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UNCOUNTABLE REAL CLOSED FIELDS WITH PA INTEGER PARTS

Published online by Cambridge University Press:  22 April 2015

DAVID MARKER ,
JAMES H. SCHMERL  and
CHARLES STEINHORN
DAVID MARKER
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO 851 S. MORGAN ST., CHICAGO, IL 60607-7045, USAE-mail: marker@math.uic.edu
JAMES H. SCHMERL
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CONNECTICUT 196 AUDITORIUM RD U-3009 STORRS, CT 06269-3009, USAE-mail: schmerl@math.uconn.edu
CHARLES STEINHORN
Affiliation:
DEPARTMENT OF MATHEMATICS VASSAR COLLEGE 124 RAYMOND AVENUE, BOX 257 POUGHKEEPSIE, NY 12604-0257, USAE-mail: steinhorn@vassar.edu
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Abstract

D’Aquino, Knight, and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of ɷ1-like models of PA.

Keywords

real closed fieldPeano arithmeticrecursively saturated modelɷ1-like model
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 490 - 502
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Barwise, J. and Schlipf, J., An introduction to recursively saturated and resplendent models, this Journal, vol. 41 (1976), no. 2, pp. 531–536.Google Scholar
Carl, M., D’Aquino, P., and Kuhlmann, S., Real closed exponential fields and models of Peano Arithmetic, preprint.Google Scholar
D’Aquino, P., Knight, J., and Starchenko, S., Real closed fields and models of Peano arithmetic, this Journal, vol. 75 (2010), no. 1, pp. 1–11.Google Scholar
D’Aquino, P., Kuhlmann, S., and Lange, K., A valuation theoretic characterization of recursively saturated real closed fields, this Journal, to appear.Google Scholar
van den Dries, L. P. D., Macintyre, A., and Marker, D., The elementary theory of restricted analytic fields with exponentiation. Annals of Mathematics (2), vol. 140 (1994), no. 1, pp. 183–205.Google Scholar
Harnik, V., ɷ 1-like recursively saturated models of Presburger’s arithmetic, this Journal, vol. 51 (1986), no. 2, pp. 421–429.Google Scholar
Kołodziejczyk, L. and Jeřábek, E., Real closures of models of weak arithmetic, Archive for Mathematical Logic, vol. 52 (2013), no. 1–2, pp. 143–157.Google Scholar
Kossak, R. and Schmerl, J., The Structure of Models of Peano Arithmetic, Oxford Logic Guides, vol. 50, Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford, 2006.Google Scholar
Kuhlmann, F.-V., Kuhlmann, S., Marshall, M., and Zekavat, M., Embedding ordered fields in formal power series fields. Journal of Pure and Applied Algebra, vol. 169 (2002), no. 1, pp. 71–90.Google Scholar
Kuhlmann, F.-V., Kuhlmann, S., and Shelah, S., Exponentiation in power series fields. Proceedings of the American Mathematical Society, vol. 125 (1997), no. 11, pp. 3177–3183.CrossRefGoogle Scholar
Marker, D., Model theory. An introduction, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002.Google Scholar
Mourgues, M.-H. and Ressaryre, J. P., Every real closed field has an integer part, this Journal, vol. 58 (1993), pp. 641–647.Google Scholar
Neumann, B. H., On ordered division rings. Transactions of the American Mathematical Society, vol. 66 (1949), pp. 202–252.Google Scholar
Scott, D., On completing ordered fields,Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967), Holt, Rinehart and Winston, New York, 1969, pp. 274–278.Google Scholar
Schmerl, J. H., Models of Peano Arithmetic and a question of Sikorski on ordered fields. Israel Journal of Mathematics, vol. 50 (1985), no. 1–2, pp. 145–159.Google Scholar
Shepherdson, J., A non-standard model for a free variable fragment of number theory. Bulletin de l’Academique Polonaise des Sciences. Série des Sciences, Mathematiques, Astronomiques et Phisiques, vol. 12 (1967), pp. 79–86.Google Scholar