In mathematics, Rathjen's ψ {\displaystyle \psi } psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals M {\displaystyle M} to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below M {\displaystyle M} is closed under M {\displaystyle M} (i.e. all normal functions closed in M {\displaystyle M} are closed under some regular ordinal < M {\displaystyle <M} ). Rathjen uses this to diagonalise over the weakly inaccessible hierarchy. It admits an associated ordinal notation T ( M ) {\displaystyle T(M)} whose limit (i.e. ordinal type) is ψ Ω ( χ ε M + 1 ( 0 ) ) {\displaystyle \psi _{\Omega }(\chi _{\varepsilon _{M}+1}(0))} , which is strictly greater than both | K P M | {\displaystyle \vert KPM\vert } and the limit of countable ordinals expressed by Rathjen's ψ {\displaystyle \psi } . | K P M | {\displaystyle \vert KPM\vert } , which is called the "Small Rathjen ordinal" is the proof-theoretic ordinal of K P M {\displaystyle {\mathsf {KPM}}} , Kripke–Platek set theory augmented by the axiom schema "for any Δ 0 {\displaystyle \Delta _{0}} -formula H ( x , y ) {\displaystyle H(x,y)} satisfying ∀ x ∃ y ( H ( x , y ) ) {\displaystyle \forall x\,\exists y\,(H(x,y))} , there exists an addmissible set z {\displaystyle z} satisfying ∀ x ∈ z ∃ y ( H ( x , y ) ) {\displaystyle \forall x\in z\,\exists y\,(H(x,y))} ". It is equal to ψ Ω ( ψ χ ε M + 1 ( 0 ) ( 0 ) ) {\displaystyle \psi _{\Omega }(\psi _{\chi _{\varepsilon _{M}+1}(0)}(0))} in Rathjen's ψ {\displaystyle \psi } function.
Definition Restrict π {\displaystyle \pi } and κ {\displaystyle \kappa } to uncountable regular cardinals < M {\displaystyle <M} ; for a function f {\displaystyle f} let dom ( f ) {\displaystyle \operatorname {dom} (f)} denote the domain of f {\displaystyle f} ; let cl M ( X ) {\displaystyle \operatorname {cl} _{M}(X)} denote X ∪ { α < M : α is a limit point of X } {\displaystyle X\cup \{\alpha <M:\alpha {\text{ is a limit point of }}X\}} , and let enum ( X ) {\displaystyle \operatorname {enum} (X)} denote the enumeration of X {\displaystyle X} . Lastly, an ordinal α {\displaystyle \alpha } is said to be to be strongly critical if φ α ( 0 ) = α {\displaystyle \varphi _{\alpha }(0)=\alpha } . For α ∈ Γ M + 1 {\displaystyle \alpha \in \Gamma _{M+1}} and β ∈ M {\displaystyle \beta \in M} :
… excerpt ends here. Continue reading the full article.
