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It is consistent that there exists an ordered real closed field which is not hyper-real. (English) Zbl 0999.12501

Hyper-real fields are non-Archimedean fields of the form \(C(X)/M\), where \(X\) is a completely regular space, \(C(X)\) the ring of continuous function from \(X\) to \(R\), and \(M\) a maximal ideal in \(C(X)\). They are \(\eta_1\)-ordered and real closed fields.
The authors show that there exists a model of ZFC in which there exists an \(\eta_1\)-ordered real closed field which is not isomorphic to any hyper-real closed field.

MSC:

12J15 Ordered fields
03C60 Model-theoretic algebra
03E05 Other combinatorial set theory
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