Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The applications of gödel's completeness theorem in the context of compactness and isomorphism of dense linear orderings. The compactness theorem for predicate calculus, the lowenheim-skolem theorem, and the isomorphism of two countable models of dense linear orderings. It also includes examples and counterexamples of dense linear orderings.
Typology: Study notes
1 / 9
This page cannot be seen from the preview
Don't miss anything!
14.1 Compactness Theorem for Predicate Calculus Let L be a first-order language and let Γ ⊆ Sent(L).
Then Γ has a model iff every finite subset of Γ has a model.
Proof: as for Propositional Calculus – Exercise sheet ♯ 4, (5)(ii).
14.2 Example Let Γ ⊆ Sent(L). Assume that for every N ≥ 1 , Γ has a model whose domain has at least N elements.
Then Γ has a model with an infinite domain.
Proof:
For each n ≥ 2 let χn be the sentence
∃x 1 ∃x 2 · · · ∃xn
∧ 1 ≤i<j≤n
¬xi =. xj
⇒ for any L-structure A =< A;... >,
A |= χn iff ♯A ≥ n
Let Γ′^ := Γ ∪ {χn | n ≥ 1 }.
If Γ 0 ⊆ Γ′^ is finite, let N be maximal with χN ∈ Γ 0. By hypothesis, Γ ∪ {χN } has a model. ⇒ Γ 0 has a model (note that ⊢ χN → χN − 1 → χN − 2 →.. .)
⇒ By the Compactness Theorem 14.1, Γ′^ has a model, say A =< A;... >
⇒ A |= χn for all n ⇒ ♯A = ∞ 2
14.5 Remark Let Γ ⊆Sent(L) be any set of L-sentences. Then TFAE:
(i) Γ is strongly maximal consistent (i.e. for each L-sentence φ, φ ∈ Γ of ¬φ ∈ Γ)
(ii) Γ |=Th(A) for some L-structure A
(iii) Γ has models, and, for any two models A and B, Th(A) =Th(B).
Proof: (i) ⇒ (ii) + (iii): Completeness Theorem Rest: clear. 2
A worked example: Dense linear orderings without endpoints
Let L = {<} be the language with just one binary predicate symbol ‘<’, and let Γ be the L-theory of dense linear or- derings without endpoints (cf. Example 10.8) consisting of the axioms ψ 1 ,... , ψ 4 :
ψ 1 : ∀x∀y((x < y ∨ x =. y ∨ y < x) ∧¬((x < y ∧ x =. y) ∨ (x < y ∧ y < x))) ψ 2 : ∀x∀y∀z(x < y ∧ y < z) → x < z) ψ 3 : ∀x∀z(x < z → ∃y(x < y ∧ y < z)) ψ 4 : ∀y∃x∃z(x < y ∧ y < z)
14.6 (a) Examples
(a, b) < (c, d) ⇔ a < c or (a = c & b < d)
Let φ(an+1) = bm, where m > 1 is minimal s.t.
for all i ≤ n : bm <B φ(ai) ⇔ an+1 <A ai,
i.e. the position of φ(an+1)
relative to φ(a 1 ),... , φ(an)
is the same as that of an+
relative to a 1 ,... , an
(possible as A, B |= Γ).
⇒ (⋆n+1) holds for a 1 ,... , an+
⇒ φ is injective
And φ is surjective, by minimality of m. 2
14.8 Corollary Γ is maximal consistent
Proof: to show: Th(A) =Th(B) for any A, B |= Γ (by Remark 14.5)
By the Theorem of L¨owenheim-Skolem (14.3), Th(A) and Th(B) have countable models, say A 0 and B 0.
⇒ Th(A 0 ) =Th(A) and Th(B 0 ) =Th(B)
Theorem 14.7 ⇒ A 0 and B 0 are isomorphic
⇒ Th(A 0 ) =Th(B 0 )
⇒ Th(A) =Th(B) 2
