The \({\mathcal L}^{<\omega}\)-theory of the class of Archimedian real closed fields
For the class \({\mathcal A}\) of Archimedian real closed fields it is shown that the Malitz quantifier \(Q^ n\) is eliminable over \({\mathcal A}\), provably in ZFC, if \(n\leq 2\). If the Continuum-Hypothesis (CH) is true, \(Q^ 3\), hence any \(Q^ n\) for \(n\geq 3\), cannot be eliminated over \({\mathcal A}\). Conversely, it is consistent with ZFC that every \(Q^ n\), for \(n\geq 1\), is eliminable over \({\mathcal A}\); the last statement is already a consequence of the Proper-Forcing-Axiom.
scientific article; zbMATH DE number 4116508