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Scales in the Point Spectrum

Tom Benhamou

Abstract

We study the Point/Tukey spectrum of a general directed set using PCF theoretic tools and uncover basic connections between the theories. In particular, we prove that if the supremum of the Tukey spectrum is singular, then its cofinality must also be a member of the Tukey spectrum.

Scales in the Point Spectrum

Abstract

We study the Point/Tukey spectrum of a general directed set using PCF theoretic tools and uncover basic connections between the theories. In particular, we prove that if the supremum of the Tukey spectrum is singular, then its cofinality must also be a member of the Tukey spectrum.
Paper Structure (4 sections, 28 theorems, 7 equations)

This paper contains 4 sections, 28 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. On the Tukey spectrum and PCF theory
  3. Two generalizations
  4. Further Directions and Open Problems

Key Result

Theorem 1.3

Let $\mathcal{U}$ be an ultrafilter and $\lambda$ be a regular cardinal. Then the following are equivalent:

Theorems & Definitions (49)

  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.6: Isbell
  • Theorem
  • Corollary
  • Corollary
  • Corollary
  • Theorem
  • Theorem
  • ...and 39 more