Well-Ordering Hereditarily Finite Sets

  • Level: Graduate 
  • Thread starter Dragonfall
  • Start date
  • Tags Tags
    Finite Sets
2 replies · 3K views · last reply
Dragonfall
Messages
1,023
Reaction score
5
Does there exist a recursive well-ordering of the hereditarily finite sets?
 

Discussion

Sure. At each step in the construction, only a finite number of sets are added; these can be ordered as desired (lexiographically, say).

The natural well-order is then ({}, {{}}, {{{}}}, {{}, {{}}}, ...)
 

Similar threads

Undergrad Applying the well-ordering principle to non-negative integers
  • matqkks
  • · Replies 3 ·
  • Set Theory, Logic, Probability, Statistics
Replies
3
Well-ordering property of ordinal subsets in theorem 1.4.3
  • Math Amateur
  • · Replies 1 ·
  • Set Theory, Logic, Probability, Statistics
Replies
1
Proof that every subset of {1,...,n} is finite using well-ordering
  • zenterix
  • · Replies 4 ·
  • Calculus and Beyond Homework Help
Replies
4
Is the Axiom of Choice Necessary to Well-Order Finite Sets?
  • Csharp
  • · Replies 3 ·
  • Set Theory, Logic, Probability, Statistics
Replies
3
Proving well ordering principle from Peano Axioms
  • issacnewton
  • · Replies 9 ·
  • Calculus and Beyond Homework Help
Replies
9
Graduate Can the Well-Ordering Theorem Prove the Axiom of Induction?
  • kostas230
  • · Replies 4 ·
  • Set Theory, Logic, Probability, Statistics
Replies
4
Proving Induction Principle using Well Ordering Principle of Natural Nos
  • solakis1
  • · Replies 18 ·
  • Set Theory, Logic, Probability, Statistics
Replies
18
Graduate Well-ordering in a first-order language
  • jgens
  • · Replies 2 ·
  • Set Theory, Logic, Probability, Statistics
Replies
2
Graduate Why Can't Well Ordering of Reals be Proven with Dedekind Cuts?
  • SW VandeCarr
  • · Replies 22 ·
  • Set Theory, Logic, Probability, Statistics
Replies
22
Graduate Why can't well-ordering be expressed in first-order logic?
  • Semo727
  • · Replies 8 ·
  • Set Theory, Logic, Probability, Statistics
Replies
8