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Assignment of ordinals to patterns of resemblance

Published online by Cambridge University Press:  12 March 2014

Gunnar Wilken
Gunnar Wilken*
Affiliation:
Department of Mathematics, The Ohio State University, 130 Math Tower, 231 W 18th Avenue, Columbus, Ohio 43210, USA, E-mail: wilkeng@math.ohio-state.edu
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Abstract

In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ1-elementarily as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 704 - 720
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Carlson, T.J., Ordinal arithmetic and Σ1-elementarity, Archive for Mathematical Logic, vol. 38 (1999), pp. 449–460.CrossRefGoogle Scholar
[2]Carlson, T.J., Elementary patterns of resemblance, Annals of Pure and Applied Logic, vol. 108 (2001). pp. 19–77.CrossRefGoogle Scholar
[3]Pohlers, W., Proof theory. An introduction, Lecture Notes in Mathematics, vol. 1407, Springer-Verlag, 1989.Google Scholar
[4]Wilken, G., Σ1-elementarity and Skolem hull operators, Ph.D. thesis, University of Münster, 2004.Google Scholar
[5]Wilken, G., The Bachmann-Howard structure in terms of Σ1-elementarity, Archive for Mathematical Logic, vol. 45 (2006), pp. 807–829.CrossRefGoogle Scholar
[6]Wilken, G., Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic, vol. 145 (2007), no. 2, pp. 130–161.CrossRefGoogle Scholar
[7]Wilken, G., Σ1-elementarity and Skolem hull operators, Annals of Pure and Applied Logic, vol. 145 (2007), no. 2, pp. 162–175.CrossRefGoogle Scholar