An Lω1ω1 axiomatization of the linear Archimedean continua as merely relational structures

Abstract

We have chosen the language Lω1ω1, in which to formulate the axioms of two systems of the linear Archimedean continua - the point-based system, Sp, and the stretch-based system, SI - for the following reasons: 1. It enables us to formulate all the axioms of each system in one and the same language; 2. It makes it possible to apply, without any modification, Arsenijević's two sets of rules for translating formulas of each of these systems into formulas of the other, in spite of the fact that these rules were originally formulated in a first-order language for systems that are not continuous but dense only; 3. It enables us to speak about an infinite number of elements of a continuous structure by mentioning explicitly only denumerably many of them; 4. In this way we can formulate not only Cantor's coherence condition for linear continuity but also express the large-scale and small-scale variants of the Archimedean axiom without any reference, either explicit or implicit, to a metric; 5. The models of the two axiom systems are structures that need not be relational-operational but only relational, which means that we can speak of the linear geometric continua directly and not only via the field of real numbers (numbers will occur as subscripts only, and they will be limited to the natural numbers).