By Robert S. Boyer

ISBN-10: 0121229505

ISBN-13: 9780121229504

In contrast to such a lot texts on common sense and arithmetic, this ebook is ready tips to turn out theorems instead of evidence of particular effects. We provide our solutions to such questions as: - while should still induction be used? - How does one invent a suitable induction argument? - while should still a definition be multiplied?

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**Sample text**

The term s u b t e r m i is a variable. The proof in this case is im mediate. Case 2. T h e term s u b t e r m i has function symbol I F . T h e n s u b t e r m i is ( I F s u b t e r m a s u b t e r m b s u b t e r m c ) for some a , b , and c all less than i . Hence we have previously proved *2 (IMPLIES testsa *3 (IMPLIES testsb (EQUAL s u b t e r m a [ F l ' ] s u b t e r m a [ G l f ]) ) , (EQUAL s u b t e r m b [ F l ' ] s u b t e r m b [ G l ' ]) ) , and *4 (IMPLIES testsc (EQUAL s u b t e r m c [ F l ' ] s u b t e r m c [ G l ' ]) ) .

If we can show that G is partially cor rect a contradiction will arise because then G would be a subset of FO by the definition of FO . The domain of G is an RM-closed subset of Dn because it was formed by adding to an RM-closed subset of Dn an RMminimal element of Dn not in that subset. Let Gf be the extension of G. We need to show that for every n-tuple (XI, . . , Xn) in the domain of G that (GX1 . . Xn) = b o d y [ G ' ] . For every (XI, . . , Xn) in the domain of G, we may apply the lemma for G, G ' , FO, FO ' , and (XI, .

For example, if f de notes the function symbol G, and t denotes the term (ADDI Y) , then ( f t X) denotes the term (G (ADDI Y) X) . When we are speaking in naive set theory we use both upperand lowercase words as variables ranging over numbers, sets, func tions, etc. Context will make clear the range of these variables. " Whenever we add a new shell or function defini tion, we insist that certain function symbols not have been men tioned in any previous axiom. We call a function symbol new until an axiom mentioning the function symbol has been added.

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