Lecture 4 Notes

# Lecture 4 Notes - PL 3014 Metalogic Proposition 4.40 1...

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PL 3014 - Metalogic 1 Proposition 4.40 Preliminaries Suppose we enlarge L by adding new constants b 0 , b 1 , .. to form L + . Let S be an extension of K . Now construct an extension S + of S by including as axioms all axioms of S and all instances of S -axioms that contain any of the new constants b 0 , b 1 , ... . Example : Axiom (K5) ( x i ) A ( x i ) A ( t ), where t is a term free for x i in A ( x i ), is an axiom of S + , as is the particular instance ( x 1 ) A 1 1 ( x 1 ) A 1 1 ( b 1 ). L emma 1 : If S is consistent, so is S + . Proof : Suppose S is consistent and S + is not. Then : There's a wf B such that S + B and S + ( B ). Note : These S + -proofs can be converted into S -proofs. Just replace all occurrences of b -constants with a - constants that do not occur in the S + -proofs. (There will always be such a -constants available since there is a countable infinity of them, and there can only be a finite number of wf s, and hence occurrences of b -constants, in any S + -proof.) Result : S B and S ( B ). But S was assumed consistent. Hence S + must also 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 b 0 , b 1 , ... . 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 .

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PL 3014 - Metalogic 2 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 S 0 , S 1 , ..., of S + . This sequence is constructed in 4 steps: 1. List all wf s of L + that contain one free variable: F 0 ( x i 0 ), F 1 ( x i 1 ), F 2 ( x i 2 ), ... 2. Choose a subset { c 0 , c 1 , ... } of the b -constants that are free for the x i 0 , x i 1 , ... in the list. Require: (i) c 0 doesn't appear in F 0 ( x i 0 ). (ii) For n > 0, c n { c 0 , ..., c n 1 } and c n doesn't appear in F 0 ( x i 0 ), ..., F n ( x in ). 3. Let G k be the wf ( x ik ) F k ( x ik ) F k ( c k ). 4. Construct the sequence S 0 , S 1 , ... as follows: (i) Let S 0 = S + .
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