COMP 2711H: Lecture 31

Date: 2024-11-11 17:53:22

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Real Numbers

  • Real numbers

    • every : w/ an infinite decimal expansion
      • i.e. non-zero entries forever:
    • interval definition: for
      • and so on for
      • only can be used for
    • theorem: if and
      • then
      • and
    • lemma: is infinite
      • consider: and , let be countable
      • from below theorems (up to "hole"), one can always add new elements in
    • and
    • theorem:
      • i.e. rationals: are dense in reals
      • both : cuts;
        • there must be a hole!
    • theorem:
    • theorem: (and all other segments)
        • then: map
      • thus:
    • size of
      • first,
        • 👨‍🏫 as all elements to : cut
      • the other way: show
        • check: whether is in
          • if so: otherwise
        • put in front of number we can thus construct a one-to-one function from
        • : one to one
      • by Schroder-Bernstein:

  • Axiom of Choice: ZFC 10

    • axioms (equivalent)
      1. for every two sets , either
        • or
      2. for any relation , exists function
        • s.t.
        • 👨‍🏫 but... how can you choose the mapping?
          • believe: there is a way to make choice
      3. for every set , there is a function
        • s.t.
      4. for every set of non-empty disjoint sets
        • set s.t.
          • : set of a point from each region
      5. Zorn's lemma: let be a set s.t. for every chain
        • we have
        • then: has a maximal element
    • a set : is chain if
      • maximal element of : an element s.t.
    • assuming ZFC 10-5: for any two sets :
      • either or
      • : all 1-to-1 function s.t. and
        • i.e. degree of vertex in : min 0, can be
      • there is a maximal function
      • claim: it either covers entire , or entire
      • finally: if Zorn's lemma holds, ZFC 10-1 also holds (all ZFC 10: equivalent)