Syntactic-Semantic Axiomatic Theories in Mathematics

Abstract

A more careful consideration of the recently introduced "Grossone Theory" of Yaroslav Sergeev, [1], leads to a considerable enlargement of what can constitute possible legitimate mathematical theories by the introduction here of what we may call the Syntactic-Semantic Axiomatic Theories in Mathematics. The usual theories of mathematics, ever since the ancient times of Euclid, are in fact axiomatic, [1,2], which means that they are syntactic logical consequences of certain assumed axioms. In these usual mathematical theories semantics can only play an indirect role which is restricted to the inspiration and motivation that may lead to the formulation of axioms, definitions, and of the proofs of theorems. In a significant contradistinction to that, and as manifestly inspired and motivated by the mentioned Grossone Theory, here a direct involvement of semantics in the construction of axiomatic mathematical theories is presented, an involvement which gives semantics the possibility to act explicitly, effectively, and altogether directly upon the usual syntactic process of constructing the logical consequences of axioms. Two immediate objections to what appears to be an unprecedented and massive expansion of what may now become legitimate mathematical theories given by the syntactic - semantic axiomatic theories introduced here can be the following: the mentioned direct role of semantics may, willingly or not, introduce in mathematical theories one, or both of the "eternal taboo-s" of inconsistency and self-reference. Fortunately however, such concerns can be alleviated due to recent developments in both inconsistent and self-referential mathematics, [1,2]. Grateful recognition is acknowledged here for long and most useful ongoing related disccussions with Yaroslav Sergeev.