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How to develop Proof‐Theoretic Ordinal Functions on the basis of admissible ordinals
Author(s) -
Rathjen Michael
Publication year - 1993
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19930390107
Subject(s) - uncountable set , mathematics , mathematical proof , basis (linear algebra) , notation , algebra over a field , calculus (dental) , discrete mathematics , pure mathematics , arithmetic , medicine , geometry , dentistry , countable set
In ordinal analysis of impredicative theories so‐called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly on admissible sets also renders a new and perspicuous approach to well‐ordering proofs possible. MSC: 03F15, 03F35.