The enumerability and invariance of complexity classes*

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Several properties of complexity classes and sets associated with them are studied. An open problem, the enumerability of complexity classes, is settled by exhibition of a measure with some nonenumerable classes. Classes for natural measures are found to occupy the same isomorphism type; and a criterion for measures comes from this finding. General results about measures and unsolvability are presented and constraints are placed on complexity classes so that they possess identical properties.

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Much of this work was done while the author was at Cornell University. Some of these results were first announced at the Second Annual Symposium on the Theory of Computing in May 1970 at Northampton, Mass. An expanded version appears in the author's thesis [10]. This research has been supported in part by National Science Foundation grants GJ-155 and GJ-579.