Patterns of resemblance

The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in BMS version 4. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS.

A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The core is the set of ordinals which occur in an isominimal pattern. The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously. For all systems currently analyzed, the core is a recursive ordinal.^{implicit in section 3corollary 6.12}

A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").^p.23p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this)

Reflection criterion

Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns. (I think is a citation)

Stability

It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS.