Real functions on the family of all well-ordered subsets of a partially ordered set

Definition 1 (Kurepa [3, p. 99]). Let E be a partially ordered set. Then σE denotes the set of all bounded well-ordered subsets of E . We consider σE as a partially ordered set with ordering defined as follows: s t if and only if s is an initial segment of t . Then σE is a tree, i.e., { s ∈ σ E ∣ s t } is well-ordered for every t ∈ σE . The trees of the form αE were extensively studied by Kurepa in [3]–[10]. For example, in [4], he used σQ and σR to construct various sorts of Aronszajn trees. (Here Q and R denote the rationals and reals, respectively.) While considering monotone mapping between some kind of ordered sets, he came to the following two questions several times: P.1. Does there exist a strictly increasing rational function on σQ ? (See [4, Probleme 2], [5, p. 1033], [6, p. 841], [7, Problem 23.3.3].) P.2. Let T be a tree in which every chain is countable and every level has cardinality ℵ 0 . Does there exist a strictly increasing real function on T ? (See [6, p. 246] and [7].) It is known today that Problem 2 is independent of the usual axioms of set theory (see [1]). Concerning Problem 1 we have the following.