On definable completeness for ordered fields

• Autorzy: Moniri Mojtaba

Identyfikatory

DOI

10.4467/20842589RM.19.005.10653

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

ordered field; 0-definably complete; real closed field

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2019
• Tom: Vol. 54
• Strony: 95--100
• Opis fizyczny: Bibliogr. 7 poz., rys.

Twórcy

autor

Moniri Mojtaba

• Department of Mathematics and Computer Science Normandale Community College 9700 France Ave S., Bloomington, MN 55431, USA

Bibliografia

• [1] J. Ax and S. Kochen, Diophantine Problems over Local Fields I, American Journal of Mathematics 87:3 (1965), 605–630.
• [2] J. Ax and S. Kochen, Diophantine Problems over Local Fields III, Decidable Fields, Annals of Mathematics (2) 83:3 (1966), 437–456.
• [3] L. B´elair, Z. Chatzidakis, P. D’Aquino, D. Marker, M. Otero, F. Point, and A. Wilkie, Open Problems in Model Theory, In: Same editors, Proceedings of the Euro-Conference on Model Theory and Applications, Ravello, Italy, May 27–June 1, 2002, Quad. Mat. 11, Aracne, Rome, 2002, pp. 459–466.
• [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.
• [6] A. Robinson and E. Zakon, Elementary Properties of Ordered Abelian Groups, Transactions of the American Mathematical Society 96:2 (1960), 222–236.
• [7] E. Zakon, Generalized Archimedean Groups, Transactions of the American Mathematical Society 99:1 (1961), 21–40.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).