Integer parts of real closed fields are especially interesting as they are known to coincide with models of a certain natural fragment of Peano Arithmetic, namely Open Induction (see [S]).
Furthermore, there are other structures associated with real closed fields that are used in the Mourgues-Ressayre construction, namely value group sections and residue field sections.
In section 5, we give some extra conditions on real closed fields under which the Axiom of Choice is not necessary for obtaining an integer part.
As a byproduct of the considerations above, we get two consequences for the valuation theory of real closed fields.
2] in K for countable real closed fields K and that this bound is strict.
Consequently, ZF does not prove that every real closed fields has a residue field section.
i] | i [member of] N) of [omega]-homogenous real closed fields whose residue field has infinite transcendence degree over Q, where [K.
Remark: Similarly, the same holds for real closed fields with finite Pfaffian chains, Pfaffian functions etc.
In fact, it follows from ZF that every supported real closed fields has an integer part:
To this end, we generalize the arguments from [KL] dealing with the case of countable real closed fields to higher cardinalities.
We have seen that there are examples of real closed fields that have neither an integer part nor a value group section or a residue field sections in some transitive model of ZF containing them.
IP] (or the existence of value group sections or residue field sections) for real closed fields really is.