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Axioms for pseudo real closed fields. (English) Zbl 0555.12009
A field \(K\) is called ’pseudo real closed’ if it is existentially closed in each extension \(L\) of \(K\) to which all orderings of \(K\) can be lifted. This notion was introduced by the reviewer in [Set theory and model theory, Bonn 1979, Lect. Notes Math. 872, 127–156 (1981; Zbl 0466.12018)]. In this paper it was also proved that the notion of pseudo real closed fields is elementary. The proof made explicit use of the elimination of quantifiers for real closed fields.
The author uses non-standard methods in order to give an alternative proof not appealing to elimination of quantifiers.

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
12L12 Model theory of fields
03C60 Model-theoretic algebra
03C65 Models of other mathematical theories
03H05 Nonstandard models in mathematics