A marriage of category theory and set theory: a finitely axiomatized nonclassical first-order theory implying ZF

Por • 4 jul, 2018 • Sección: Educacion

Marcoen J.T.F. Cabbolet

Abstract: The main purpose of this paper is to introduce a finitely axiomatized theory that might be applicable as a foundational theory for mathematics. For that matter, some twenty axioms in a formal language are introduced, which are to hold in a universe consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function on a set. One of the axioms is nonclassical: it states that, given a family of ur-functions – i.e. functions on a singleton – with disjunct domains, there exists a uniquely determined sum function on the union of these domains. This ‘sum function axiom’ is so powerful that it allows to derive ZF from a finite axiom scheme. In addition, it is shown that the Loewenheim-Skolem theorem does not hold for the present theory, which therefore can be considered stronger than ZF. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

arXiv:1806.05538v1 [math.GM]

General Mathematics (math.GM)

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