Rathjen's psi function

<p>In mathematics, Rathjen's 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  
 psi function is an <a href="/facts/Ordinal_collapsing_function/YEjT7QzV">ordinal collapsing function</a> developed by Michael Rathjen. It collapses <a href="/facts/Mahlo_cardinal/NbxaHMef">weakly Mahlo cardinals</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 to generate <a href="/facts/Large_countable_ordinal/u0Ea19nd">large countable ordinals</a>. A weakly Mahlo cardinal is a cardinal such that the set of <a href="/facts/Regular_cardinal/wrGvPz2c">regular cardinals</a> below 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is closed under 
  
    
      
        M
      
    
    {\displaystyle M}
  
 (i.e. all <a href="/facts/Normal_function/GCXHAu3H">normal functions</a> closed in 
  
    
      
        M
      
    
    {\displaystyle M}
  
 are closed under some regular ordinal 
  
    
      
        <
        M
      
    
    {\displaystyle <M}
  
). Rathjen uses this to diagonalise over the <a href="/facts/Inaccessible_cardinal/EUydsllM">weakly inaccessible</a> hierarchy.
</p><p>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.
</p>

Rathjen's psi function open-in-new

Rathjen's psi function