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Axiom Beta Implies Elementary Transfinite Recursion

Emanuele Frittaion, Giorgio G. Genovesi

Abstract

We show that $\mathbf{C}$, a weak theory of sets with Axiom Beta, proves the scheme of Elementary, or $Δ_0$ Transfinite Recursion and can generate, for every set, the corresponding relativized constructible hierarchy. We show that the theory $\mathbf{C}$ corresponds to Simpson's system $\mathbf{ATR}_0^\text{set}$ without the Axiom of Countability. In fact, $\mathbf{C}$ proves the totality of the Veblen function and of all primitive recursive set functions. In particular, this means our system $\mathbf{C}$ is equivalent to $\mathbf{PRS}ω+$ Axiom Beta. We also establish an upper bound, though not a sharp one, for the $Σ_1$-definable functions of $\mathbf{C}$. Finally, we show that the variant of $\mathbf{C}$ in which the Finite Powerset Axiom is replaced by the closure under the rudimentary functions is a strictly weaker theory and no longer ensures the existence of the relativized constructible hierarchy.

Axiom Beta Implies Elementary Transfinite Recursion

Abstract

We show that , a weak theory of sets with Axiom Beta, proves the scheme of Elementary, or Transfinite Recursion and can generate, for every set, the corresponding relativized constructible hierarchy. We show that the theory corresponds to Simpson's system without the Axiom of Countability. In fact, proves the totality of the Veblen function and of all primitive recursive set functions. In particular, this means our system is equivalent to Axiom Beta. We also establish an upper bound, though not a sharp one, for the -definable functions of . Finally, we show that the variant of in which the Finite Powerset Axiom is replaced by the closure under the rudimentary functions is a strictly weaker theory and no longer ensures the existence of the relativized constructible hierarchy.
Paper Structure (12 sections, 27 theorems, 94 equations)

This paper contains 12 sections, 27 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary diagrams
  4. Transfinite recursion
  5. Clopen Games
  6. The constructible universe
  7. Primitive recursive set functions
  8. Veblen Hierarchy
  9. Stability
  10. Primitive recursion and Axiom Beta
  11. Stabilizing functions
  12. Interpretation theorem

Key Result

Theorem 3.1

Let $\langle A,\prec\rangle$ be a well founded relation and $X$ be a set. Given a $\Delta_0$ formula $\vartheta[x,y,z]$, we can define trees $T_s\subseteq X^{<\omega}$ for every $s\in A$ by Moreover, suppose that for every $s\in A$ there exists $l\in\omega$ such that $\vartheta[\sigma,q,s]$ with $q\prec s$ implies $|\sigma|< l$. Then every $T_s$ is well founded.

Theorems & Definitions (63)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1: $\mathbf{B}$
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • ...and 53 more