
TL;DR
This paper proves that the structure ${ m{f C}}_2$, related to proof-theoretic ordinals and elementary substructure relations, is elementary recursive, clarifying its computability and foundational properties.
Contribution
It demonstrates that ${ m{f C}}_2$ is an elementary recursive structure, providing a detailed analysis of its connectivity components and building on ordinal arithmetic techniques.
Findings
${ m{f C}}_2$ is elementary recursive.
Disentanglement of connectivity components of $ ext{ m{f C}}_2$.
Enhanced ordinal arithmetic analysis for structure ${ m{f C}}_2$.
Abstract
The structure , introduced and first analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary recursive. Here, denotes the proof-theoretic ordinal of the fragment - of second order number theory, or equivalently the set theory , which axiomatizes limits of models of Kripke-Platek set theory with infinity. The partial orderings and denote the relations of - and -elementary substructure, respectively. In a subsequent article we will show that the structure comprises the core of the structure of pure elementary patterns of resemblance of order . In Carlson and Wilken 2012 (APAL) the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order is . However, it is not…
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