Model Theory

1
Model Theory
Max Euwe

• Jouko Väänänen

2
Isomorphism preserves truth

• fA ? B
• ??L??
• A ?a iff B ?fa
• Proof Exercise.

3
Definable sets and relations

• ?(A,b) a?A A ?a,b
• These are definable subsets of A.
• Definable relations are defined in the same way.
• Subset is X-definable if it is ?(A,b) with b from
  X.

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Few definable subsets

• L signature, A L-structure, X?A
• Y is X-definable
• f is an automorphism which fixes X pointwise
  (f(a)a for a ? X)
• Then f fixes Y setwise (a?Y iff f(a)?Y).
• Proof Isomorphism preserves truth! QED

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Example

• L?, X?A
• Y is X-definable
• Then Y ? X or A-Y ? X.

Y
X
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Concepts (page 30)

• Language L, first order language L??
• Sentence ?
• Theory T
• Model A truth A ?, A T model of ?, T
• Mod(T)A A T
• Axiomatizing a class, a theory, a structure
• Theory of a class ThL(K), of a structure ThL(A)
• L-definable class, first order definable

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Two worlds
Models
Sentences
Mod(T)
T
K
ThL(K)
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Examples of axioms

• Axioms of
• Term algebras
• Peano
• Groups
• Fields
• Lattices
• Boolean algebras
• Linear orderings, dense without endpoints

9
Some notions from logic

• ? is a logical consequence of T, T ? if every
  model of T is a model of ?
• Valid, theorem
• Consistent
• Equivalent theories
• ? equivalent to ? modulo T, logical equivalence
• Boolean combination, disjunctive normal form,
  prenex normal form

10
More concepts from logic

• n-type
• Structure realizing the type, omitting
• L-equivalent models, elementarily equivalent
• L-theory of a structure ThL(A)
• Complete theory
• Categorical theory, ?-categorical theory
• Decidable theory

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Hintikka sets

• A Hintikka set imitates a theory of the form
• T ? A ?
• A theory T is a Hintikka set if it satisfies

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• Atomic If ??T, then not ???T for atomic ?
• Identity tt?T if t is a closed term
• Identity If ts?T, then ?(s)?T if and only if
  ?(t)?T for atomic formulas ?(x)
• Negation If ????T, then ??T
• Conjunction If ???T, then ??T. If ????T, then
  ???T for some ???
• Disjunction If ???T, then ??T for some ???. If
  ????T, then ???T for all ???
• Universal quantifier If ?x? ?T, ? is ?(x), then
  ?(t)?T for all closed terms t. If ??x? ?T, then
  ??(t)?T for some closed term t
• Existential quantifier If ?x? ?T, ? is ?(x),
  then ?(t)?T for some closed term t. If ??x? ?T,
  then ??(t)?T for all closed terms t

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Theorem 2.3.3

• If T is a Hintikka set, then T has a model.
• Easy! The canonical model of the atomic part of T
  is a model of all of T.
• Let A be the canonical model
• CLAIM
• If ??T, then A ?
• If ???T, then A ??

We prove these two claims simultaneously.
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Proof
Induction on ?
? atomic
??
???
??
???
??
???
?x?
??x?
?x?
??x?
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Finding Hintikka sets (2.3.4)

• Let T be a first order theory
• Every finite subset of T has a model
• For every sentence ? either ??T or ???T
• For every ?x?(x)?T there is a closed term t such
  that ?(t)?T.
• Then T is a Hintikka set.
• Proof (?) T is closed under entailment If U?T
  finite and U ?, then ??T.

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() ?blue, (d) ?green, (c)(b)?red, (?)
green?orange
tt?T
If ts?T, then ?(s)?T if and only if ?(t)?T
If ????T, then ??T
If ???T, then ??T.
If ????T, then ???T for some ???
If ???T, then ??T for some ???.
If ????T, then ???T for all ???
If ?x? ?T, then ?(t)?T for all closed terms t
If ??x? ?T, then ??(t)?T for some closed term t
If ?x? ?T, then ?(t)?T for some closed term t
If ??x? ?T, then ? ?(t)?T for all closed terms t
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Application Compactness (page 124)

• If every finite subset of a first order T has a
  model, then T has a model.
• Countable signature L (later more general).
• c0,c1,c2, new constant symbols, witnesses
• L the larger signature
• ?0, ?1, ?2, all first order sentences in L
• T0 ?T1 ? T2 ?

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• T0T
• Tn1Tn??n if every finite subset of this has
  a model, o/w Tn1Tn
• Tn1Tn1 ??(ci), if ?n ? x?(x) is in Tn1
  and i is the least i such that ci does not occur
  in Tn1, o/w Tn1Tn1.
• Note Every finite subset of Tn1 has a model.
• Claim T?nTn is a Hintikka set.
• Every finite subset of T has a model
• For every sentence ? either ??T or ???T
• For every ?x?(x)?T there is a closed term t such
  that ?(t)?T.
• So T has a model. Hence T has a model.

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Application

• If T ?, then U ? for some finite U?T.
• Otherwise every finite subset of T??? has a
  model, hence T??? itself has a model,
  contradiction. QED

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Applications in fields

• If a first order sentence of field theory is true
  in fields of arbitrary high characteristic, it is
  true in a field of characteristic zero.
• The ordered real field (R,,.,0,1,lt) is
  elementarily equivalent to a non-Archimedean
  field.