Bulletin of the Belgian Mathematical Society - Simon Stevin
- Bull. Belg. Math. Soc. Simon Stevin
- Volume 23, Number 3 (2016), 401-419.
Structures Associated with Real Closed Fields and the Axiom of Choice
An integer part $I$ of a real closed field $K$ is a discretely ordered subring of $K$ with minimal positive element $1$ such that, for every $x\in K$, there is $i\in I$ with $i\leq x<i+1$. Mourgues and Ressayre showed in [MR] that every real closed field has an integer part. Their construction implicitly uses the Axiom of Choice. We show that $AC$ is actually necessary to obtain the result by constructing a transitive model of $ZF$ 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. On the way, we demonstrate that a class of questions containing the question whether the Axiom of Choice is necessary for the proof of a certain $ZFC$-theorem is algorithmically undecidable. We further apply the methods to show that it is independent of $ZF$ whether every real closed field has a value group section and a residue field section. This also sheds some light on the possibility to effectivize constructions of integer parts and value group sections which was considered e.g. in [DKKL] and [KL]
Bull. Belg. Math. Soc. Simon Stevin, Volume 23, Number 3 (2016), 401-419.
First available in Project Euclid: 6 September 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03D65: Higher-type and set recursion theory 03C55: Set-theoretic model theory 03C64: Model theory of ordered structures; o-minimality 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05]
Secondary: 03E30: Axiomatics of classical set theory and its fragments 03E75: Applications of set theory 12J20: General valuation theory [See also 13A18]
Carl, Merlin. Structures Associated with Real Closed Fields and the Axiom of Choice. Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 3, 401--419. doi:10.36045/bbms/1473186514. https://projecteuclid.org/euclid.bbms/1473186514