Non-Computable, Indiscernible and Uncountable Mathematical Constructions. Sub-Cardinals and Related Paradoxes

Abstract

One of the most important achievements of the last century is the knowledge of the existence of non-countable
sets. The proof by Cantor’s diagonal method requires the assumption of actual infinity. By two paradoxes we show that
this method sometimes proves nothing because of it can involve self-referential definitions. To avoid this inconvenient,
we introduce another proving method based upon the information in the involved object definitions. We also introduce
the concepts of indiscernible mathematical construction and sub-cardinal. In addition, we show that the existence of
indiscernible mathematical constructions is an unavoidable consequence of uncountability.