Axiom Beta Implies Elementary Transfinite Recursion

Emanuele Frittaion; Giorgio G. Genovesi

mathematical logic

Axiom Beta Implies Elementary Transfinite Recursion

arXiv:2603.13913

summary

The paper introduces a weak set theory called C that includes Axiom Beta and proves elementary (Δ₀) transfinite recursion, allowing the construction of relativized constructible hierarchies and showing equivalence to known systems like ATR₀^set and PRS⁺Axiom Beta.

Abstract

We show that $C$ , a weak theory of sets with Axiom Beta, proves the scheme of Elementary, or $I^_{0}$ Transfinite Recursion and can generate, for every set, the corresponding relativized constructible hierarchy. We show that the theory $C$ corresponds to Simpson's system $ATR_{0}^{set}$ without the Axiom of Countability. In fact, $C$ proves the totality of the Veblen function and of all primitive recursive set functions. In particular, this means our system $C$ is equivalent to $PRS I ¨ + Axiom Beta$ . We also establish an upper bound, though not a sharp one, for the $I^£_{1}$ -definable functions of $C$ . Finally, we show that the variant of $C$ 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.

43 pages

Topics & keywords

#set theory #proof theory #transfinite recursion #axiom beta #constructible hierarchy #primitive recursive set functionsAxiom BetaElementary Transfinite Recursiontheory CATR₀^setVeblen functionΣ₁-definable functionsrudimentary functions