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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.13913 |
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| _version_ | 1866915892199161856 |
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| author | Frittaion, Emanuele Genovesi, Giorgio G. |
| author_facet | Frittaion, Emanuele Genovesi, Giorgio G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13913 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Axiom Beta Implies Elementary Transfinite Recursion Frittaion, Emanuele Genovesi, Giorgio G. Logic 03E30, 03B30 (Primary) 03D60, 03F05 (Secondary) 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. |
| title | Axiom Beta Implies Elementary Transfinite Recursion |
| topic | Logic 03E30, 03B30 (Primary) 03D60, 03F05 (Secondary) |
| url | https://arxiv.org/abs/2603.13913 |