On the divergence of extension procedures in isol theory

McLaughlin, T. G.

doi:10.1090/S0002-9939-1981-0630052-3

On the divergence of extension procedures in isol theory
HTML articles powered by AMS MathViewer

by T. G. McLaughlin

Proc. Amer. Math. Soc. 83 (1981), 769-773

DOI: https://doi.org/10.1090/S0002-9939-1981-0630052-3

PDF | Request permission

Abstract:

We show that the Myhill and Nerode extensions begin to disagree on the domain of the Nerode extension at a point in the arithmetical hierarchy $\leqslant \Delta _3^0$. This disagreement, at level $\Delta _3^0$, goes hand in hand with a certain way in which the Myhill extension fails, at $\Delta _3^0$, to commute with composition.

References

J. Myhill, Recursive equivalence types and combinatorial functions, Bull. Amer. Math. Soc. 64 (1958), 373–376. MR 101194, DOI 10.1090/S0002-9904-1958-10241-4
Anil Nerode, Extensions to isols, Ann. of Math. (2) 73 (1961), 362–403. MR 131363, DOI 10.2307/1970338
Erik Ellentuck, The positive properties of isolic integers, J. Symbolic Logic 37 (1972), 114–132. MR 392536, DOI 10.2307/2272554
Erik Ellentuck, On the degrees of universal regressive isols, Math. Scand. 32 (1973), 145–164 (1974). MR 332462, DOI 10.7146/math.scand.a-11451
Stephen Cole Kleene, Introduction to metamathematics, D. Van Nostrand Co., Inc., New York, N. Y., 1952. MR 0051790
J. C. E. Dekker, Infinite series of isols, Proc. Sympos. Pure Math., Vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 77–96. MR 0142447