The structure of infinite Friedman trees (Q1908484)

A partial order \(A\) is called a quasi-wellordering if \(A\) has no infinite descending chain and no infinite antichain. Here \(A\) is allowed to be a class. In this paper a tree is a partial order with a least element such that for each element the set of its predecessors is a finite linear order. An embedding of trees is a map \(\varphi\) that preserves infima of nonempty sets. A Friedman tree is a tree \(t\) together with a map \(g : V_t \to \text{Ord}\). For \(s,t\) Friedman trees, an embedding \(\varphi : s \to t\) is a Friedman embedding subject to gap conditions if for all \(x \in s\), \(y \in t\) with \(\varphi (x) \geq y\) we have \(g_t (y) \geq g_s (\inf \{x : \varphi (x) \geq y\})\). Here the author considers the class of all Friedman trees partially ordered by Friedman embedding subject to the gap condition. He shows that Friedman trees are quasi-wellordered with respect to Friedman embedding. This generalizes a result of the author for finite trees to the infinite case [``Well-quasi-ordering finite trees with gap-condition. Proof of Harvey Friedman's conjecture'', Ann. Math., II. Ser. 130, 215-226 (1989; Zbl 0684.05016)].