Unbounded and dominating reals in Hechler extensions

TL;DR

This paper investigates the properties of unbounded and dominating reals in Hechler extensions, revealing differences between standard and tree Hechler models and providing a representation theorem for dominating reals.

Contribution

It establishes a representation theorem for dominating reals in standard Hechler extensions and clarifies their relationship with unbounded reals, settling conjectures and questions in the field.

Findings

Existence of an unbounded real dominated by all dominating reals in standard Hechler extension

Failure of this property in the tree Hechler extension

Representation theorem for dominating reals as sandwich compositions

Abstract

We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated by every dominating real, but that this fails to hold in the tree Hechler extension. We prove a representation theorem for dominating reals in the standard Hechler extension: every dominating real eventually dominates a sandwich composition of the Hechler real with two ground model reals that monotonically converge to infinity. We apply our results to negatively settle a conjecture of Brendle and L\"owe. We also answer a question due to Laflamme.