By Robert S. Boyer, J. Strother Moore

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D. 5 Shells Thus far the theory is somewhat impoverished in that it does not have any “interesting” objects. It would be convenient, for example, if we could refer to the natural numbers 0, 1, 2, . . and ordered pairs from within our theory (as we have several times in discussions of our theory). We could invent appropriate axioms for each individual “type” of object. However, we want to ensure that no natural number is T, F, or an ordered pair. 1 Because of considerations such as these, we address the general problem of extending the theory by adding a new “type” of object.

For example, we want (NOT P) to be T if P is F and to be F if P is not F. Similarly, we want (AND P Q) to be T if both P and Q are non-F, and F otherwise. Thus, we define the functions NOT, AND, OR, and IMPLIES as follows: Definition (NOT P) = (IF P F T) Definition (AND P Q) = (IF P (IF Q T F) F) Definition (OR P Q) = (IF P T (IF Q T F)) Definition (IMPLIES P Q) = (IF P (IF Q T F) T). (We adopt the notational convention of treating AND and OR as though they took an arbitrary number of arguments.

Vz ) = (R (M U1 . . Uz) (M V1 . . Vz)). Note that RM is well-founded. If p is not a theorem there must exist a z-tuple X1, . . , Xz such that (P X1 . . Xz)=F. Let X1, . . , Xz be an RM-minimal such z-tuple. We now consider the cases on which, if any, of the qi are true on the chosen z-tuple. , suppose (Q1 X1 . . Xz)=F, (Q2 X1 . . Xz)=F, . . , and (Qk X1 . . Xz)=F. Then by the base case (P X1 . . Xz)=F, contradicting the assumption that (P X1 . . Xz)=F. Case 2: Suppose that at least one of the qi is true.