Vol. 43, No. 3, 1972

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The decidability of a class of AE sentence in the isols

Erik Maurice Ellentuck and Alfred Berry Manaster

Vol. 43 (1972), No. 3, 573–584

The isols are a recursive analogue of the Dedekind finite cardinals originally developed by Dekker. In this paper a metatheorem is proved which shows that for certain sentences 𝒜 about addition in which no existential quantifier precedes a universal quantifier, the truth of 𝒜 in the natural numbers is sufficient to ensure the truth of 𝒜 in the isols. A more general class of sentences is also considered and it is seen that the applicability of the metatheorem is also necessary for the truth of any sentence in this class. It follows that there exists a decision procedure for that class of sentences. Extensions of these results to the case of the cosimple isols are also considered.

Mathematical Subject Classification
Primary: 02F40
Received: 21 June 1971
Revised: 11 April 1972
Published: 1 December 1972
Erik Maurice Ellentuck
Alfred Berry Manaster