2Model theory

III Logic



2.1 Universal theories
Recall the following definition:
Definition
(Universal theory)
.
A universal theory is a theory that can be
axiomatized in a way such that all axioms are of the form
···
(stuff not involving quantifiers) ()
For example, groups, rings, modules etc. are universal theories. It is easy to
see that if we have a model of a universal theory, then any substructure is also a
model.
It turns out the converse is also true! If a theory is such that every sub-
structure of a model is a model, then it is universal. This is a very nice result,
because it gives us a correspondence between syntax and semantics.
The proof isn’t very hard, and this is the first result we will prove about
model theory. We begin with some convenient definitions.
Definition
(Diagram)
.
Let
L
be a language and
M
a structure of this language.
The diagram of
M
is the theory obtained by adding a constant symbol
a
x
for
each
x M
, and then taking the axioms to be all quantifier-free sentences that
are true in M. We will write the diagram as D(M).
Lemma.
Let
T
be a consistent theory, and let
T
be the set of all universal
consequences of
T
, i.e. all things provable from
T
that are of the form (
). Let
M be a model of T
. Then T D(M) is also consistent.
Proof.
Suppose
T D
(
M
) is not consistent. Then there is an inconsistency
that can be derived from finitely many of the new axioms. Call this finite
conjunction
ψ
. Then we have a proof of
¬ψ
from
T
. But
T
knows nothing
about the constants we added to
T
. So we know
T `
x
¬ψ
. This is a universal
consequence of T that M does not satisfy, and this is a contradiction.
Theorem.
A theory
T
is universal if and only if every substructure of a model
of T is a model of T .
Proof.
is easy. For
, suppose
T
is a theory such that every substructure of
a model of T is still a model of T .
Let
M
be an arbitrary model of
T
. Then
T D
(
M
) is consistent. So it
must have a model, say
M
, and this is in particular a model of
T
. Moreover,
M is a submodel of M
. So M is a model of T .
So any model of
T
is also a model of
T
, and the converse is clearly true. So
we know T
is equivalent to T .