Open AccessOn definable completeness for ordered fields
Author(s) -
Morteza Moniri
Publication year - 2019
Publication title -
reports on mathematical logic
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.101
H-Index - 6
eISSN - 2084-2589
pISSN - 0137-2904
DOI - 10.4467/20842589rm.19.005.10653
Subject(s) - completeness (order theory) , dedekind cut , mathematics , ordered field , pure mathematics , field (mathematics) , discrete mathematics , mathematical analysis
We show that there are 0-definably complete ordered fields which are not real closed. Therefore, the theory of definably with parameters complete ordered fields does not follow from the theory of 0-definably complete ordered fields. The mentioned completeness notions for ordered fields are the definable versions of completeness in the sense of Dedekind cuts. In earlier joint work, we had shown that it would become successively weakened if we just required nonexistence of definable regular gaps and then disallowing parameters. The result in this note shows reducing in the opposite order, at least one side is sharp.