R. Gurevic. Equational theory of positive numbers with exponentiation. American Mathmatical Society, 94(1):135--141, May 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... sum, product, exponentiation, and the constant 1, however, the answer is negative, witness an equation due to Wilkie that holds true in N but that is not provable with the usual arithmetic identities [46] Furthermore, Gurevic has shown that in that case, equalities are not finitely axiomatizable [30]. To this end, he exhibited an infinite number of equalities in N such that for every finite set of axioms, one of them can be shown not to follow. 2.2 Tarski s high school algebra problem, type theoretically If one replaces sums, product, and exponentiation respectively by the sum, product, and ....

R. Gurevic. Equational theory of positive numbers with exponentiation. Proceedings of the American Mathematical Society, 94(1):135--141, May 1985.

....not provable in E . Gurevic later gave an argument by an ad hoc countermodel [18] and, more importantly, showed that there is no finite axiomatisation for the valid equations in the standard model hN; 1; i of positive natural numbers with one, multiplication, exponentiation, and addition [19]. He did this by producing an infinite family of equations such that for every sound finite set of axioms one of the equations can be shown not to follow. Gurevic s identities are the following x (n 3 odd) 3) A = 1 x Bn = 1 x x Cn = 1 x ....

....advance in the study of type isomorphisms in the presence of empty and sum types. Many questions still remain open, as for instance whether there are arithmetic equations in the language of 1, 0 or of 1, that do not correspond to type isomorphisms. We conjecture that Gurevic s result [19] for establishing the non finite axiomatisability of the equational theory of the model of positive natural numbers hN; 1; i can be generalised to the case of the model of natural numbers hN 0 ; 1; 0; i, and hence, by the results of this paper, that the equational theory of type ....

R. Gurevic. Equational theory of positive numbers with exponentiation is not finitely axiomatizable. Annals of Pure and Applied Logic, 49:1--30, 1990.

....was the first to establish Tarski s conjecture in the negative. Indeed, by a proof theoretic analysis, he showed that the identity A = 1 x ; B = 1 x x C = 1 x 3 ; D = 1 x 4 is not provable in E . Gurevic later gave an argument by an ad hoc countermodel [18] and, more importantly, showed that there is no finite axiomatisation for the valid equations in the standard model hN; 1; i of positive natural numbers with one, multiplication, exponentiation, and addition [19] He did this by producing an infinite family of equations such that for every ....

R. Gurevic. Equational theory of positive numbers with exponentiation. Proceedings of the American Mathematical Society, 94(1):135--141, 1985.

....then reasonable to 1 If addition is also included, the question whether the usual axioms are equationally complete for arithmetic is known as Tarski s high school algebra problem. A. J. Wilkie found a counter example [Wil81] and R. Gurevic showed recently that no finite axiomatization suffices [Gur90]. interpret a query A as asking for all identifiers e in the library for which e : A holds. These are, in fact, the identifiers whose most general types are at least as general as A. One can compare the situation with a search by specification, which would be ideal but cannot be automated. If ....

R. Gurevic. Equational theory of positive numbers with exponentiation is not finitely axiomatizable. Ann. of Pure and Appl. Logic, 49(1):1--30, 1990.

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R. Gurevic. Equational theory of positive numbers with exponentiation. American Mathmatical Society, 94(1):135--141, May 1985.

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