On Definable Completeness for Ordered Fields
On Definable Completeness for Ordered Fields

Summary/Abstract: 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.

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