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 .