Lecture 4 Notes

Prop 440 let s be a consistent extension of k then

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Unformatted text preview: o be consistent. Prop. 4.40. Let S be a consistent extension of K. Then there is an interpretation of L in which every theorem of S is true. Outline of Proof: I. Enlarge L to L+ by adding new constants b0, b1, ... . Extend S to S+ as above. Construct a particular consistent extension S∞ of S+ . Then, by Prop. 4.39, there must be a complete consistent extension of S ∞, call it T. II. Use T to construct an interpretation I of L+ . Prove that for every closed wf A of L+ , T A iff I A. III. Show that for any (open or closed) wf B of L, if S B, then I B. PL 3014 - Metalogic Part I. Let S be a consistent extension of K. S∞ will be the extension of S+ that has as its axioms the union of the sets of axioms of a particular sequence of extensions S0, S1, ..., of S+ . This sequence is constructed in 4 steps: 1. 2. 3. 4. 2 List all wfs of L+ that contain one free variable: F0(xi0), F1(xi1), F2(xi2), ... Choose a subset {c0, c1, ... } of the b-constants that are free for the xi0, xi1, ... in the list. Require: (i) c 0 doesn't a...
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