Extremality Assumptions in the Foundations of Mathematics

Abstract
What are here called extremality conditions limit the models of a first-order language to those that are minimal (in a certain sense), maximal (in a certain sense) or both (in different respects). It is indicated how the requisite senses of maximality and minimality can be defined. By restricting models to those that satisfy these extremality conditions, complete axiomatizations can be given to elementary number theory and to the theory of the continuum. The same extremality conditions can also be used to limit the models of axiomatic set theory to certain "standard" ones.
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