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
Full Text: