45.12 - My Rendition of Hunter’s Proof of Metatheorem...

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Unformatted text preview: My Rendition of Hunter’s Proof of Metatheorem 45.12 Branden Fitelson 03/22/07 Theorem . Let K be a consistent first order theory. And, let K = K +{ α } , where α is an arbitrary (particular) well-formed formula of the following form (added as a new proper axiom to K to form K ): ( α ) Ac/v ⊃ ^ vA where (i) A is a well-formed formula of K , (ii) c does not occur in any proper axiom of K , (iii) c does not occur in A , and (iv) V vA is a closed formula of K (as a result, v is the only variable that possibly occurs free in A ). Then, K is also a consistent first order theory, and one which is an extension of K . Proof. First, we will show that K is a first order theory that extends K . Then, we will show that K is consistent. To see that K is a first order theory that extends K , note that any WFF of form α will be a closed formula. This is because (a) its consequent V vA is closed by hypothesis (iv) of the theorem, and (b) its antecedent Ac/v is closed (since, again by hypothesis (iv) of the theorem, v is the only variable that possibly occurs free in A , and we are replacing all of its free occurrences in A with a constant c ). And, any conditional with a closed antecedent and a closed consequent is itself closed. Thus, since K is just K plus a closed formula α , K is just K with one additional proper axiom. So, since K is a first order theory, K is a first order theory which extends K . That was the easy part. Now, we have to show that K is consistent ....
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This note was uploaded on 08/01/2008 for the course PHIL 140A taught by Professor Fitelson during the Spring '07 term at Berkeley.

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45.12 - My Rendition of Hunter’s Proof of Metatheorem...

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