Truth, Deduction,
Computation
Lecture I
Set Theory, ZFC axioms
Vlad Patryshev
SCU
2013
Sets, Formally (ZFC)
Abraham

Fraenkel

Ernst Friedrich Ferdinand

Zermelo
(pronounced [tsermelo])

Axiom of

Choice
ZFC Axioms
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

Extensionality Axiom
Axiom of Empty Set
Comprehension Axiom Schema (filter)
Axi...
Extensionality
A=B ≡ ∀x ((x∈A) ↔ (x∈B))

That’s it.
We can prove symmetry, associativity, reflexivity.
Axiom of Empty Set
∃A ∀x (¬x∈A)
denoted as ∅
Why do we need it?

Because all other axioms assume existence of
sets; this o...
Comprehension Axiom Schema
A is a set ⊢ {x∈A | P(x)} is a set

● Note that we have no rule allowing to build something
lik...
Axiom of Pairing
∀x ∀y ∃A (x∈A ∧ y∈A)
● Notation: {a,b} = {x∈?|x=a v x=b}
(here we made a choice of an A… not very pure)

...
Axiom of Union
∀A ∃B ∀y ∀x((x∈y ∧ y∈A) → x∈B)
∪{y∈A} ≡ {x∈? | ∃y (y∈A)}
X ∪ Y ≡ ∪{a∈{X,Y}} = {x∈? | x∈X v x∈Y}
E.g.
● A={{...
Replacement Axiom Schema
∀A (∀x(x∈A → (∃!y P(x,y))) ⊢
∃B ∀x(x∈A → (∃y (P(x,y)∧ y∈B))
In other words, if we have a function...
Axiom of Infinity
∃X (∅∈X ∧ ∀y (y∈X → S(y)∈X))
where S(y) ≡ y ∪ {y}

Do you recognize Peano Natural Numbers?
The axiom say...
Axiom of Powerset
∀x∃y∀z (z⊂x → z∈y)
where A⊂B ≡ ∀x ((x∈A)→(x∈B))

P(X) ≡ {z∈?/*maybe an y from above*/|z⊂X}
● A⊂B here me...
Well-founded Relationship
R ⊂ A×A is well-founded iff
∀S⊂A (S=∅ v (∃x∈S ∀y∈S ¬xRy))

●
●
●
●
●

xRy means (x,y) ∈ R
E.g. p...
Foundation Axiom
∀x (x=∅ v (∃y∈x (¬∃z
(z∈y∧z∈x))

●
●
●
●

No set is an element of itself (Quine atoms outlawed)
No infini...
Axiom of Choice (AC)
∀x (∅∉x v (∃f:(x→∪x) (∀y∈x f(y)
∈y))
What it means
● every set can be fully ordered
● f:X→Y is surjec...
There’s a Java implementation
https://gist.github.com/vpatryshev/7711870
That’s it.
Thanks for listening.
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set theory, ZFC

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Truth, deduction, computation lecture i (last one)

  1. 1. Truth, Deduction, Computation Lecture I Set Theory, ZFC axioms Vlad Patryshev SCU 2013
  2. 2. Sets, Formally (ZFC) Abraham Fraenkel Ernst Friedrich Ferdinand Zermelo (pronounced [tsermelo]) Axiom of Choice
  3. 3. ZFC Axioms 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Extensionality Axiom Axiom of Empty Set Comprehension Axiom Schema (filter) Axiom of Pair Axiom of Union Replacement Axiom Schema (map) Axiom of Infinity Powerset Axiom Foundation Axiom Axiom of Choice
  4. 4. Extensionality A=B ≡ ∀x ((x∈A) ↔ (x∈B)) That’s it. We can prove symmetry, associativity, reflexivity.
  5. 5. Axiom of Empty Set ∃A ∀x (¬x∈A) denoted as ∅ Why do we need it? Because all other axioms assume existence of sets; this one states an existence.
  6. 6. Comprehension Axiom Schema A is a set ⊢ {x∈A | P(x)} is a set ● 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
  7. 7. Axiom of Pairing ∀x ∀y ∃A (x∈A ∧ y∈A) ● 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}}
  8. 8. Axiom of Union ∀A ∃B ∀y ∀x((x∈y ∧ y∈A) → x∈B) ∪{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
  9. 9. Replacement Axiom Schema ∀A (∀x(x∈A → (∃!y P(x,y))) ⊢ ∃B ∀x(x∈A → (∃y (P(x,y)∧ y∈B)) In other words, if we have a function f defined on A, we can say that its image B is a set.
  10. 10. Axiom of Infinity ∃X (∅∈X ∧ ∀y (y∈X → S(y)∈X)) where S(y) ≡ y ∪ {y} Do you recognize Peano Natural Numbers? The axiom says: we have natural numbers.
  11. 11. Axiom of Powerset ∀x∃y∀z (z⊂x → z∈y) where A⊂B ≡ ∀x ((x∈A)→(x∈B)) P(X) ≡ {z∈?/*maybe an y from above*/|z⊂X} ● A⊂B here means non-strict subset. E.g. A⊂A ● Now we have ℵ1 ● Can |x| = |P(x)|? No (see Russell paradox)
  12. 12. Well-founded Relationship R ⊂ A×A is well-founded iff ∀S⊂A (S=∅ v (∃x∈S ∀y∈S ¬xRy)) ● ● ● ● ● 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)
  13. 13. Foundation Axiom ∀x (x=∅ v (∃y∈x (¬∃z (z∈y∧z∈x)) ● ● ● ● 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
  14. 14. Axiom of Choice (AC) ∀x (∅∉x v (∃f:(x→∪x) (∀y∈x f(y) ∈y)) 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
  15. 15. There’s a Java implementation https://gist.github.com/vpatryshev/7711870
  16. 16. That’s it. Thanks for listening.

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