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Unformatted text preview: Gödel’s Metatheorem (45.17) and the Strong Completeness Theorem for FOTs (46.2) Branden Fitelson 04/12/07 Before getting to the salient proofs, it’s important to understand Hunter’s terminology “consistent set Γ of WFFs of a first order theory K ”. For Hunter, Γ is a consistent set of WFFs of K iff there is no WFF A of K such that Γ ‘ K A and Γ ‘ K ∼ A . As a result, this definition of “consistent set of WFFs Γ of K ” implies that K is itself a consistent first-order theory! That is, an inconsistent first order theory K does not have any consistent sets of WFFs on this definition. This sounds a bit odd, but it’s crucial for the proofs below. In this handout, I will go through the proper proofs of 45.17 and 46.2. To this end, I will begin with the background ingredients of the proof of 45.17: metatheorem 45.16, and Lemmas 1 and 2. 45.16 . If Γ is a consistent set of closed WFFs of a first order theory K , then Γ has a denumerable model. Proof. Assume Γ is a consistent set of closed WFFs of a first order theory K . Then, by Hunter’s definition (above), there is no WFF A of K such that Γ ‘ K A and Γ ‘ K ∼ A . Therefore, it follows that the first order theory K + Γ is a consistent first order theory. If K + Γ were inconsistent, then there would have to be a WFF A of K + Γ such that both A and ∼ A were theorems of K + Γ . That would imply the existence of an A such that Γ ‘ K A and Γ ‘ K ∼ A , which contradicts Hunter’s definition of “consistent set of WFFs Γ of K .” Since K + Γ is a consistent first order theory, it must have a denumerable model [this is implied by theorems 45.10–45.14]. Thus, Γ itself has a denumerable model ( Γ is a subset of the set of theorems of K + Γ ). Lemma 1 for 45.17 . If Γ is a consistent set of WFFs of a first order theory K , then Γ is also a consistent set of WFFs of of the first order theory K , where K is the first order theory one gets when one adds denumerably many new constant symbols with an effective enumeration h c 1 ,...c n ,... i to K . Proof. Assume Γ is a consistent set of WFFs of a first order theory K , and assume that K is K with the new constant symbols h c 1 ,...c n ,... i added to it. Now, assume, for reductio , that Γ is an inconsistent set of K . Then, by definition, this means that there is a WFF B of K such that Γ ‘ K B and Γ ‘ K ∼ B . So, since derivations are finite, there is a finite subset Δ ⊆ Γ such that Δ ‘ K B and Δ ‘ K ∼ B . These derivations in K can be converted into derivations in K , as follows. Let X = Xv i /c i , where v i is the i th variable in our enumeration that does not occur in either of the derivations Δ ‘ K B or Δ ‘ K ∼ B , and c i is the i th constant symbol in our enumeration of new symbols added to K to yield K . Then, Δ = Δ , since Δ ⊆ Γ and Γ is a set of WFFs of K (and so do not contain any c i ’s). Moreover, Δ ‘ K B and Δ ‘ K ∼ B . Why? Think about Δ ‘...
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- Spring '07
- Logic, Mathematical logic, Model theory, First-order logic, Metalogic, Proof theory