Logic: Lecture 7, 25 August 2015
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Free variables:

  Notation:
  phi(x1,...,xn) means that FV(phi) is contained in {x1,...,xn}

Expressing properties in FOL

  Groups

    Axioms G1-G3
    (Left) cancellation
    Abelian groups

    Element of order 2 ... element of order k
    All elements have a finite order
       "Infinite" formula
       Cannot express even with infinite set of finite formulas

  Equivalence relations
    Ax r(x,x)
    Ax Ay r(x,y) <=> r(y,x)
    Ax Ay Az (r(x,y) and r(y,z)) => r(x,z)

  Strict linear orders
    Ax ~(x < x)
    Ax Ay Az (x < y and y < z) => x < z
    Ax Ay (x < y or x = y or y < x)

  Peano's axioms for arithmetic

    R = {<}, F = {succ,+,*}, C = {0}
    Interpret over Nat = {0,1,2,...}
   

    ~Ex succ(x) = 0
    Ax Ay (succ(x) = succ(y) => x = y)
    Ax Az (x < y <=> Ez (x + succ(z) = y)
    Ax (x + 0) = x
    Ax Ay (x + succ(y) = succ(x+y)
    Ax (x * 0) = 0
    Ax Ay (x * succ(y)) = (x*y + x)

    These axioms on their own allow "non standard" models of
    arithmetic (which we will see later).  To fully characterize
    Nat, we need an induction axiom.

      Any set X that contains 0 and is closed with respect to succ
      (i.e. x in X implies succ(x) in X) must contain all of Nat

    To formulate this in logic, need a second order quantifier
    AX, where X ranges over all subsets.

    To express this in first order logic:

    - A formula phi(x) with one free variable defines a subset:
      those values of x for which phi is true.

    - Induction axiom can be formulated as an infinite set of
      sentences, one for each formula phi(x) in the language with
      none free variable:

        (phi(0) and Ax (phi(x) => phi(succ(x))) => Ax phi(x)

Cardinality

  At least two elements: Ex Ey ~(x = y)
  At least k elements:   Ex1 ... Exn And_{i =/= j} ~(xi = xj)

  At most one element:   Ax Ay (x = y)
  At most k-1 elements:  Ax1 ... Axn Or_{i =/= j} (xi = xj)

  Structure is not finite: Infinite family
    At least two
    At least three
    ...
    At least k
    ...

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