Order

Abstract

So far, we have only been considering equational theories, that is, theories whose only primitive predicate symbol is =. We have been looking at how function symbols such as + and × behave inconsistently in such theories. In this chapter a second primitive predicate < is added to =, interpreted as ‘less than’. In the first section, function symbols are omitted. The main result of this section is as follows. There is a well-known metatheorem of classical model theory to the effect that the theory of dense order without endpoints is N₀-categorical, that is, that all classical models of that theory of cardinality N₀ are isomorphic. We see that this breaks down for RM3-theories, given a natural extension of the notion of isomorphism of models to the more general case. It follows that the classical metatheorem depends on the assumption of classical logic. Such is not always the case: various other results in this book are invariant with respect to changes in background logic away from the classical case. In the next section, function symbols are re-introduced and several results about these are summarised.