Axiom Beta Implies Elementary Transfinite Recursion

We show that $\mathbf{C}$, a weak theory of sets with Axiom Beta, proves the scheme of Elementary, or ${\mathrm{\Delta }}_0$ 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 ${\mathbf{A}\mathbf{T}\mathbf{R}}_0^{\text{set}}$ 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 $\mathbf{P}\mathbf{R}\mathbf{S}\omega +\text{Axiom Beta}$. We also establish an upper bound, though not a sharp one, for the ${\mathrm{\Sigma }}_1$-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.