Construction of the Transnatural Numbers in Transset Theory
journal contributionposted on 25.10.2016, 16:23 by James AndersonJames Anderson
We assert that the universe of all classes is partitioned by the universe of all sets and the universe of all antinomies. The usual set operations are applied to all classes, giving a total transset-theory that contains a set theory and an antinomy theory. Transset theory is just naive set-theory with antinomies so it is pedagogically simple.
We define that a transset that contains only itself is an atom and that all other transsets, including the empty transset, are molecules. Transset theory can model all set-theories, including those with atoms, and the whole of category theory, using atoms as category objects and molecules as category relations.
We construct the transnatural numbers such that: the natural numbers are von Neumann ordinals; transnatural infinity, being the greatest ordinal, is built from the universal set; and transnatural nullity, being the only unordered, transnatural number, is built from the universal antinomy.
Comment and collaboration is sought on the second version of this paper, now entitled "Construction of the Transnatural Numbers in Transset Theory."
We extend transnatural arithmetic to transordinal arithmetic and show that the classical paradoxes of set theory are dissolved in transset theory, with counting that is consistent with transordinal arithmetic.
We use the fact that all antinomies have subsets to provide a foundation for paraconsistent logics and to explain how scientific theories can be useful, despite having both internal contradictions in explanations and external contradictions with observations.