Lecture 4 Notes

N 0 1 0 1 i i i pl 3014 metalogic 4 case 3 a

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Unformatted text preview: d T ∼C. So: T B and T C. (T is consistent.) Hence: I B and I ∼C. (Inductive Hypothesis.) Thus: I (B → C). So I A. this process until we exhaust the list of wfs. T is then the extension of S that includes as axioms all axioms of sequence members. Recall: These are terms with no variables: a , a , ..., b , b , ..., f ( a , b , ...), e tc . ∞ † n 0 1 0 1 i i i PL 3014 - Metalogic 4 Case 3: A has form (∀xi)B (xi), for B(xi) shorter than A. A. Suppose xi does not occur free in B. Then: B is closed (since A is closed). So: T B iff I B. (Inductive Hypothesis.) Note: T B iff T (∀xi)B (xi). (Proof: 1. "⇒": Gen on xi. 2. "⇐": Use (K4) and MP.) Note: I B iff I (∀xi)B (xi). (Prop. 3.27.) So: T (∀xi)B (xi) iff I (∀xi)B (xi). Thus T A iff I A. B. Suppose xi occurs free in B. Then: x i is the only free variable in B (since A is closed). So: B (xi) occurs in the sequence F0(xi0), F1(xi1), ..., say as Fm(xim). Then: A has form (∀xim)Fm(xim). 1. "⇐&quo...
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This document was uploaded on 03/25/2014 for the course PL 3014 at NYU Poly.

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