By Robert S. Boyer

ISBN-10: 0121229505

ISBN-13: 9780121229504

Not like so much texts on good judgment and arithmetic, this booklet is set how one can end up theorems instead of facts of particular effects. We supply our solutions to such questions as: - while may still induction be used? - How does one invent a suitable induction argument? - while may still a definition be multiplied?

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**Example text**

D. That concludes the proof that the definition principle is sound. No constructivist would be pleased by the foregoing justification of recur- J. LEXICOGRAPHIC RELATIONS / 51 sive definition because of its freewheeling, set-theoretic character. The truth is that induction and inductive definition are more basic than the truths of high-powered set theory, and it is slightly odd to jus tify a fundamental concept such as inductive definition with set the ory. We have presented this proof only to provide the careful reader with some clear talk about our definition principle.

Case XI . case Xz ) 1. , suppose ( QI XI . . Xz) = F, ( Q2 . Xz ) = F, . . , and ( Qk XI . . Xz ) = F . Then by the base ( P X 1 . . Xz) ^ F , contradicting the assumption that ( P XI . . = F. Case 2. Suppose that at least one of the q^ is true. Without loss of gen erality we can assume that ( QI XI . . Xz) ^ F . By condition (g) above we have (R (M d lflel . . d l a , n ) (R (M d 1 A 1 . . d 1 A n ) (M XI . . (M XI . . Xn)), Xn)), and (R (M du,,,, . . d 1)h „ n ) (M XI . . Xn)). Thus, by the definition of RM, we have (RM

FLATTEN, FLATTEN and APPEND, to: (TRUE). Case 2. FLATTEN X ANS) (APPEND (FLATTEN X) ANS))). FLATTEN (CDR X) ANS)) (APPEND (APPEND (FLATTEN (CAR X)) (FLATTEN (CDR X))) ANS))). Appealing to the lemma CAR/CDR. ELIM, we now replace X 24 / II. A SKETCH OF THE THEORY AND TWO SIMPLE EXAMPLES by (CONS Z V) to eliminate (CAR X) and (CDR X). FLATTEN V ANS)) (APPEND (APPEND (FLATTEN Z) (FLATTEN V)) ANS))). FLATTEN V ANS)) and throwing away the equality. FLATTEN V ANS)) (APPEND (APPEND (FLATTEN Z) (FLATTEN V)) ANS))).

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