Reverse mathematics and well-ordering principles: A pilot study |
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Authors: | Bahareh Afshari Michael Rathjen |
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Affiliation: | aDepartment of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK |
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Abstract: | The larger project broached here is to look at the generally sentence “if X is well-ordered then f(X) 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 (deduction chains) [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of . |
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Keywords: | Reverse mathematics Well-ordering principles Schü tte deduction chains Countable coded ω -model |
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