Solutions of Extension and Limits of Some Cantorian Paradoxes

Josué-Antonio Nescolarde-Selva, José-Luis Usó-Doménech, Lorena Segura-Abad, Kristian Alonso-Stenberg and Hugh Gash
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Josué-Antonio Nescolarde-Selva: Department of Applied Mathematics, University of Alicante, 03690 Alicante, Spain
José-Luis Usó-Doménech: Department of Applied Mathematics, University of Alicante, 03690 Alicante, Spain
Lorena Segura-Abad: Department of Mathematics, University of Alicante, 03690 Alicante, Spain
Kristian Alonso-Stenberg: Department of Applied Mathematics, University of Alicante, 03690 Alicante, Spain
Hugh Gash: Institute of Education, Dublin City University, D09 Y18 Dublin, Ireland

Mathematics, 2020, vol. 8, issue 4, 1-11

Abstract: Cantor thought of the principles of set theory or intuitive principles as universal forms that can apply to any actual or possible totality. This is something, however, which need not be accepted if there are totalities which have a fundamental ontological value and do not conform to these principles. The difficulties involved are not related to ontological problems but with certain peculiar sets, including the set of all sets that are not members of themselves, the set of all sets, and the ordinal of all ordinals. These problematic totalities for intuitive theory can be treated satisfactorily with the Zermelo and Fraenkel (ZF) axioms or the von Neumann, Bernays, and Gödel (NBG) axioms, and the iterative conceptions expressed in them.

Keywords: cantorian paradoxes; classes; inconsistent totalities; sets; solutions of extension; solutions of limitation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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