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Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

David Fernández-Duque, Andreas Weiermann

Abstract

We prove that Buchholz's system of fundamental sequences for the $\vartheta$ function enjoys various regularity conditions, including the Bachmann property. We partially extend these results to variants of the $\vartheta$ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along $\vartheta(\varepsilon_{Ω+1})$.

Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

Abstract

We prove that Buchholz's system of fundamental sequences for the function enjoys various regularity conditions, including the Bachmann property. We partially extend these results to variants of the function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along .
Paper Structure (10 sections, 36 theorems, 23 equations)

This paper contains 10 sections, 36 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Ordinals below $\varepsilon_{{\Omega}+1}$
  3. Parametrized theta functions
  4. Approximating $\mathbb X$-limits
  5. Regular Bachmann systems
  6. Buchholz systems of fundamental sequences
  7. Comparing $\vartheta$ and $\sigma$
  8. The Bachmann property
  9. Notation systems and norms
  10. The Hardy hierarchy

Key Result

Lemma 2.3

Suppose that ${\Omega}^{\alpha}({\beta}+\delta)+\gamma <\varepsilon_{{\Omega}+1}$ is in ${\Omega}$-normal form, $\delta$ is a successor or zero, and ${\Omega}^{\alpha} \delta +\gamma>0$. Then, for all $\theta$, $({\Omega}^{\alpha}({\beta}+\delta)+\gamma ) [\theta]={\Omega}^{\alpha}{\beta}+ ({\Omega}

Theorems & Definitions (80)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5: BuchholzOrd
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 70 more