Models of Peano arithmetic and a question of Sikorski on ordered fields
Let \(\kappa\) be a regular cardinal. An ordered field F is said to have the Bolzano-Weierstrass property for \(\kappa\)-sequences (briefly: F is BW(\(\kappa)\)) if every bounded \(\kappa\)-sequence has a convergent \(\kappa\)-subsequence. \textit{R. Sikorski} [C.R. Soc. Sci. Lett. Varsovie, Cl. III 41, 69-96 (1950; Zbl 0039.276)] asked about the existence of BW(\(\kappa)\) fields F where \(| F| >\kappa >\aleph_ o\). Some results had been obtained by \textit{L. Manevitz} and \textit{A. W. Miller} [Isr. J. Math. 45, 209-218 (1983; Zbl 0526.03017)], \textit{S. B. Todorčević} [Ann. Math. Logic 20, 233-268 (1981; Zbl 0469.03036)] and \textit{D. J. Velleman} [Ann. Math. Logic 23, 199-281 (1982; Zbl 0521.03034)], including the result that for all regular \(\kappa >\aleph_ o\) there were such fields, assuming \(V=L\). \textit{J. Cowles} and \textit{R. LaGrange} [Notre Dame J. Formal Logic 24, 133-140 (1983; Zbl 0488.03027)] observed that the existence of such a field implies the existence of a \(\kappa\)-Kurepa tree with no \(\kappa\)-Aronszajn subtrees. The main result of this paper, proved using models of Peano Arithmetic, is a converse to this. Specifically, it is Theorem: If \(\lambda \geq \kappa >\aleph_ o\) are cardinals with \(\kappa\) regular, then there is a real-closed BW(\(\kappa)\) field of cardinality \(\lambda\) iff there is a \(\kappa\)-tree having exactly \(\lambda\) branches but no \(\kappa\)-Aronszajn subtree.
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