Lecture Notes, 25 Oct 2010

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§2.1: Sets

1. Where We've Been:
   1. I've taught you a language (FOL),
      • which we'll "speak" from now on.

   2. Now we need something to talk about.

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2. Logicism:
   1. Many mathematicians, logicians, and philosophers believe that:
      • Math = logic + set theory

   2. i.e., "sets" are the basic data type of mathematical objects
   3. All other mathematical objects can be constructed from,
      —i.e., defined in terms of—
      sets

   4. For a graphic history of logicism, see:
      Doxiadis, Apostolos; Papadimitriou, Christos H.; Papadatos, Alecos; & Di Donna, Annie (2009), Logicomix (Bloomsbury USA).

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3. What is a set?
   1. Def: A set is_def … ?

      ∃ 2 possible ways to continue this definition:

      1. Take "object" as primitive & undefined
         • Then define "set" as a "collection" of objects.
           • But what's a "collection"?

         • This was Cantor's definition, but …

         • … it led to Russell's paradox:
           R = {x | x ∉ x}

      2. Take "set" & "member" as primitive & undefined
         • But give axioms for them.

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   2. So: ∃ 2 primitive, undefined types of things:
      1. sets
      2. members (or: elements)

      & one 2-place relation between them ("set membership"):

      e ∈ S

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   3. Notation:
      1. "e ∉ S" for: ¬(e ∈ S)
      2. "S = {e₁, e₂, … , e_n}" for: the set S such that (hereafter: "s.t.") ∀x[x ∈ S ↔ (x=e₁) ∨ … ∨ (x=e_n)]
         • This is called an "extensional" description of S;
           it shows you all its members.

      3. A special case of an extensional description:
         {} or ∅ for: the empty set
         • i.e., the set S s.t. ¬∃x[x ∈ S]

      4. An "intensional" description of S;
         it describes all its members:

         {x | P(x)} for: the set of all x (in the domain) s.t. P(x)

         • i.e., the set S s.t. ∀x[x ∈ S ↔ P(x)]

         • Note: This gets us in trouble if we let P(x) = x ∉ x:
           • See Rosen, p. 121, #38.
           • Bertrand Russell's way out:
             sets come in "types";
             every set must be of a "higher" type than its members.
             (So P(x) is simply ungrammatical.)

         • E.g.: The set S consisting of all natural numbers n ≤ 4:
           0 ∈ S, 1 ∈ S, 2 ∈ S, 3 ∈ S, 4 ∈ S

           S = {0,1,2,3,4} (extensional description)
              = {x | (x ∈ N) ^ (x ≤ 4)} (intensional description)

         • E.g.: The set USP of all US presidents:
           USP = {GW, JA, TJ, …, GWB, BO, …} (extensionally)
                   = {x | POTUS(x)} (intensionally)
           (where "POTUS(x)" = x is President Of The US)

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   4. Def: Set Equality:
      Let A,B be sets.
      Then A = B =def ∀x[x ∈ A ↔ x ∈ B]
      1. I.e., a set is "determined by" its members.
      2. ∴ teams and clubs are not sets!

      3. Sometimes this is presented as an axiom rather than a definition:
         • Axiom of Extensionality:
           A = B ↔ ∀x[x ∈ A ↔ x ∈ B]

      4. E.g.: {1,2,3} = {3,1,2} = {1,1,1,2,2,3,3,3,3}

      ---

   5. Def: Subset
      Let A,B be sets.
      Then A is a subset of B =def ∀x[x ∈ A → x ∈ B]
      1. I.e., all A's are Bs.

      2. Notation: A ⊆ B

      3. E.g.: Recall that:
         W = the whole numbers = {1,2,3,…}
         N = the natural numbers = {0,1,2,3,…}
         N⁺ = the positive natural numbers = {1,2,3,…}
         Z = the integers = {…–3,–2,–1,0,1,2,3,…}
         Q = the rational numbers
         R = the real numbers
         C = the complex numbers

         ∴ W ⊆ N⁺ ⊆ N ⊆ Z ⊆ Q ⊆ R ⊆ C.

      4. Thm:
         (∀ set S)[(∅ ⊆ S) ∧ (S ⊆ S)]
         • Proof of (∅ ⊆ S) is in text.
         • Proof of (S ⊆ S):
           Show (S ⊆ S):
           
           Show ∀x[x ∈ S → x ∈ S]:
           Choose arb. e ∈ S, & show e ∈ S → e ∈ S:
           Supp. e ∈ S, & show e ∈ S:
           Trivial! QED

      5. Def: Proper Subset
         Let A,B be sets.
         Then A is a proper subset of B =def ∀x[x ∈ A → x ∈ B] ∧ ∃x[x ∈ B ∧ x ∉ A]
         1. Notation: A ⊂ B
         2. ∴ A ⊂ B ↔ A ⊆ B ∧ A ≠ B
         3. Also: A ⊄ A
         4. E.g.: N⁺ ⊂ N ⊂ Z ⊂ Q ⊂ R ⊂ C

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   6. Def: Power Set
      Let S be a set.
      Then the power set of S =def {A | A ⊆ S}
      1. Notation: ℘(S)

      2. E.g.: Let S = {0,1,2}.
         Then ℘(S) = {∅, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, S}
         Notes:
         • {0} ⊆ S, and 0 &isin S.
           These are related, but distinct, facts.

         • {0} &isin ℘(S), but 0 ∉ ℘(S)

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