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Unformatted text preview: Part (d) of Hunter’s Proof of Henkin’s Completeness Theorem for PS Branden Fitelson 02/21/07 The Lindenbaum Construction. We assume that we have an enumeration h A 1 ,A 2 ,...A n ,... i of all the formulas A i of P . [This is part (c) of Hunter’s proof. He does a pretty good job explaining parts (b) and (c), so I won’t rehearse those parts here.] Now, let Γ be an arbitrary pconsistent set of formulas. Using our enumeration of P formulas, we construct (Lindenbaumstyle) an infinite sequence of sets of formulas Γ =h Γ , Γ 1 ,... Γ n ,... i — using Γ as the starting point of our construction — in the following way: Γ Γ , and for n ≥ 1, Γ n Γ n 1 ∪{ A n } if this set is pconsistent. Γ n 1 otherwise. This sequence Γ has several important properties, which I will now prove by induction. ① For all n ≥ 0, Γ n ∈ Γ is pconsistent. Basis Step . Γ = Γ is pconsistent, by assumption. Inductive Step . Assume (IH) that Γ i is pconsistent, for all i such that 0 ≤ i < n . And, use this to prove that Γ n is pconsistent. So, the (IH) tells us that Γ n 1 is pconsistent. But, Γ n = Γ n 1 ∪{ A n } if this set is pconsistent. Γ n 1 otherwise. So, either ( i ) Γ n = Γ n 1 ∪{ A n } , which is pconsistent by construction, or ( ii ) Γ n = Γ n 1 , which is pconsistent by the inductive hypothesis (IH). Either way, Γ n is pconsistent. ② For all n ≥ 1, Γ n Γ ∪ Γ 1 ∪ Γ 2 ∪···∪ Γ n = Γ n . Basis Step . By the construction of Γ and the definition of Γ n , we have: Γ 1 = Γ ∪ Γ 1 = Γ ∪ Γ ∪{ A 1 } if this set is pconsistent....
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 Spring '07
 FITELSON
 Set Theory, Natural number, Mathematical logic, n1 n1, Lindenbaum Construction

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