Much of mathematics is concerned with the study of “standard” mathematical systems such as the natural numbers, the rationals, the real numbers and the complex numbers, each of which is regarded as a unique system. When we attempt to study one of these systems by axiomatising it within the first-order predicate calculus, we find that our axiomatisation cannot be categorical, and that there exist models of our axiomatic theory not isomorphic to the system we wish to study. Such models have been constructed as ultrapowers in Chapter VII. In this chapter, we investigate ways of exploiting such models in the study of a standard system. We begin by considering elementary systems, i.e., systems in which relations between elements, but not properties of subsets, can be studied.
KeywordsStandard System Relation Symbol Concurrent Relation Real Number System Definable Subset
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