Ordered families of Baire-2-functions (Q1813309)

A (partially) ordered set \((X,<)\) is called Baire-\(\alpha\)-representable if it is possible to associate a Baire-\(\alpha\)-function \(f^ x\) to each element \(x\in X\) such that \(x<y\) iff \(f^ x<f^ y\). Here, for two real functions \(f\), \(g\), \(f<g\) denotes that \(f(x)\leq g(x)\) for all \(x\) and \(f(x)<g(x)\) for some \(x\). The author proves the following results: Theorem 1. No Suslin-line is Baire-1-representable. Theorem 3. If \(V\) is a model of ZFC+CH, \((X,<)\in V\) is an ordered set of size \(\aleph_ 2\), and if we generically add \(\kappa\geq\omega_ 2\) Cohen reals, then \((X,<)\) is not Baire-\(\alpha\)-representable in the enlarged model, for any \(\alpha<\omega_ 1\). Corollary 4. It is consistent with ZFC that \({\mathfrak c}=\aleph_ 2\) and (a) \((\omega_ 2,<)\) is not Baire-\(\alpha\)-representable for \(\alpha<\omega_ 1\); (b) there is an ordered set of cardinality \(\aleph_ 2\), not containing subsets of type \(\omega_ 2\), \(\omega^*_ 2\), which is not Baire-\(\alpha\)-representable for any \(\alpha<\omega_ 1\).