Ordinal collapsing function: Difference between revisions
→Use of nonrecursive countable ordinals
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It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref>
The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023.</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" /><sup>p.5</sup>
== Quantifier complexity ==
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