Logic: Lecture 13, 15 September 2015
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Recall: Model theory
X a set of FO sentences. Mod(X) = { M | M |= X }
Elementary and Delta-Elementary classes of structures
C is elementary if C = Mod(phi) for some FO sentence phi.
C is Delta-elementary if C = Mod(X) for some set of FO sentences X.
Examples:
1. A torsion group is one in which every element is of finite
order
Ax (x = 0 or x + x = 0 or ....)
Not Delta-elementary
Suppose X is a set of sentences characterizing torsion.
Let Y = X U { ~(x + x + ... + x = 0) | n >= 1 }
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n-times
Every finite subset Y0 of Y is satisfiable. The second set is
bounded by some n0. Choose a cyclic group of order > n0 and
assign x some generator of the group.
Hence, Y is satisfiable in (M,sigma). But sigma(x) cannot have
finite order, so the underlying group is torsion free.
2. Connected graphs are not Delta-elementary
Suppose X characterizes connectedness
Let psi_n = ~(x = y) and
~Ex1..Exn( x1 = x and xn = y and
E(x1,x2) and ... and E(xn-1,xn) )
Let Y = X U { psi_n | n >= 2}
Every finite subset Y0 is satisfiable. There is a bound n0
for psi_n. Choose a cyclic polygon with 2n0 vertices.
Y is satisfiable in (M,sigma) but sigma(x) and sigma(y) are
not connected.
Elementary equivalence
M == M' wrt L if, for every sentence phi in L, M |= phi iff M' |= phi
Th(M) = { phi | M |= phi)
Lemma
M == M' iff M' |= Th(M)
Observation
If two structures are isomorphic, they are elementary equivalent.
Theorem
For any structure M, C = { M' | M' == M } is Delta-elementary,
given by Mod(Th(M)). C is the smallest Delta-elementary
elementary class containing M.
Proof
If M in Mod(X), then M |= X, so M' == M also satisfies M' |= X
so M' in Mod(x). So C subset of Mod(X).
Theorem
If M is infinite, the class of structures isomorphic to M is
not Delta-elementary.
Proof
By Upward Lowenheim-Skolem theorem. Can find a model of size
powerset of M.
Categorical theory: all models are isomorphic. No FO theory can
be categorical (by upward and downward LS theorems)
Peano's axioms for (N,s,0)
1. 0 is not the successor of any element
2. successor is injective
3. For all X, if 0 in X and n in X implies s(n) in X, then X = N
Second order axioms. This is categorical in second order logic
but not in FO.
Nonstandard model of arithmetic:
Structure that is elementary equiv but not isomorphic to
(N,s,0) satisfying Th(N) --- all FO sentences that hold by
virtue of Peano's axioms.
Theorem (Skolem)
There exists a countable nonstandard model of arithmetic
X = Th(N) U { ~(x = 0), ~(x = 1) .... }
Again, by compactness, there is a (countable) model M' for X in
which sigma(x) not in N.
What does the model look like?
sigma(x) = a, but a is outside N
a must be closed under predecessor and successor (Th(N))
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0 1 2 ... ... a-2 a-1 a a+1 a+2 ...
a+a must give us another copy of the same to the right (if a+a
is in the second copy, then a+a = a+n for some n, but then a =
n by cancellation)
By midpoint theorem (Am An Ep m+n = 2p or m+n = 2p + 1).
Midpoint of a in one copy and b in another copy must be
in a different copy in between (otherwise a and b would lie in
the same copy)
So, Skolem's theorem implies there is a model of Th(N) that
looks like Q with each rational replaced by a copy of N
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