Foundations of Transmathematics

Abstract

Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics.

Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic.

In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known.

As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.