A Note on Integer Parts of Real Closed Fields and the Axiom of Choice

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CARL, Merlin, 2014. A Note on Integer Parts of Real Closed Fields and the Axiom of Choice

@unpublished{Carl2014Integ-29881, title={A Note on Integer Parts of Real Closed Fields and the Axiom of Choice}, year={2014}, author={Carl, Merlin} }

Carl, Merlin 2014 An integer part I of a real closed field K is a discretely ordered subring with minimal element 1 such that, for every x in K, I contains some i such that x is between i and i+1 in the ordering of K. Mourgues and Ressayre showed that every real closed field has an integer part. Their construction implicitely uses the axiom of choice.<br /><br />We show that the axiom of choice is actually necessary to obtain the result by constructing a transitive model of ZF (i.e. set theory without the axiom of choice) which contains a real closed field without an integer part. Then we analyze some cases where the axiom of choice is not necessary for obtaining an integer part. This also sheds some light on the possibility to effectivize constructions of integer parts. Carl, Merlin 2015-02-19T09:14:23Z eng 2015-02-19T09:14:23Z A Note on Integer Parts of Real Closed Fields and the Axiom of Choice

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