FOM: Query on first-order logic

[Joe Shipman]

Given: a sentence S of first-order logic (may include function symbols
and relation symbols of arbitrary arity, constants, and the equality
symbol).

Consider the set T(S) = {complete, consistent theories in the language
of S which include S}.  |T(S)|=0 iff S is inconsistent; |T(S)|=1 iff S
axiomatizes a complete theory.

Is it provable that the only possible cardinalities for T are
{0,1,2,...,aleph_0, 2^aleph_0}?

Is there a nice characterization of the S for which T(S) is finite?

-- Joe Shipman