Normal forms for elementary patterns. (English) Zbl 1275.03154

In this paper the authors develop an ordinal notation system for the proof-theoretic ordinal of \(\mathrm{KP}\l_0\) based on patterns of resemblance of order one. A pattern of resemblance of order one (henceforth: pattern) is a finite structure in the language \((0,+,\leq,\leq_1)\) that is isomorphic to a substructure of \(R_1=(\mathrm{ORD},0,+,\leq,\leq_1)\) where \(\leq_1\) is defined inductively by specifying that \(a \leq_1 b\) iff \((a,0,+,\leq,\leq_1)\) is a \(\Sigma_1\) elementary substructure of \((b,0,+,\leq,\leq_1)\).
Patterns provide notations for ordinals in the following way. To a given pattern one associates the unique substructure \(X\) of \(R_1\) that is minimal in the pointwise ordering among all substructures of \(R_1\) isomorphic to \(P\) (such a structure is called isominimal). The pair \((P,a)\) where \(a\in P\) is then a notation for the corresponding ordinal in \(X\). Notations of this form are not unique as a given ordinal may appear in many different isominimal sets. The central result of the article is that amongst all isominimal structures containing a given ordinal there exists a minimal such one (Theorem 4.6). Moreover, the minimal candidate can be computed and thus a notation systems for all notatable ordinals constructed.
The resulting notation system plays an important role in the analysis of patterns of higher order; see [the authors, Ann. Pure Appl. Logic 163, No. 1, 23–67 (2012; Zbl 1273.03159)]. Although the article assumes familiarity with the subject, the authors do provide an accessible summary of the important definitions and theorems necessary for the results up to the uniqueness of minimal isominimal sets (Section 4). The remainder of the paper is devoted to computing the sets and refers heavily to [the second author, J. Symb. Log. 72, No. 2, 704–720 (2007; Zbl 1121.03081)].


03F15 Recursive ordinals and ordinal notations
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