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Normal forms for elementary patterns

Published online by Cambridge University Press:  12 March 2014

Timothy J. Carlson  and
Gunnar Wilken
Timothy J. Carlson
Affiliation:
Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA, E-mail: carlson@math.ohio-state.edu
Gunnar Wilken
Affiliation:
Mathematical Biology Unit, Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-Son, 904-0412 Okinawa, Japan, E-mail: wilken@oist.jp
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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].

Keywords

patterns of resemblanceordinal notationsordinal arithmeticproof theoryalgorithms
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 174 - 194
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

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