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Cardinal Characteristics and Computability

Logan McDonald

TL;DR

This work connects set-theoretic cardinal characteristics of the continuum with computability theory by framing them as mass problems in the Muchnik/Medvedev lattices and, via Weihrauch problems, analyzes how dominating functions, independent families, and ultrafilter bases correspond to highness and related notions. A core result is the Medvedev equivalence between dominating functions and maximal independent families, and this equivalence extends to maximal ideal independent families and to the setting of ω-computably approximable sets. The Gamma question is integrated through IOE/AED mass problems, showing that Gamma-values below 1/2 collapse to 0, with concrete mass-problem representations D(p) and B(p). The paper further demonstrates that combinatorial and ideal-structure mass problems collapse to dominating-function analogues across several Boolean algebras, yielding a unified perspective on the computability content of continuum characteristics and suggesting broad avenues for future explorations in higher-cardinal or alternative-algebra settings.

Abstract

Cardinal characteristics of the continuum represent the boundaries in size between the countable and the continuum with respect to certain properties of sets. They are often defined as the minimum sizes of families of reals that meet some criteria. Taking these families and considering their analogues in the setting of computability theory provides a rich hierarchy of properties of oracles, which can be studied in terms of the Muchnik/Medvedev lattices of mass problems. We provide more detail to the proof of the Medvedev equivalence between dominating functions and maximal independent families given by Lempp et al. (2023) and adapt their construction of maximal almost disjoint families to the setting of $ω$-computably approximable sets. We then extend the theory to include correspondents of maximal ideal independent families and show they behave similarly to the maximal independent families.

Cardinal Characteristics and Computability

TL;DR

This work connects set-theoretic cardinal characteristics of the continuum with computability theory by framing them as mass problems in the Muchnik/Medvedev lattices and, via Weihrauch problems, analyzes how dominating functions, independent families, and ultrafilter bases correspond to highness and related notions. A core result is the Medvedev equivalence between dominating functions and maximal independent families, and this equivalence extends to maximal ideal independent families and to the setting of ω-computably approximable sets. The Gamma question is integrated through IOE/AED mass problems, showing that Gamma-values below 1/2 collapse to 0, with concrete mass-problem representations D(p) and B(p). The paper further demonstrates that combinatorial and ideal-structure mass problems collapse to dominating-function analogues across several Boolean algebras, yielding a unified perspective on the computability content of continuum characteristics and suggesting broad avenues for future explorations in higher-cardinal or alternative-algebra settings.

Abstract

Cardinal characteristics of the continuum represent the boundaries in size between the countable and the continuum with respect to certain properties of sets. They are often defined as the minimum sizes of families of reals that meet some criteria. Taking these families and considering their analogues in the setting of computability theory provides a rich hierarchy of properties of oracles, which can be studied in terms of the Muchnik/Medvedev lattices of mass problems. We provide more detail to the proof of the Medvedev equivalence between dominating functions and maximal independent families given by Lempp et al. (2023) and adapt their construction of maximal almost disjoint families to the setting of -computably approximable sets. We then extend the theory to include correspondents of maximal ideal independent families and show they behave similarly to the maximal independent families.

Paper Structure

This paper contains 16 sections, 9 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Cardinal characteristics of the continuum
  4. Preliminaries on families in Boolean algebras
  5. Recursive analogues of characteristics
  6. Highness classes and Weihrauch problems
  7. Mass problems in the Medvedev lattice
  8. Other work on cardinal characteristics in computability
  9. The Gamma question and mass problems
  10. Combinatorial Mass Problems
  11. Independent families and dominating functions
  12. The Boolean algebra of $\omega$-c.a. sets
  13. Ideal Independent Families
  14. Ideal independent families and dominating functions
  15. The Boolean algebra of $K$-trivial sets
  16. ...and 1 more sections

Key Result

Lemma 2.1

$\mathrm{DomFcn}$ is Medvedev equivalent to the mass problem of approximations to $\mathrm{Tot}$.

Figures (5)

  • Figure 1: Hasse diagram of many combinatorial cardinal characteristics
  • Figure 2: An effective morphism
  • Figure 3: Relations between some cardinal characteristics and their analogues
  • Figure 4: Placement of $\mathfrak s_{\mathrm{mm}}$ among other cardinal characteristics
  • Figure 5: The collapse of relations between several analogues of cardinal characteristics

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • ...and 24 more