Von Neumann cardinal assignment

The <a href="/facts/John_von_Neumann/aUk6urof">von Neumann</a> cardinal assignment is a <a href="/facts/Cardinal_assignment/NFNyIewW">cardinal assignment</a> that uses <a href="/facts/Ordinal_number/GelFLjJt">ordinal numbers</a>. For a <a href="/facts/Well-order/GrWEBWlW">well-orderable</a> set U, we define its <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a> to be the smallest ordinal number <a href="/facts/Equinumerosity/7hMvasXx">equinumerous</a> to U, using the von Neumann definition of an ordinal number. More precisely:

|
        
        U
        
          |
        
        =
        
          c
          a
          r
          d
        
        (
        U
        )
        =
        inf
        {
        α
        ∈
        
          O
          N
        
         
        
          |
        
         
        α
        
          =
          
            c
          
        
        U
        }
        ,
      
    
    {\displaystyle |U|=\mathrm {card} (U)=\inf\{\alpha \in \mathrm {ON} \ |\ \alpha =_{c}U\},}

where ON is the <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of ordinals. This ordinal is also called the initial ordinal of the cardinal.
That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the <a href="/facts/Axiom_schema_of_replacement/dmJn0BaS">axiom of replacement</a>. With the full <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, <a href="/facts/Well-ordering_theorem/teE1cYkm">every set is well-orderable</a>, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤c. This is a well-ordering of cardinal numbers.

Von Neumann cardinal assignment open-in-new

Von Neumann cardinal assignment