Download course notes - Theory and Logic Group

The completeness theorem can also be stated in a more general form w.r.t. a theory T .
Theorem 1.3. If T
Proof. Suppose T
( A then T $ A.
& A, then ClpT Y t Auq is consistent for suppose T,
T r Asi
.
A $ K, then by
..
.
K raai
A
* A.
Theorem 1.4 (Compactness). Let Γ be a set of sentences. If every finite Γ0 „ Γ is satisfiable,
also T
$ A. So T Y t Au has a model M, i.e. M ( T
and M * A, so T
then Γ is satisfiable.
Proof. Suppose Γ is unsatisfiable, then ClpΓq … Γ is unsatisfiable and by Lemma 1.9, ClpΓq is
inconsistent, i.e. there is a proof π of Γ $ K. Letting Γ0 „ Γ be the finite set of assumptions
used in π, we have a proof of Γ0 $ K and therefore Γ0 is unsatisfiable.
The compactness theorem is a very useful tool for the construction of models as it allows to
“pass to the limit” after one has carried out a model construction on all finite subsets. For
example, we have seen in Section 1.2 that there is an infinite set of sentence Γ tLn | n ¥ 1u
in the empty language s.t. M ( Γ iff M is infinite. This set Γ is an infinite set of sentences –
a natural question is now whether it is possible to strenghten this characterisation by finding
a finite set having the same property. It turns out that such a set does not exist and the
compactness theorem enables us to proves this:
Corollary 1.1. There is no finite set of sentences Γ in the empty language s.t. M ( Γ iff M
is infinite.
™
Proof. Suppose that such a Γ exists and let I Γ. We have M ( I iff M is finite. Consider
∆ t I u Y tLn | n ¥ 1u. Let ∆0 be a finite subset of ∆, then ∆0 „ t I u Y tLn | 1 ¤ n ¤ mu
for some m and every structure of size m 1 is a model of ∆0 . So by the compactness theorem
∆ would have a model which is impossible as it could be neither finite nor infinite.
Another of the important limitative theorems about first-order logic is the following:
Theorem 1.5 (Löwenheim-Skolem). Every satisfiable set of sentences has a countable model.
Proof. Let Γ be a set of sentences. First, if Γ is satisfiable then ClpΓq is consistent, for suppose
ClpΓq would not be consistent, then there would be a proof of Γ0 Ñ K for Γ0 „ Γ, so Γ0 would
not be satisfiable and hence also Γ. Then, by Lemma 1.9 we obtain a countable model M of
ClpΓq hence of Γ.
The above-mentioned set tLn | n ¥ 1u has exactly the infinite structures as models. Another
natural generalisation of this characterisation would be a set of sentences having only uncountable models. However, the Löwenheim-Skolem theorem shows that such a set does not exist.
Corollary 1.2. There is no set of sentences having only uncountable models.
Theorem 1.6 (Upward Löwenheim-Skolem). Every set of sentences that has an infinite model
has an uncountable model.
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Without Proof.
Short Digression: Set theory and the Skolem paradox
A motivation for the above Skolem-Löwenheim theorem is the following argument, known as
Skolem paradox: Consider an axiom system for set theory such as ZFC. Such axiom systems
work in a simple language which contains P as only predicate symbol and every element of the
domain is a set. ZFC is a very strong theory and allows the formalisation of (most) mathematical
proofs, in particular, let F be a sentence of the form Dϕpϕ : N Ñ R ^ ϕ bijectionq which
expresses the fact that R is uncountable, then ZFC $ F .
Now, by the Löwenheim-Skolem theorem ZFC has a countable model, say M and therefore also
M ( F , and so M * Dϕpϕ : N Ñ R ^ ϕ bijectionq. However, by countability of M there are
only countable many real numbers in M (real numbers are represented as certain kinds of sets),
so there exists a bijection f between N and the reals in M.
The solution to this paradoxical situation is the insight that while this f does indeed exist it
does not exist in M, i.e. there is no set in the domain of M which represents f .
End of Digression
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