Generalizing Theorems in Real Closed Fields

Jan Kraj cek posed the following problem: Is there is a generalization result in the theory of real closed elds of the form: If A(1 + + 1) (n occurrences of 1) is provable in length k for all n 2 !, then (8x)A(x) is provable? It is argued that the answer to this question depends on the particular formulation of the \theory of real closed elds." Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive answer to Kraj cek's question for (1) the axiom system RCF of Artin{Schreier with Gentzen's LK as underlying logical calculus, (2) RCF with the variant LKB of LK allowing introduction of several quantiiers of the same type in one step, (3) LKB and the rst-order schemata corresponding to Dedekind cuts and the supremum principle. A negative answer is given for (4) any system containing the schema of extensionality.