Fraenkel

Axiom of Choice

ZFC Axioms

1. Extensionality Axiom2. Axiom of Empty Set3. Comprehension Axiom Schema (filter)4. Axiom of Pair5. Axiom of Union6. Replacement Axiom Schema (map)7. Axiom of Infinity8. Powerset Axiom9. Foundation Axiom

10. Axiom of Choice

Extensionality

That’s it. We can prove symmetry, associativity, reflexivity.

A=B ≡ ∀x ((x∈A) ↔ (x∈B))

Axiom of Empty Set

Why do we need it?

Because all other axioms assume existence of sets; this one states an existence.

∃A ∀x (¬x∈A)denoted as ∅

Comprehension Axiom Schema

● Note that we have no rule allowing to build something like {x | x=x} or even {x | x=1}

● Do you see that {x∈A | true} = A?● It does not give us ∅● Also, it does not give us union of two sets

A is a set ⊢ {x∈A | P(x)} is a set

Axiom of Pairing

● Notation: {a,b} = {x∈?|x=a v x=b}

(here we made a choice of an A… not very pure)

● x∈{a,b} ↔ (x=a v x=b)● Now we can, given an x, build {x}● We can also build an unlimited number of sets: ∅,

{∅}, {{∅}}, {{{∅}}}...● Introduce ordered pair: {x, {x,y}} (problems?)● Kuratowski pair: {{x}, {x,y}}

∀x ∀y ∃A (x∈A ∧ y∈A)

Axiom of Union

∪{y∈A} ≡ {x∈? | ∃y (y∈A)}

X ∪ Y ≡ ∪{a∈{X,Y}} = {x∈? | x∈X v x∈Y}

E.g.● A={{a,b}, {b,c,d}} ∪A = {a,b,c,d} ● ∪∅ = ∅● ∪{x} = x

∀A ∃B ∀y ∀x((x∈y ∧ y∈A) → x∈B)

Replacement Axiom Schema

In other words, if we have a function f defined on A, we can say that its image B is a set.

∀A (∀x(x∈A → (∃!y P(x,y))) ⊢ ∃B ∀x(x∈A → (∃y (P(x,y)∧ y∈B))

Axiom of Infinity

Do you recognize Peano Natural Numbers?The axiom says: we have natural numbers.

∃X (∅∈X ∧ ∀y (y∈X → S(y)∈X))where S(y) ≡ y ∪ {y}

Axiom of Powerset

P(X) ≡ {z∈?/*maybe an y from above*/|z⊂X}

● A⊂B here means non-strict subset. E.g. A⊂A● Now we have ℵ

1● Can |x| = |P(x)|? No (see Russell paradox)

∀x∃y∀z (z⊂x → z∈y)where A⊂B ≡ ∀x ((x∈A)→(x∈B))

Well-founded Relationship

● xRy means (x,y) ∈ R● E.g. partial order on N, a>b● Counterexample: a<b on N● Counterexample: aRb, bRc, cRa● Can use generalized induction (Noether)

R ⊂ A×A is well-founded iff∀S⊂A (S=∅ v (∃x∈S ∀y∈S ¬xRy))

Foundation Axiom

● No set is an element of itself (Quine atoms outlawed)● No infinite descending sequence of sets exists● No need for Kuratowski pairs● For every two sets, only one can be an element of the

other

∀x (x=∅ v (∃y∈x (¬∃z(z∈y∧z∈x))

Axiom of Choice (AC)

What it means● every set can be fully ordered● f:X→Y is surjection ⊢ f has an inverse● Banach-Tarski paradox

Alternatively: Axiom of Determinacy

∀x (∅∉x v (∃f:(x→∪x) (∀y∈x f(y)∈y))

There’s a Java implementation

https://gist.github.com/vpatryshev/7711870

That’s it. Thanks for listening.