Rathjen's psi function
In mathematics, Rathjen's  psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals  to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below  is closed under  (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 -formula  satisfying , there exists an addmissible set  satisfying ". It is equal to  in Rathjen's  function. Definition 
Restrict  and  to uncountable regular cardinals ; for a function  let  denote the domain of ; let  denote , and let  denote the enumeration of . Lastly, an ordinal  is said to be to be strongly critical if . For  and :

 

If  for some , define  using the unique . Otherwise if  for some , then define  using the unique , where  is a set of strongly critical ordinals  explicitly defined in the original source. For :

Explanation 

 Restrict  to uncountable regular cardinals. is a unique increasing function such that the range of  is exactly . is the closure of , i.e. , where  denotes the class of non-zero limit ordinals. 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. The original  functions used in Rathjen's original OCF are also not so easy to understand, and differ from the  functions defined above. Rathjen's  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  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former one is a published OCF, while the latter one is just a function symbol in an ordinal notation associated to an unpublished OCF. References

Mathematical logic
Ordinal numbers
Cardinal numbers