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Truth, deduction, computation lecture h

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My logic lectures at SCU
set theory (informal)

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Truth, deduction, computation lecture h

  1. 1. Truth, Deduction, Computation Lecture H Set Theory, Informally Vlad Patryshev SCU 2013
  2. 2. Sets, Informally 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 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. 3. Examples of Sets 1. 2. 3. 4. 5. 6. 7. 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. 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. 5. Powerset P(A) aka 2A ≡ {x|x⊂A} ● 2∅ ={∅} ● 2{a,b} = {∅,{a},{b},{a,b}} ● 2A ∪ B = 2 A × 2A
  6. 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. 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. 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. 9. Kinds of Binary Relationships R ⊂ A × B ● ● ● ● ● ● ● ● 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. 10. Infinities A set is infinite if it is in bijection with its proper subset ● ● ● ● ● ● ● 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. 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. 12. That’s it for today

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