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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
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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.
 Spring '06
 JonathanBain

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