1. COMPLETNESS
CATEGORICITY
COMPLETENESS
SEP
Erik A. Andrejko
University of Wisconsin - Madison
Summer 2007
ERIK A. ANDREJKO COMPLETENESS

2. COMPLETNESS
CATEGORICITY
GÖDEL
ERIK A. ANDREJKO COMPLETENESS

3. COMPLETNESS
CATEGORICITY
COMPLETENESS THEOREM
THEOREM (COMPLETENESS)
|= =⇒
THEOREM (SOUNDNESS)
=⇒ |=
COROLLARY
T is consistent if and only if T is satisfiable.
ERIK A. ANDREJKO COMPLETENESS

4. COMPLETNESS
CATEGORICITY
ELEMENTARY SUBMODELS
DEFINITION
Let M ⊆ N. Then M is an elementary submodel of N if for all a ∈ M
¯
we have
M |= ϕ(¯ ) ⇐⇒ N |= ϕ(¯ )
a a
for all L -formulas ϕ.
M N
FACT
Suppose that M ⊆ N. Let a ∈ M and suppose that ϕ(v ) is quantifier
¯ ¯
free. Then M |= ϕ(¯ ) if and only if N |= ϕ(¯ ).
a a
ERIK A. ANDREJKO COMPLETENESS

5. COMPLETNESS
CATEGORICITY
TARSKI VAUGHT TEST
THEOREM (TARKSI VAUGHT TEST)
Let M ⊆ N. Then M N if for all ϕ of the form
∃¯ ψ(¯ )
aa
with ψ quantifier free,
M |= ϕ(¯ ) ⇐⇒ N |= ϕ(¯ )
a a
ERIK A. ANDREJKO COMPLETENESS

6. COMPLETNESS
CATEGORICITY
LOWENHEIM SKOLEM TARSKI
THEOREM (DOWNWARD LOWENHEIM-SKOLEM-TARSKI)
Let M be an L -structure,
let κ be an infinite cardinal with |L | ≤ κ ≤ |M|,
1
let A ⊆ M be any set with |A| = κ.
2
Then there exists a N ⊆ M with
A ⊆ N,
1
|N| = κ,
2
N M.
3
ERIK A. ANDREJKO COMPLETENESS

7. COMPLETNESS
CATEGORICITY
LOWENHEIM SKOLEM TARSKI
THEOREM (UPWARD LOWENHEIM-SKOLEM-TARSKI)
Let M be an L -structure and let κ ≥ |L | + |M| be a cardinal. Then
there is an L -structure N with
|N| = κ
1
M N.
2
ERIK A. ANDREJKO COMPLETENESS

8. COMPLETNESS
CATEGORICITY
DEFINABILITY
DEFINITION
Let M be an L -structure. Let ϕ(¯ ) be a formula. Then
a
A = {x : M |= ϕ(¯ )}
¯ a
is definable in M.
ERIK A. ANDREJKO COMPLETENESS

9. COMPLETNESS
CATEGORICITY
CATEGORICITY
DEFINITION
Let κ be an infinite cardinal. Let Σ be an L -theory. Then Σ is
κ-categorical if for every M, N with |M| = |N| = κ
∼
M |= Σ and N |= Σ =⇒ M = N
ERIK A. ANDREJKO COMPLETENESS

10. COMPLETNESS
CATEGORICITY
COMPLETE THEORIES
THEOREM
Let Σ be an L -theory. Suppose for some κ ≥ |L | for all M, N of size
κ
M≡N
Then Σ is complete.
COROLLARY
Let Σ be κ-categorical for any κ ≥ |L |, then Σ is complete.
ERIK A. ANDREJKO COMPLETENESS