psi function is an ordinal collapsing function developed by Michael Rathjen.
It collapses weakly Mahlo cardinals
to generate large countable ordinals.
(i.e. all normal functions closed in
are closed under some regular ordinal
Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.
It admits an associated ordinal notation
whose limit (i.e. ordinal type) is
, which is strictly greater than both
and the limit of countable ordinals expressed by Rathjen's
, which is called the "Small Rathjen ordinal" is the proof-theoretic ordinal of
, Kripke–Platek set theory augmented by the axiom schema "for any
, there exists an addmissible set
to uncountable regular cardinals
is a limit point of
{\displaystyle X\cup \{\alpha Lastly, an ordinal is said to be to be strongly critical if is a set of strongly critical ordinals explicitly defined in the original source. : Rathjen originally defined the function in more complicated a way in order to create an ordinal notation associated to it. Therefore, it is not certain whether the simplified OCF above yields an ordinal notation or not. functions used in Rathjen's original OCF are also not so easy to understand, and differ from the and the simplification provided above are not the same OCF. This is partially because the former is known to admit an ordinal notation, while the latter isn't known to admit an ordinal notation. [citation needed] Rathjen's is often confounded with another of his OCFs which also uses the symbol , but they are distinct notions. The former one is a published OCF, while the latter one is just a function symbol in an ordinal notation associated to an unpublished OCF.