1.
Truth, Deduction,
Computation
Lecture I
Set Theory, ZFC axioms
Vlad Patryshev
SCU
2013
2.
Sets, Formally (ZFC)
Abraham
Fraenkel
Ernst Friedrich Ferdinand
Zermelo
(pronounced [tsermelo])
Axiom of
Choice
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.
Extensionality
A=B ≡ ∀x ((x∈A) ↔ (x∈B))
That’s it.
We can prove symmetry, associativity, reflexivity.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
There’s a Java implementation
https://gist.github.com/vpatryshev/7711870
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