Normal forms for elementary patterns

Journal of Symbolic Logic 77 (1):174-194 (2012)
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Abstract

A notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. This provides a capstone to the results in [2, 6, 8, 9, 7], using further refinement of ordinal arithmetic developed in [8] which then both allows for a simple characterization of normal forms for patterns of order one and will play a key role in the arithmetical analysis of pure patterns of order two, [5]

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Citations of this work

Pure Σ2-elementarity beyond the core.Gunnar Wilken - 2021 - Annals of Pure and Applied Logic 172 (9):103001.
Pure patterns of order 2.Gunnar Wilken - 2018 - Annals of Pure and Applied Logic 169 (1):54-82.
Tracking chains of Σ 2 -elementarity.Timothy J. Carlson & Gunnar Wilken - 2012 - Annals of Pure and Applied Logic 163 (1):23-67.
Arithmetical algorithms for elementary patterns.Samuel A. Alexander - 2015 - Archive for Mathematical Logic 54 (1-2):113-132.

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References found in this work

Ordinal arithmetic and $\Sigma_{1}$ -elementarity.Timothy J. Carlson - 1999 - Archive for Mathematical Logic 38 (7):449-460.
Elementary patterns of resemblance.Timothy J. Carlson - 2001 - Annals of Pure and Applied Logic 108 (1-3):19-77.
Patterns of resemblance of order 2.Timothy J. Carlson - 2009 - Annals of Pure and Applied Logic 158 (1-2):90-124.
The Bachmann-Howard Structure in Terms of Σ1-Elementarity.Gunnar Wilken - 2006 - Archive for Mathematical Logic 45 (7):807-829.

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