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  1.  7 DLs
    Gunnar Wilken (2007). Assignment of Ordinals to Patterns of Resemblance. Journal of Symbolic Logic 72 (2):704 - 720.
    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-elementarity 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 (...)
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  2.  6 DLs
    Andreas Weiermann & Gunnar Wilken (2011). Ordinal Arithmetic with Simultaneously Defined Theta‐Functions. Mathematical Logic Quarterly 57 (2):116-132.
    This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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  3.  2 DLs
    Gunnar Wilken (2007). Ordinal Arithmetic Based on Skolem Hulling. Annals of Pure and Applied Logic 145 (2):130-161.
    Taking up ordinal notations derived from Skolem hull operators familiar in the field of infinitary proof theory we develop a toolkit of ordinal arithmetic that generally applies whenever ordinal structures are analyzed whose combinatorial complexity does not exceed the strength of the system of set theory. The original purpose of doing so was inspired by the analysis of ordinal structures based on elementarity invented by T.J. Carlson, see [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (...)
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  4.  1 DLs
    Timothy J. Carlson & Gunnar Wilken (2012). Normal Forms for Elementary Patterns. Journal of Symbolic Logic 77 (1):174-194.
    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. (...)
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  5.  1 DLs
    Gunnar Wilken (2006). The Bachmann-Howard Structure in Terms of Σ1-Elementarity. Archive for Mathematical Logic 45 (7):807-829.
    The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.
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  6.  1 DLs
    Timothy J. Carlson & Gunnar Wilken (2012). Tracking Chains of Σ2-Elementarity. Annals of Pure and Applied Logic 163 (1):23-67.
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  7.  1 DLs
    Gunnar Wilken (2007). Σ1-Elementarity and Skolem Hull Operators. Annals of Pure and Applied Logic 145 (2):162-175.
    The exact correspondence between ordinal notations derived from Skolem hull operators, which are classical in ordinal analysis, and descriptions of ordinals in terms of Σ1-elementarity, an approach developed by T.J. Carlson, is analyzed in full detail. The ordinal arithmetical tools needed for this purpose were developed in [G. Wilken, Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic 145 130–161]. We show that the least ordinal κ such that κ<1∞ 19–77] and described below) is the proof theoretic (...)
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  8.  0 DLs
    Andreas Weiermann & Gunnar Wilken (2013). Goodstein Sequences for Prominent Ordinals Up to the Ordinal Of. Annals of Pure and Applied Logic 164 (12):1493-1506.
    We introduce strong Goodstein principles which are true but unprovable in strong impredicative theories like IDn.
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