On Equality and Natural Numbers in Cantor-Lukasiewicz Set Theory

Abstract

Two equality predicates in Cantor-Lukasiewicz set theory (with full comprehension, over Lukasiewicz predicate logic) are investigated: extensional =e and Leibniz equality =. It is proved that there are many pairs of sets x,y such that x =e y & x =/= y is true. In particular, x may be the set omega of natural numbers, defined together with ternary predicates for addition and multiplication. The main result says that the Cantor-Lukasiewicz set theory is essentially undecidable and essentially incomplete. The proof is difficult since it is not supposed that the set omega is crisp (non-fuzzy).