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arXiv · 2603.13913

Axiom Beta Implies Elementary Transfinite Recursion

Emanuele Frittaion·Giorgio G. Genovesi
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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 $\mathbf{PRS}ω+\text{Axiom Beta}$. 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.

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Emanuele Frittaion, Giorgio G. Genovesi. 2026-07-14. Axiom Beta Implies Elementary Transfinite Recursion. https://arxiv.org/abs/2603.13913

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