On Equality and Natural Numbers in Cantor-Lukasiewicz Set Theory

0343863 - ÚI 2013 RIV GB eng J - Článek v odborném periodiku
Hájek, Petr
On Equality and Natural Numbers in Cantor-Lukasiewicz Set Theory.
Logic Journal of the IGPL. Roč. 21, č. 1 (2013), s. 91-100. ISSN 1367-0751. E-ISSN 1368-9894
Grant CEP: GA MŠMT(CZ) 1M0545
Výzkumný záměr: CEZ:AV0Z10300504
Klíčová slova: Lukasiewicz logic * Cantor set theory * full comprehension
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 0.530, rok: 2013

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).
Trvalý link: http://hdl.handle.net/11104/0186240