Logic: Lecture 8, 30 August 2012
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FO satisfiability

Theorem: Let X be a set of *sentences* over L. 
         The following are equivalent.

1. There is an L-structure M = (S,i) that satisfies X
2. There is an L_H-structure M' = (S,i') that satisfies X
3. X U Phi_H U Phi_Q U Phi_Eq is propositonally satisfiable

(2 => 1)

  Immediate (restrict i' over C U C_H to i over C)

(1 => 3)

  Need to extend M = (S,i) to M = (S,i') to cover L_H

  i'(c) = i(c) for c in C

  For c = c_phi in C_H, 
     if M |= Ex phi, map i'(c_phi) to a s.t. I |= phi(a)
        otherwise map i'(c_phi) arbitrarily
     (need to do this in stages for C_1, C_2, ...)

  Now, from M' we derive v such that for every prime formula phi,
    v(phi) = tt iff M' |= phi

  It follows that v |= X U Phi_H U Phi_Q U Phi_Eq

(3 => 2)

  We have a valuation v over all prime formulas in L_H such that
  v |= X U Phi_H U Phi_Q U Phi_Eq

  Need to define an L_H structure M = (S,i) that satisfies X

  First S:

  Define and equivalence relation on terms, 
     t ~ t'  iff  v |= t = t'

     v |= Phi_Eq guarantees that ~ is an equivalence relation

  Elements of S are equivalence classes [c] for c in C U C_H

  Define i as follows:

   For C : i(c) = [c]

   For R : ([c1],...,[cn]) in i(r) iff v |= r(c1,...,cn)

   For F :
     
     Need to identify c such that i(f)([t1],...,[tn]) = [c]
     To show that such a constant c must exist

       Define phi(x) to be the formula f(c1,...,cn) = x

       If v |= Ey phi(y), then v |= f(c1,...,cn) = c_phi (Phi_H)

       If not, the quantifier axiom says
          phi(f(c1,...,cn)) -> Ey phi(y)

       Since not(v |= Ey phi(y)), we must have
         not(v |= phi(f(c1,...,cn)))

       But phi(f(c1,...,cn)) is just f(c1,...,cn) = f(c1,...,cn)
         which is an instance of the equality axiom t = t!
     
  The fact that interpretations of F and R are compatible
  with ~ follows from Phi_Eq

  Claim: For all sentences phi in L_H, M |= phi iff v |= phi

    Construction of S ensures that M |= phi iff v |= phi for
    all atomic formulas.

    Boolean connectives are easy.

    phi = Ex psi

     M |= Ex psi => M |= psi(a) for some element a of s.
       Every a in S is [c] for some constatnt c.  Use
       quantifier axiom to show that v |= Ex psi

     v |= Ex psi => v |= psi(c_phi) by Henkin axiom, 
       so M |= psi(c_phi) so M |= Ex psi. 

  From the claim, it follows that M |= X


Lemma: Finite satisfiability
   X a set of FO sentences over L.  X is satisfiable iff every
   finite subset Y of X is satisfiable

(<==)
   Every finite subset Y of X is FO satisfiable.  To show that X
   is FO satisfiable.

   Sufficient to show that X U Phi_H U Phi_Q U Phi_Eq is
   propositionally satisfiable.

   Sufficient to show that every finite subset of 
   X U Phi_H U Phi_Q U Phi_Eq is propositionally satisfiable

   If Y is FO satisfiable, then Y U Phi_H U Phi_Q U Phi_Eq is
   propositionally satisfiable

   Each finite subset of X U Phi_H U Phi_Q U Phi_Eq is contained
     in some Y U Phi_H U Phi_Q U Phi_Eq, hence satisfiable.

   So the result follows.

Theorem: Compactness
   X |= phi iff Y |= phi for some finite Y subset of X

   X U {~phi} is not satisfiable, so there is finite Y such that
   Y U {~phi} is not satisfiable, so Y |= A

Theorem: Lowenheim-Skolem

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

Proof:
1. From the size of the witnessing expansion
   Countable union of countable sets is countable.

2. Similar argument for larger cardinalities

Corollary:
  No finite/countable language can axiomatize the real numbers
  (upto isomorphism).  i.e., any countable set of axioms that is
  satisfied over the reals will also be satisfied in some
  countable structure. 

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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