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Some theorems on R-maximal sets and major subsets of recursively enumerable sets

  • Manuel Lerman (a1)

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In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many хĀ, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

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[1]Feferman, S., Scott, D. S. and Tennenbaum, S., Models of arithmetic through function rings, 556–31, Notices of the American Mathematical Society, vol. 6 (1959), pp. 159160.
[2]Friedberg, R. M., Three theorems on recursive enumeration, this Journal, vol. 23 (1958), pp. 309316.
[3]Kleene, S. C., Introduction to metamathematics, D. van Nostrana, Inc., New York, 1952.
[4]Lachlan, A. H., The lattice of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 130 (1968), pp. 137.
[5]Lerman, M., Recursive functions modulo co-r-maximal sets, Transactions of the American Mathematical Society, vol. 148 (1970), pp. 429444.
[6]Lerman, M., Turing degrees and many-one degrees of maximal sets, this Journal, vol. 35 (1970), pp. 2940.
[7]Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295310.
[8]Robinson, R. W., Two theorems on hyperhypersimple sets, Transactions of the American Mathematical Society, vol. 128 (1967), pp. 531538.
[9]Sacks, G. E., Degrees of unsolvability, Annals of Mathematics Study number 55, Princeton 1963.

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

  • Manuel Lerman (a1)

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