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Notes on the coexistence of limit notions

Takashi Yamazoe

Abstract

We summarize the current knowledge on the three limit notions: ultrafilter-limits, closed-ultrafilter-limits and FAM-limits. Also, we consider the possibility to perform an iteration which has all the three limits and clarify the problem we face.

Notes on the coexistence of limit notions

Abstract

We summarize the current knowledge on the three limit notions: ultrafilter-limits, closed-ultrafilter-limits and FAM-limits. Also, we consider the possibility to perform an iteration which has all the three limits and clarify the problem we face.
Paper Structure (10 sections, 19 theorems, 8 equations, 2 figures, 1 table)

This paper contains 10 sections, 19 theorems, 8 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Cichoń's maximum
  3. Limit notions
  4. Structure of the article
  5. Relational systems and preservation theory
  6. Limit notions
  7. Ultrafilter-limit and closed-ultrafilter-limit
  8. FAM-limit
  9. The forcing poset $\widetilde{\mathbb{E}}$
  10. Iteration

Key Result

Theorem 1.3

(GMS16) Ultrafilter-limits keep $\mathfrak{b}$ small.

Figures (2)

  • Figure 1: Cichoń's diagram and other cardinal invariants.
  • Figure 2: Constellation of $\mathbb{P}^7_\mathrm{fin}$, if the construction is possible.

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Corollary 2.6
  • ...and 38 more