Nonsets

Daniel Parrochia

pp. 409-421

in: Arnold Koslow, Arthur Buchsbaum (eds), The road to universal logic I, Basel, Birkhäuser, 2015

Abstract

Our topics are set theory and the problem of the definition of a set. With the existence of non-collectivizing relations and inconsistent multiplicities, the usual definition meets some limits. But, what is a nonset? Making a difference between sets and collections could appear as a subtle concern. In fact, this does not work very well. Indeed, philosophers show a lot of examples when naive sets and collectivizing relations fail, and modern mathematicians, from Cantor and Dedekind to Aczel suspect themselves that rising objections to the uncritical use of collectivizing relations is not unreasonable. A solution of the problem may be found in a rational model (logic but not Logic, as Beziau would say) that can formalize the notion of uncomparability between objects. An example is given with the ethical relation of "absolute alterity" developed by the French philosopher Emmanuel Levinas. Then, non-transitivity, trellis and weakly associative structures can be used to formalize such a situation.

Cited authors

Levinas Emmanuel