Ordinal collapsing function: Difference between revisions
→Quantifier complexity
(→List) |
|||
Line 15:
== Remarks ==
Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref>
== Use of nonrecursive countable ordinals ==
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.<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>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.
== Quantifier complexity ==
| |||