Using nets in Dedekind, monotone, or Scott incomplete ordered fields and definability issues

TL;DR

This paper explores the properties of Dedekind, monotone, and Scott incomplete ordered fields, focusing on definability issues, nets, and completeness criteria, revealing new connections between gaps, continuous functions, and definability in these fields.

Contribution

It introduces new criteria for Dedekind completeness, constructs examples of functions with specific properties, and establishes independence results related to definable gaps and Cauchy functions in ordered fields.

Findings

A new criterion for Dedekind completeness involving continuous one-to-one functions.

Existence of continuous, non-uniformly continuous, unbounded functions with non-closed ranges.

Construction of an ordered field with parametrically definable gaps but no definable divergent Cauchy functions.

Abstract

Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps which are respectively increasing or decreasing to the same point is used to obtain a further equivalent criterion for Dedekind completeness of ordered fields: Every continuous one-to-one function defined on a closed bounded interval maps interior of that interval to the interior of the image. Next, it is shown that over all closed bounded intervals in any monotone incomplete ordered field, there are continuous not uniformly continuous unbounded functions whose ranges are not closed, and continuous 1-1 functions which map every interior point to an interior point (of the image) but are not open. These are achieved using appropriate nets cofinal in gaps or coinitial in their complements. In our third main theorem, an ordered field is constructed which has parametrically definable regular gaps but no $\emptyset$-definable divergent Cauchy functions (while we show that, in either of the two cases where parameters are or are not allowed, any definable divergent Cauchy function gives rise to a definable regular gap). Our proof for the mentioned independence result uses existence of infinite primes in the subring of the ordered field of generalized power series with rational exponents and real coefficients consisting of series with no infinitesimal terms, as recently established by D. Pitteloud.