Logic: Lecture 9, 4 September 2012
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FO Satisfiability Lemma:
  Let X be s set of sentences.

  Then X is FO-satisfiable iff 
       X u Phi_H U Phi_Q U Phi_Eq is propositionally satisfiable

Consequences:

Finite Satisfiability Lemma:

  Let X be a set of sentences.  X is satisfiable iff every finite
  subset of X is satisfiable.

Compactness Theorem:

  X |= phi iff Y |= phi for some finite subset Y of X

Both results can be proved by appealing to the FO satisfiability
lemma and then invoking the corresponding result for
propositional satisfiability.

Also the FO satisfiability lemma construction establishes the
following

Lowenheim-Skolem Theorem:

1. If L is finite or countable, then X is satisfiable iff X is
   satisfiable in a countable structure.

2. If L is not countable, X is satisfiable in a structure of
   cardinality bounded by cardinality of L

Corollary:
  No finite/countable language can axiomatize the real numbers
  (upto isomorphism).


Finiteness of structures cannot be captured in FO.

Theorem: If X has arbitrarily large finite models (for all n in N
  there is a model of at least size n), then X has an infinite
  model.

Proof
  Let Y = X U {phi_>=n | n >= 2}.
  Every finite subset Y_0 of Y is satisfiable by assumption.
  By finite satisfiability, Y has a model.  This must be
  infinite, so X has an infinite model.

Theorem: Upward Lowenheim-Skolem
  If X has an infinite model, then for any set A, X has a model
  of cardinality at least the size of A (i.e., there exists an
  injection from A to the underlying set of the structure.)

Proof

  Add constants c_a for a in A 

  Y = X U { ~(c_a = c_b) | a =/= b}
  
  Every finite subset Y_0 of Y is satisfiable.  Y_0 has a finite
  set of inequalities ~(c_a = c_b).  Map these constants to
  distinct elements in the structure and map the remaining
  constants arbitrarily.

  By finite satisfiability, Y is satifiable.  Each constant c_a,
  a in A, is mapped to a distinct element of the underlying
  structure for Y.  The interpreretation for the constants c_a, a
  in A, defines an injection from A to the underlying set.

A complete axiomatization

(A1)  All tautologies of PC

(A2a) x = x
(A2b) x = y => phi(x) = phi(y), where phi(u) is an atomic formula

(A3)  phi(t) => Ex phi(x)

      alpha, alpha => beta
(MP)  --------------------
             beta

          phi(x) => psi
(G)   --------------------      x not in FV(psi)
        Ex phi(x) => psi

(A1 at the level of prime formulas)

Define X |- phi as usual.

Can derive equality axioms (see notes)

(Fill in some details of PC in some of the derivations.)

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