Model Theory

1
Model Theory
Max Euwe

• Jouko Väänänen

2
Classifying formulas

• Quantifier-free, ?0 , ?0
• Positive, no negation
• Universal, i.e. ?1 quantifier-free, ?, ?, ?
• Existential, i.e. ?1 quantifier-free, ?, ?, ?
• Universal-existential ?2 ?x?, where ? is ?1
• fA?B preserves ? if for all a A ?(a) ?
• B ?(fa)

3
Embeddings preserve existential formulas

• Let fA?B be an embedding and let ? be
  existential.
• Claim A ?(a) ? B ?(fa)
• Proof
• First A ?(a) ? B ?(fa) for
  quantifier-free ?
• Then A ?(a) ? B ?(fa) for existential ?
• Application There are at most 10 elements
  cannot be written with an existential sentence.

4
Universal formulas are preserved by substructures

• Follows from the previous!
• Application There are at least 10 elements
  cannot be written with a universal sentence.

5
Chains and their unions

• (Ai ilt?) chain
• Ai?Aj for iltjlt?
• Union of a chain B ?ilt? Ai
• dom(B) is ?ilt?dom(Ai)
• Constants as in A0
• Relations union
• Functions union
• Note Ai ? B for all ilt?

6
Preservation by unions of chains

• A formula ?(x) is preserved by unions of chains
  if for all unions of chains ?ilt?Ai and all a in
  A0
• Ai ?(a) for all ilt
• implies
• ?ilt?Ai ?(a).

7
Universal-existential formulas are preserved by
unions of chains

• Suppose B ?ilt?Ai
• Suppose Ai ?x ?(x,a) for all i, where ?(x,a)
  is existential.
• Claim B ?x ?(x,a)
• Application Has a last element cannot be
  written with a universal-existential sentence.

8
Classifying maps

• A homomorphism that preserves all first order
  formulas is an elementary embedding.
• Elementary substructure A B
• Elementary extension
• Tarski-Vaught Criterion A is an elementary
  substructure of B iff for every formula ?(x,y)
  and all a from A, if B ?y ?(a,y), then
• B ?(a,b) for some b in A.

9
Tarski-Vaught Criterion
B
?y?(a,y)
A
b
a
b
?(a,b)