Reverse mathematics and well-ordering principles: A pilot study

Annals of Pure and Applied Logic 160 (3):231-237 (2009)
Michael Rathjen
University of Leeds
The larger project broached here is to look at the generally sentence “if X is well-ordered then f is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory Tf whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we prove in this paper that the statement “if X is well-ordered then εX is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a fragment of.
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DOI 10.1016/j.apal.2009.01.001
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References found in this work BETA

Elementary Descent Recursion and Proof Theory.Harvey Friedman & Michael Sheard - 1995 - Annals of Pure and Applied Logic 71 (1):1-45.
Reverse Mathematics and Ordinal Exponentiation.Jeffry L. Hirst - 1994 - Annals of Pure and Applied Logic 66 (1):1-18.
Bar Induction and Ω Model Reflection.Gerhard Jäger & Thomas Strahm - 1999 - Annals of Pure and Applied Logic 97 (1-3):221-230.
The Role of Parameters in Bar Rule and Bar Induction.Michael Rathjen - 1991 - Journal of Symbolic Logic 56 (2):715-730.
Descending Sequences of Degrees.John Steel - 1975 - Journal of Symbolic Logic 40 (1):59-61.

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Citations of this work BETA

Reverse Mathematics: The Playground of Logic.Richard A. Shore - 2010 - Bulletin of Symbolic Logic 16 (3):378-402.
The Veblen Functions for Computability Theorists.Alberto Marcone & Antonio Montalbán - 2011 - Journal of Symbolic Logic 76 (2):575 - 602.
Derivatives of Normal Functions and $$\Omega $$ Ω -Models.Toshiyasu Arai - 2018 - Archive for Mathematical Logic 57 (5-6):649-664.

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