WEEK 7
M: Holiday.
W: Completion of the proof of Goodstein Theorem. Equinumerosity.
      1.59. Definition. Equinumerosity, \preceq
      1.60. Remark. Comparing the cardinalities using injections and surjections. 
      1.61. Remark. Defining cardinalities using the foundation axiom. 
      1.62. Remark. Basic cardinal arithmetic: ({0}× A) U ({1}× B), A × B, ^BA, infinite disjoint unions and products.
      1.63. Proposition. Basic formulas in cardinal arithmetic:
                (a) Addition and Multiplication are commutative and associative.
                (b) Multiplication is distributive w.r.t. addition/unions.
                (c) ^BuCA ~ ^BA × ^CA  if B,C are disjoint.
                (d) ^C(A × B) ~ ^CA × ^CB.
                (e) ^B×CA ~ ^B(^CA).
      1.64. Notation. The characteristic funtion of a subset of a set. Observation: P(A) ~ ^A{0,1}.  
F: No lecture. Make up will be on Fri Nov 3 at 8:00am.