Giorgio G. Genovesi | Searcharxiv

arXiv2026

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 $Δ_0$ 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.

math.LO↗

arXiv2026

Reverse mathematics of regular countable second countable spaces

We study the reverse mathematics of characterization theorems of regular countable second countable spaces (or $CSCS$ for short). We prove that arithmetic comprehension is equivalent over $\textbf{RCA}_0$ to every $T_3$ $CSCS$ being metrizable, and we characterize the $T_3$ spaces which are metrizable over $\textbf{RCA}_0$. We show that Lynn's theorem for $CSCS$ can be carried out in $\textbf{ACA}_0$, namely that every zero dimensional separable space is homeomorphic to the order topology of a linear order. We also show that arithmetic comprehension is equivalent to every $T_2$ compact $CSCS$ being well-orderable. From general topology, we know that the locally compact $T_2$ $CSCS$ are the well-orderable $CSCS$, and that the $T_3$ scattered $CSCS$ are the completely metrizable $CSCS$. We show that these characterizations and a few others are equivalent to arithmetic transfinite recursion over $\textbf{RCA}_0$. We also find a few statments that are equivalent to $Π^1_1$ comprehension. In particular we show that every $T_3$ $CSCS$ has a Cantor Bendixson rank and that every $T_3$ $CSCS$ is the disjoint union of a scattered space and dense in itself space are equivalent to $Π^1_1$ comprehension over $\textbf{RCA}_0$.

math.LO↗