completeness

COMPLETNESSCATEGORICITY

COMPLETENESSSEP

Erik A. Andrejko

University of Wisconsin - Madison

Summer 2007

ERIK A. ANDREJKO COMPLETENESS


GÖDEL



COMPLETENESS THEOREM

THEOREM (COMPLETENESS)

|= =⇒ `

THEOREM (SOUNDNESS)

`=⇒ |=

COROLLARY

T is consistent if and only if T is satisfiable.



ELEMENTARY SUBMODELS

DEFINITION

Let M⊆N. Then M is an elementary submodel of N if for all a ∈Mwe have

M |= ϕ(a) ⇐⇒ N |= ϕ(a)

for all L -formulas ϕ .

M≺N

FACT

Suppose that M⊆N. Let a ∈M and suppose that ϕ(v) is quantifierfree. Then M |= ϕ(a) if and only if N |= ϕ(a).



TARSKI VAUGHT TEST

THEOREM (TARKSI VAUGHT TEST)Let M⊆N. Then M≺N if for all ϕ of the form

∃aψ(a)

with ψ quantifier free,

M |= ϕ(a) ⇐⇒ N |= ϕ(a)



LOWENHEIM SKOLEM TARSKI

THEOREM (DOWNWARD LOWENHEIM-SKOLEM-TARSKI)Let M be an L -structure,

1 let κ be an infinite cardinal with |L |≤ κ ≤ |M |,2 let A⊆M be any set with |A| = κ .

Then there exists a N ⊆M with1 A⊆ N,2 |N | = κ ,3 N ≺M.



LOWENHEIM SKOLEM TARSKI

THEOREM (UPWARD LOWENHEIM-SKOLEM-TARSKI)

Let M be an L -structure and let κ ≥ |L |+ |M| be a cardinal. Thenthere is an L -structure N with

1 |N| = κ

2 M≺N.



DEFINABILITY

DEFINITION

Let M be an L -structure. Let ϕ(a) be a formula. Then

A = {x : M |= ϕ(a)}

is definable in M.



CATEGORICITY

DEFINITION

Let κ be an infinite cardinal. Let Σ be an L -theory. Then Σ isκ-categorical if for every M,N with |M| = |N| = κ

M |= Σ and N |= Σ =⇒ M ∼= N



COMPLETE THEORIES

THEOREM

Let Σ be an L -theory. Suppose for some κ ≥ |L | for all M,N of sizeκ

M≡N

Then Σ is complete.

COROLLARY

Let Σ be κ-categorical for any κ ≥ |L |, then Σ is complete.


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