My logic lectures at SCU set theory (informal)

1. Truth, Deduction,
Computation
Lecture H
Set Theory, Informally
Vlad Patryshev
SCU
2013

2. Sets, Informally
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a ∈ A : a is an element of A
a ∉ A ≡ ¬(a∈A)
empty set ∅ ≡ ¬∃x (x∈∅)
A ⊂ B ≡ ∀x ((x∈A)→(x∈B))
A ⊄ B ≡ ¬(A⊂B)
{a,b,c…} : a set consisting of a,b,c…
{x∈A | P(x)} : set comprehension (like in Python)
A ∩ B = {x | (x∈A) ∧ (x∈B)}
A ∪ B = {x | (x∈A) ∨ (x∈B)}
A B = {x | (x∈A) ∧ ¬(x∈B)}
Do we have two monoids?
powerset P(A) ≡ {B | B ⊂ A}
e.g. P(∅)={∅}; P(P(∅)={∅,{∅}}

3. Examples of Sets
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empty set ∅ - size 0
finite sets e.g. {∅, {∅, {∅}}, {{∅}}}
set of natural numbers ℕ - size ℵ0
set of integer numbers ℤ - size ℵ0
set of rational numbers ℚ - size ℵ0
set of real numbers ℝ - size ℵ1
set of complex numbers ℂ - size ℵ1

4. Sets Equality
A=B ≡ ∀x ((x∈A) ↔ (x∈B))
● {a,b} = {b,a}
● {a,a,a} = {a}
● A = B iff {A} = {B} iff {{A}} = {{B}}

5. Powerset
P(A) aka 2A ≡ {x|x⊂A}
● 2∅ ={∅}
● 2{a,b} = {∅,{a},{b},{a,b}}
● 2A ∪ B = 2 A × 2A

6. Natural Numbers in Sets
∅, {∅}, {{∅}}, {{{∅}}}, etc…
Meaning, 0≡∅; S(n)≡{n}. Just count the curlies.
Or, better, ∅, {∅}, {∅, {∅}}, {∅, {∅}, {{∅}}}
Where’s Universal Property? Oops, something’s missing.
But first, introduce pairs.

7. Define Pair
(a,b), for a and b of any nature.
No, {a,b} won’t work, it’s the same as {b,a}
How about {a,{a,b}}?
Almost there… do you see the problem?
Will fix it later.

8. Have Pairs, define Relationships
A × B ≡ {(a,b) | (a∈A)∧(b∈B)}
● A × ∅ = ∅
● A × (B∪C) = (A×B) ∪ (A×C)
● {a,b} × {x,y} = {(a,x),(a,y),(b,x),(b,y)}
Do you see a monoid?

9. Kinds of Binary Relationships
R ⊂ A × B
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xRy ≡ (x,y)∈R
reflexive: ∀x (xRx)
symmetric: ∀x∀y (xRy → yRx)
antisymmetric: ∀x∀y ¬(xRy ∧ yRx)
transitive: ∀x∀y∀z ((xRy ∧ yRz) → xRz)
equivalence: reflexive, symmetric, transitive
partial order: antisymmetric, reflexive
functional: ∀x ∃!y (xRy)

10. Infinities
A set is infinite if it is in bijection with its
proper subset
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finite: in bijection with a natural number ({∅, {∅}, {{∅}}...})
ℵ0 - in bijection with ℕ
ℵ1 - in bijection with 2ℕ, which is in bijection with
ℵ2 = 2ℵ1
etc, ℵk+1 = 2ℵk
ℵx= ∪ {ℵy| y < x}
ℵ = the power of all sets; ℵ=2ℵ (?!)
Good Reading on Aleph
ℝ

11. Many Definitions, Many Paradoxes
● Are there as many natural numbers as their squares?
● Points of a square vs points on the edge of the square same amount?!
● The diary of Tristram Shandy problem
● Ross-Littlewood-Achilles (take 1 ball, add 10, each
time twice faster)
● Russell paradox
● König's paradox (the first real number not finitely
defineable)
● etc

12. That’s it for today