Maximal Eventually Different Families of Computable Functions

Abstract

Cardinal characteristics of the continuum are the cardinalities of interesting families of reals. A well-studied example is that of maximal almost disjoint (MAD) families of sets of natural numbers. Significant work has been done investigating computability-theoretic analogues of cardinal characteristics. By considering encodings of MAD families as a single `universal' set, Lempp, Miller, Nies, and Soskova (2023) studied the class of encoded MAD families. Such a class is referred to as a mass problem; one can study the relative complexity between mass problems. In Section 2, we build on the study of mass problems as analogues of cardinal characteristics. We define mass problems of maximal eventually different (MED) families of computable functions in the same way and compare them against the mass problems defined by Lempp et al. In Section 3, we survey work by Greenberg, Kuyper, and Turetsky (2019) that provides an abstract framework for cardinal characteristics and their effective counterparts. We show that this framework is suitable for obtaining results in the setting of mass problems. In Section 4, we showcase a construction by Schrittesser (2018) of an effectively closed MED family in set theory. We show that the construction is sufficiently effective that the computable members of the constructed family are MED relative to computable functions.