well-ordering property

mathematics

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July 15, 2026

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The well-ordering property, also known as the well-ordering principle, is a fundamental concept in mathematics, particularly in set theory. It states that every non-empty set of positive integers can be arranged in a specific order such that every subset has a least element. This property is crucial for proving theorems using transfinite induction, a generalization of mathematical induction.12

The axiom of choice is closely related to the well-ordering property. It asserts that for any collection of non-empty sets, it's possible to choose exactly one element from each set. This axiom is essential for proving that any set can be well-ordered. While some mathematicians have debated its use due to its non-constructive nature, it is now widely accepted as an indispensable tool in mathematics.1

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axiom of choice
…order to prove the “well-ordering theorem” (every set can be given an order relationship, such as less than, under which it is well ordered; i.e., every subset has a first element [see set theory: Axioms for infinite and ordered sets]). Subsequently, it was shown that making any one of…
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…order to prove the “well-ordering theorem” (every set can be given an order relationship, such as less than, under which it is well ordered; i.e., every subset has a first element [see set theory: Axioms for infinite and ordered sets]). Subsequently, it was shown that making any one of…
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continuum hypothesis
…cardinal numbers of larger “well-orderable sets” are ℵ₁, ℵ₂, …, ℵ_α, …, indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2^ℵ₀; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.
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…cardinal numbers of larger “well-orderable sets” are ℵ₁, ℵ₂, …, ℵ_α, …, indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2^ℵ₀; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.
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