In mathematics, Rathjen's formula_1 psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals formula_2 to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below formula_2 is closed under formula_2 (i.e. all normal functions closed in formula_2 are closed under some regular ordinal formula_6). Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.
It admits an associated ordinal notation formula_7 whose limit (i.e. ordinal type) is formula_8, which is strictly greater than both formula_9 and the limit of countable ordinals expressed by Rathjen's formula_1. formula_9, which is called the "Small Rathjen ordinal" is the proof-theoretic ordinal of formula_12, Kripke–Platek set theory augmented by the axiom schema "for any formula_13-formula formula_14 satisfying formula_15, there exists an addmissible set formula_16 satisfying formula_17". It is equal to formula_18 in Rathjen's formula_1 function.
Definition.
Restrict formula_20 and formula_21 to uncountable regular cardinals formula_6; for a function formula_23 let formula_24 denote the domain of formula_23; let formula_26 denote formula_27, and let formula_28 denote the enumeration of formula_29. Lastly, an ordinal formula_30 is said to be to be strongly critical if formula_31.
For formula_32 and formula_33:
 formula_34
If formula_35 for some formula_36, define formula_37 using the unique formula_38. Otherwise if formula_39 for some formula_32, then define formula_41 using the unique formula_30, where formula_43 is a set of strongly critical ordinals formula_6 explicitly defined in the original source.
For formula_32:
 formula_46
 formula_47
Explanation.
Rathjen originally defined the formula_1 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. The original formula_65 functions used in Rathjen's original OCF are also not so easy to understand, and differ from the formula_65 functions defined above.
Rathjen's formula_1 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. Rathjen's formula_1 is often confounded with another of his OCFs which also uses the symbol formula_1, 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.
References.
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