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Maximal towers and ultrafilter bases in computability

Steffen Lempp, Joseph S. Miller, Andre Nies, Mariya Soskova

Abstract

The tower number $\mathfrak t$ and the ultrafilter number $\mathfrak u$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of~$ω$ and the almost inclusion relation $\subseteq^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory. We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin's characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e.\ set computes a maximal tower but no 1-generic $Δ^0_2$-set does so. We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively.

Maximal towers and ultrafilter bases in computability

Abstract

The tower number and the ultrafilter number are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of~ and the almost inclusion relation between such subsets. We consider analogs of these cardinal characteristics in computability theory. We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin's characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e.\ set computes a maximal tower but no 1-generic -set does so. We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively.

Paper Structure

This paper contains 16 sections, 15 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Comparing the complexity of the analogs in computability
  3. The mass problem T_B of maximal towers
  4. The mass problem U_B of ultrafilter bases
  5. The Boolean algebra of computable sets
  6. Basics of the mass problems T_B
  7. The equivalent mass problems T_B and A_B
  8. Descriptive complexity and index complexity for maximal towers
  9. Complexity of T and of U
  10. Maximal towers, non-lowness, and index guessability
  11. Ultrafilter bases and highness
  12. Maximal independent families in computability
  13. The case of computably enumerable complements
  14. Computably enumerable MAD sets, and co-c.e. towers
  15. Co-c.e. ultrafilter bases
  16. ...and 1 more sections

Key Result

Proposition 1.6

Given a Turing ideal $\mathcal{K}$, let $\mathbb B$ be the Boolean algebra of sets with degree in $\mathcal{K}$. Then for each $\mathbb B$-ultrafilter base $F$ and associated function $p$ in the sense of Definition df:assoc_fcn, the function $p$ is not dominated by a function with Turing degree in

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • proof
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Definition 2.1
  • proof
  • proof
  • ...and 39 more