On definable completeness for ordered fields

Mojtaba Moniri

Abstrakt

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.

AMS subject classification: Primary 03C64; Secondary 12L12

Słowa kluczowe: ordered field, 0-definably complete, real closed field
References

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[4] J.S. Eivazloo and M. Moniri, Expansions of Ordered Fields without Definable Gaps, Mathematical Logic Quarterly 49:1 (2003), 72–82.

[5] M. Moniri and J.S. Eivazloo, Using Nets in Dedekind, Monotone, or Scott Incomplete Ordered Fields and Definability Issues, In: P. Simon, editor, Proceedings  of the Ninth Prague Topological Symposium, Prague, August 19–25, 2001, Topology Atlas, North Bay, ON, 2002, pp. 195–203. Available at: http://at.yorku.ca/p/p/a/e/00.htm.

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