Extremal axioms. Logical, mathematical and cognitive aspects

Abstract

The book is devoted to extremal axioms in mathematics and is addressed mainly to cognitive scientists. As far as I am aware it is the first monograph to tackle this issue. The author relativizes extremal axioms to models in the metamathematical sense. Such an approach is the book’s main theoretical virtue, especially with respect to the so-called intended models. It should be stressed that intended models are distinguished on the basis of pragmatic criteria and as such are not reducible to purely semantic or syntactic constructions. In this context the role of extremal axioms is fundamental. Such axioms are formulated not necessarily at the very beginning of a theory but rather only after theoretical constructions have already been present for a long time in the research practice of mathematicians. A question that arises is how extremal axioms distinguish between standard and non-standard models, and whether they do so in a complete way.