Analytical hierarchy

<p>In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a> and <a href="/facts/Descriptive_set_theory/ek0tvUXb">descriptive set theory</a>, the analytical hierarchy is an extension of the <a href="/facts/Arithmetical_hierarchy/rQptcBFK">arithmetical hierarchy</a>. The analytical hierarchy of formulas includes formulas in the language of <a href="/facts/Second-order_arithmetic/abBxQsaH">second-order arithmetic</a>, which can have quantifiers over both the set of <a href="/facts/Natural_numbers/u65og8JE">natural numbers</a>, 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
, and over functions from 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
 to 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the <a href="/facts/Lightface/k6fMNJa7">lightface</a> version of the <a href="/facts/Projective_hierarchy/Ip9qupJn">projective hierarchy</a>.
</p>

Analytical hierarchy open-in-new

Analytical hierarchy