We show that $\mathbf{ C}$, a weak theory of sets with Axiom Beta, proves the scheme of Elementary, or $\Delta_0$ 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 $\mathbf{ ATR}_0^\text{ set}$ 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 $\mathbf{ PRS}\omega+$ Axiom Beta. We also establish an upper bound, though not a sharp one, for the $\Sigma_1$-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.
Categories: math.LO