section_path: Generalizations
section_name: Generalizations
paragraph_text: Generalizations
  
  Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.
  
  The least ordinal of cardinality ℵ (that is, the initial ordinal of ℵ) is ω but many well-ordered sets with cardinal number ℵ have an ordinal number greater than ω.
  
  For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.
  
  A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction. Other generalizations are discussed in Number § Extensions of the concept.
  
  Georges Reeb used to claim provocatively that "The naïve integers don't fill up N {\displaystyle \mathbb {N} } ".