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**1 - 7**of**7**### A Gap Tree Theorem for Quasi-Ordered Labels 1

"... Given a quasi-ordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greater-or-equivalent labels, and which preserves the order of children. We sh ..."

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Given a quasi-ordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greater-or-equivalent labels, and which preserves the order of children. We show that finite trees are well-quasiordered with respect to gap embedding when labels are taken from an arbitrary well-quasi-ordering provided every tree path can be partitioned into a bounded number of subpaths of comparable nodes. This extends Kˇríˇz’s combinatorial result [7] to partially ordered labels.

### Strong WQO phase transitions Preliminary report 1

"... We elaborate Weiermann-style phase transitions for Gordeev well-quasi-ordering (called well-partial-ordering below, abbr.: wpo) results with respect to nested finite sequences and nested finite trees under the homeomorphic embedding with symmetrical gap condition. For every nested partial ordering i ..."

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We elaborate Weiermann-style phase transitions for Gordeev well-quasi-ordering (called well-partial-ordering below, abbr.: wpo) results with respect to nested finite sequences and nested finite trees under the homeomorphic embedding with symmetrical gap condition. For every nested partial ordering in question, �, we fix a natural extension of Peano Arithmetic, T, that proves the corresponding 2-order sentence SPQ(�). Furthermore, we consider the appropriate parametrized 1-order slow well-partial-ordering sentence SWPO (�,r) with r ranging over computable reals and show that for some computable real α, the following holds. 1. If r < α then SWPO (�,r) is provable in PA. 2. If r> α then SWPO (�,r) is not provable in T. In the limit cases we replace computable reals r by computable functions f: N → R and prove analogous theorems. In whole generality, these results strengthen both Kruskal-Friedman-Kriz wqo theorems and Weiermann phase transition results concerning Kruskal-Friedman-Schütte-Simpson cases. 2 Preliminaries (1-D case) 2.1 Partial and linear well orderings • By � and ≤ we denote partial and linear countable well orderings (abbr.: wpo and wo), respectively. A wo O = (W, ≤) is called a linearization of a wpo W = (W,�) iff (∀x,y ∈ W) (x � y → x ≤ y). A wpo W = (W,�) is called enumerated iff it is supplied with a bijection, also called enumeration, ν: N → W. For any enumerated wpo W = (W,ν,�) we fix its lexicographical linearization Wν = (W, ≤ν) that is defined as follows W × W ∋ x ≤ν y: ⇔ (∀i ∈ N) (ν (i) � x ← → ν (i) � y) ∨ (∃i ∈ N)

### www.math.ohio-state.edu/~friedman/ TABLE OF CONTENTS

"... this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem ..."

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this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem of ZFC; in fact, ZF does not prove the existence of a standard model of Zermelo set theory with the axiom of choice. Thus in this paper, we relate our axiomatizations to extensions of ZF by large cardinal axioms. In each case, we have chosen an appropriate version of the large cardinal axiom so that if ZF is replaced by ZFC then the resulting system is equivalent to a system which is familiar in the set theory literature. But one would like to know the relationship between the system with ZF and the system with ZFC. This relationship cannot be gauged by considering standard models. {PAGE } The normal way of gauging this relationship is through

### Kruskal-Friedman Gap . . .

, 2002

"... We investigate new extensions of the Kruskal-Friedman theorems concerning well-quasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique ..."

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We investigate new extensions of the Kruskal-Friedman theorems concerning well-quasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique path in t with greater-or-equal labels. We show that finite trees are well-quasi-ordered with respect to the gap embedding when the labels are taken from an arbitrary well-quasi-ordering and each tree path can be partitioned into k ∈ N or less comparable sub-paths. This result generalizes both [Kˇrí89] and [OT87], and is also optimal in the sense that unbounded partiality over tree paths yields a counter example.

### Phase transitions in Proof Theory

"... Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program. ..."

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Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program.