Strong WQO Tree Theorems

Abstract

Ordinal (vertex- and/or edge-) labeled finite trees are well-quasi-ordered by homeomorphic embeddability with sound gap-conditions. Such strong generalizations of Harvey Friedman’s tree theorem (abbr.: FT) on trees whose vertices are labeled by bounded natural numbers are (a) provable in second-order arithmetic $\Pi _{1}^{1}$-TR$_{0}$ (also designated ITR$_{0}$ below) that extends ACA$_{0}$ by transfinite iteration of $\Pi _{1}^{1}$-comprehension along arbitrary countable ordinals but (b) not provable in a subsystem thereof that arises by weakening $\Pi _{1}^{1}$-transfinite recursion axiom to $\Pi _{1}^{1}$-transfinite recursion rule. In particular, I. Křiž’s tree theorem (abbr.: KřT) referring to ordinal edge-labeled trees [9] is provable in $\Pi _{1}^{1}$-TR$_{0}$ (that is weaker than theory $\Pi _{2}^{1}$-CA implicitly used in [9]), which is the main result of the paper. Moreover KřT is proof-theoretically equivalent to the author’s analogous theorem (abbr.: GT) referring to ordinal vertex-labeled trees under symmetric gap-condition [6]. Namely, both theorems characterize ITR$_{0}$ in the sense of ordinal provability over ACA$_{0}$. That is, the supremum of proof-theoretic ordinals provable in ACA$_{0}$ extended by GT and/or KřT is the proof-theoretic ordinal of ITR$_{0}$ [in symbols: $\left| \mathbf {ACA}_{0}+GT\right| =\left| \mathbf {ACA}_{0}+K\check{r}T\right| =\left| \Pi _{1}^{1}\text {-}\mathbf {TR}_{0}\right| =\psi 0\left( \Phi 10\right)$ (see [12] for the last equality)]. By contrast, the restricted GT and KřT referring to ordinal labeled intervals (i.e. non-branching trees) both yield analogous characterizations of a weaker (predicative) theory ATR$_{0}$, instead of ITR$_{0}$ (cf. [5]).

Basic version of this paper was presented at 14th British Combinatorial Conference held in University of Keele, 5–9 July, 1993.