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There exist two regressive sets whose intersection is not regressive

Published online by Cambridge University Press:  12 March 2014

K. I. Appel
K. I. Appel*
Affiliation:
Institute for Defense Analyses and University of Illinois
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Extract

§1. In [2], as part of an analogy between the concepts of recursive emimerability and regressiveness, Dekker showed that the intersection of any two regressive sets which are recursively equivalent is a regressive set. In [1], McLaughlin and the author showed that the intersection of two regressive sets, if infinite, has an infinite regressive subset. However, the following theorem shows that the analogy is not complete in this instance.

Theorem. There exist two regressive sets whose intersection is not regressive.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 32 , Issue 3 , 09 October 1967 , pp. 322 - 324
Copyright
Copyright © Association for Symbolic Logic 1967

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References

[1]Appel, K. I. and McLaughlin, T. G., On properties of regressive sets, Transactions of the American Mathematical Society, vol. 115 (1965), pp. 83–92.CrossRefGoogle Scholar
[2]Dekker, J. C. E., Infinite series of isols, Proceedings of symposia in pure and applied mathematics, vol. 5, American Mathematical Society, Providence, R.I., 1962, pp. 77–96.CrossRefGoogle Scholar
[3]McLaughlin, T. G., Re traceable sets and recursive permutations, Proceedings of the American Mathematical Society, vol. 17 (1966), pp. 427–429.CrossRefGoogle Scholar