Pure patterns of order 2

Abstract

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order $2$, showing that the latter characterize the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order $2$ implies transfinite induction up to the ordinal of $\mathrm{KPl}_0$. We expect that our approach will facilitate analysis of more powerful systems of patterns.Comment: corrected Theorem 4.2 with according changes in section 3 (mainly Definition 3.3), results unchanged. The manuscript was edited, aligned with reference [14] (moving former Lemma 3.5 there), and argumentation was revised, with minor corrections in (the proof of) Theorem 4.2; results unchanged. Updated revised preprint; to appear in the APAL (2017

Tracking Chains Revisited

“…[2], over the language (≤, ≤ 1 , ≤ 2 ). In a subsequent article [11] we will establish the converse by providing an effective assignment of isominimal realizations in C 2 to arbitrary respecting forests. We will provide an algorithm to find pattern notations for the ordinals in C 2 , and will conclude that the union of isominimal realizations of respecting forests is indeed the core of R 2 , i.e.…”

“…The partial orderings ≤1 and ≤2 denote the relations of Σ1and Σ2-elementary substructure, respectively. In a subsequent article [11] we will show that the structure C2 comprises the core of the structure R2 of pure elementary patterns of resemblance of order 2. In [5] the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order 2 is 1 ∞ .…”

“…With these preparations out of the way we will be able to provide, in a subsequent article [11], an algorithm that assigns an isominimal realization within C 2 to each respecting forest of order 2, thereby showing that each such respecting forest is in fact (up to isomorphism) a pure pattern of order 2. The approach is to formulate the corresponding theorem flexibly so that isominimal realizations above certain relativizing tracking chains are considered.…”

“…The mutual order isomorphisms between hull and pattern notations that will be given in [11] enable classification of a new independence result for KPℓ 0 : We will demonstrate that the result by Carlson in [3], according to which the collection of respecting forests of order 2 is well-quasi ordered with respect to coverings, cannot be proven in KPℓ 0 or, equivalently, in the restriction Π 1 1 −CA 0 of second order number theory to Π 1 1 -comprehension and set induction. On the other hand, we know that transfinite induction up to the ordinal 1 ∞ of KPℓ 0 suffices to show that every pattern is covered [5].…”

Patterns of resemblance and Bachmann-Howard fixed points

“…They also offer a new approach to the large computable ordinals considered in ordinal analysis, including a conjectured characterization of the prooftheoretic ordinal of full second order arithmetic (see [5,Section 13]). This viewpoint has been clarified and expanded in subsequent work of G. Wilken (see [25][26][27][28] and the joint paper [6] with Carlson). Furthermore, Carlson has pointed out similarities with the core models studied in set theory, and expressed the hope that proof-theoretic and settheoretic approaches "will find common ground someday" (see again [5,Section 13]).…”