properties of ranks of sets

A set A is said to be grounded, if AVα in the cumulative hierarchy for some ordinalMathworldPlanetmathPlanetmath α. The smallest such α such that AVα is called the rank of A, and is denoted by ρ(A).

In this entry, we list derive some basic properties of groundedness and ranks of sets. Proofs of these properties require an understanding of some of the basic properties of ordinals.

  1. 1.

    is grounded, whose rank is itself. This is obvious.

  2. 2.

    If A is grounded, so is every xA, and ρ(x)<ρ(A).


    AVρ(A), so xVρ(A), which means xVβ for some β<ρ(A). This shows that x is grounded. Then ρ(x)β, and hence ρ(x)<ρ(A). ∎

  3. 3.

    If every xA is grounded, so is A, and ρ(A)=sup{ρ(x)+xA}.


    Let B={ρ(x)+xA}. Then B is a set of ordinals, so that β:=B=supB is an ordinal. Since each xVρ(x)+, we have xVβ. So AVβ, showing that A is grounded. If α<β, then for some xA, α<ρ(x)+, which means xVα, and therefore AVα. This shows that ρ(A)=β. ∎

  4. 4.

    If A is grounded, so is {A}, and ρ({A})=ρ(A)+. This is a direct consequence of the previous result.

  5. 5.

    If A,B are grounded, so is AB, and ρ(AB)=max(ρ(A),ρ(B)).


    Since A,B are grounded, every element of AB is grounded by property 2, so that AB is also grounded by property 3. Then ρ(AB)=sup{ρ(x)+xAB}=max(sup{ρ(x)+xA},sup{ρ(x)+xB})=max(ρ(A),ρ(B)). ∎

  6. 6.

    If A is grounded, so is BA, and ρ(B)ρ(A).


    Every element of B, as an element of the grounded set A, is grounded, and therefore B is grounded. So ρ(B)=sup{ρ(x)+xB}sup{ρ(x)+xA}=ρ(A). Since ρ(B) and ρ(A) are both ordinals, ρ(B)ρ(A). ∎

  7. 7.

    If A is grounded, so is P(A), and ρ(P(A))=ρ(A)+.


    Every subset of A is grounded, since A is by property 6. So P(A) is grounded. Furthermore, P(A)=sup{ρ(x)+xP(A)}. Since ρ(B)ρ(A) for any BP(A), and AP(A), we have P(A)=ρ(A)+ as a result. ∎

  8. 8.

    If A is grounded, so is A, and ρ(A)=sup{ρ(x)xA}.


    Since A is grounded, every xA is grounded. Let B={ρ(x)xA}. Then β:=B=supB is an ordinal. Since ρ(x)β, Vρ(x)=Vβ or Vρ(x)Vβ. In either case, Vρ(x)Vβ, since Vα is a transitive set for any ordinal α. Since xVρ(x), xVβ for every xA. This means AVβ, showing that A is grounded. If α<β, then α<ρ(x) for some ρ(x)β, which means xVα, or AVα as a result. Therefore ρ(A)=β. ∎

  9. 9.

    Every ordinal is grounded, whose rank is itself.


    If α=0, then apply property 1. If α is a successor ordinal, apply properties 4 and 5, so that ρ(α)=ρ(β+)=ρ(β{β})=max(ρ(β),ρ({β}))=max(ρ(β),ρ(β)+)=ρ(β)+. If α is a limit ordinal, then apply property 8 and transfinite inductionMathworldPlanetmath, so that ρ(α)=ρ(α)=sup{ρ(β)β<α}=sup{ββ<α}=α. ∎


  • 1 H. Enderton, Elements of Set TheoryMathworldPlanetmath, Academic Press, Orlando, FL (1977).
  • 2 A. Levy, Basic Set Theory, Dover Publications Inc., (2002).
Title properties of ranks of sets
Canonical name PropertiesOfRanksOfSets
Date of creation 2013-03-22 18:50:31
Last modified on 2013-03-22 18:50:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Derivation
Classification msc 03E99
Defines grounded
Defines grounded set