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THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

Published online by Cambridge University Press:  01 August 2018

ERIC P. ASTOR
ERIC P. ASTOR*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CONNECTICUT 341 MANSFIELD RD U-1009 STORRS, CT 06269-1009, USAE-mail:eric.astor@uconn.eduURL: http://www.math.uconn.edu/∼astor
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Abstract

In a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.

We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

Keywords

03D28intrinsic densityTuring degreesreverse mathematics

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 817 - 828
Copyright
Copyright © The Association for Symbolic Logic 2018 

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