Comparing hierarchies of total functionals

TL;DR

This paper investigates two hierarchies of hereditarily total and continuous functionals over the reals, analyzing conditions under which they coincide and exploring their topological properties.

Contribution

It introduces and compares two hierarchies of functionals based on different real number representations and proves their potential equivalence under certain assumptions.

Findings

Hierarchies may coincide under specific assumptions

Kleene-Kreisel functionals embed into both hierarchies

Topological properties relate to hierarchy equivalence

Abstract

In this paper we consider two hierarchies of hereditarily total and continuous functionals over the reals based on one extensional and one intensional representation of real numbers, and we discuss under which asumptions these hierarchies coincide. This coincidense problem is equivalent to a statement about the topology of the Kleene-Kreisel continuous functionals. As a tool of independent interest, we show that the Kleene-Kreisel functionals may be embedded into both these hierarchies.