In mathematics, the well-ordering principle, also called the well-ordering property or least natural number principle, states that every non-empty subset of the nonnegative integers contains a least element, also called a smallest element. In other words, if ${\displaystyle A}$ is a nonempty subset of the nonnegative integers, then there exists an element of ${\displaystyle A}$ which is less than, or equal to, any other element of ${\displaystyle A}$. Formally, ${\displaystyle \forall A\left[\left(A\subseteq \mathbb {Z} _{\geq 0}\land A\neq \varnothing \right)\rightarrow \left(\exists m\in A\,\forall a\in A\,(m\leq a)\right)\right]}$. Most sources state this as an axiom or theorem about the natural numbers, but the phrase "natural number" was avoided here due to ambiguity over the inclusion of zero. The statement is true about the set of natural numbers ${\displaystyle \mathbb {N} }$ regardless whether it is defined as ${\displaystyle \mathbb {Z} _{\geq 0}}$ (nonnegative integers) or as ${\displaystyle \mathbb {Z} ^{+}}$ (positive integers), since one of Peano's axioms for ${\displaystyle \mathbb {N} }$, the induction axiom (or principle of mathematical induction), is logically equivalent to the well-ordering principle. Since ${\displaystyle \mathbb {Z} ^{+}\subseteq \mathbb {Z} _{\geq 0}}$ and the subset relation ${\displaystyle \subseteq }$ is transitive, the statement about ${\displaystyle \mathbb {Z} ^{+}}$ is implied by the statement about ${\displaystyle \mathbb {Z} _{\geq 0}}$.

Experience with numbers favors this principle. For instance, the set T = {5, 8, 3, 11} has 3 as its least element, and 2 is the least element in the set of even positive numbers. It is a deceptively obvious principle because in many cases it is not clear what the least number actually is.

The standard order on ${\displaystyle \mathbb {N} }$ is well-ordered by the well-ordering principle, since it begins with a least element, regardless whether it is 1 or 0. By contrast, the standard order on ${\displaystyle \mathbb {R} }$ (or on ${\displaystyle \mathbb {Z} }$) is not well-ordered by this principle, since there is no smallest negative number. According to Deaconu and Pfaff, the phrase "well-ordering principle" is used by some (unnamed) authors as a name for Zermelo's "well-ordering theorem" in set theory, according to which every set can be well-ordered. This theorem, which is not the subject of this article, implies that "in principle there is some other order on ${\displaystyle \mathbb {R} }$ which is well-ordered, though there does not appear to be a concrete description of such an order."

The well-ordering principle is logically equivalent to the principle of mathematical induction, according to which ${\displaystyle \forall n\in \mathbb {Z} _{\geq 0}\left[\left(P(0)\land \left(\forall k\in \mathbb {Z} _{\geq 0}[P(k)\rightarrow P(k+1)]\right)\right]\rightarrow P(n)\right]}$. In other words, if one takes the principle of mathematical induction as an axiom, one can prove the well-ordering principle as a theorem (as done in ), and conversely, if one takes the well-ordering principle as an axiom, one can prove the principle of mathematical induction as a theorem (as done in ). The former is more common due to tradition, since the principle of mathematical induction was one of Peano's axioms for the natural numbers, and Peano was an influential mathematician.

The principle of mathematical induction and the well-ordering principle are each also equivalent to the principle of strong induction (also called the principle of complete induction), according to which ${\displaystyle \left[\left(P(0)\land \forall k\left(\left(\forall j\,(0\leq j\leq k)\rightarrow P(j)\right)\rightarrow P(k+1)\right)\right)\right]\rightarrow \forall n\in \mathbb {Z} _{\geq 0},\,P(n)}$. Accordingly, one can also use the principle of strong induction as an axiom to prove the well-ordering principle as a theorem (as done in ), or take the well-ordering principle as an axiom to prove the principle of strong induction as a theorem (as in ).

This also means that, in axiomatic set theory, the definition of the natural numbers as the smallest inductive set, ${\displaystyle \mathbb {N} =\{x\in S\mid 0\in S\land \forall n\in S,\;n+1\in S\}}$, is equivalent to the statement that the well-ordering principle is true for it.

Although the equivalence between induction and well-ordering is a common result, Lars-Daniel Öhman has argued that "proofs" of induction based on well-ordering silently assume that all nonzero naturals have a unique immediate predecessor, which does not follow from the noninductive Peano axioms and the well-ordering principle; in fact, the set of ordinal numbers less than ω+ω serves as a countermodel. Hence, induction is stronger than well-ordering vis-à-vis the Peano axioms.

If one knows, as an axiom or theorem, that the real numbers are complete, then one can use this to prove the well-ordering principle for nonnegative integers. This is because the completeness property implies that every bounded-from-below subset of ${\displaystyle \mathbb {R} }$ has an infimum, which means that, since ${\displaystyle \mathbb {Z} _{\geq 0}}$ is a bounded-from-below subset of ${\displaystyle \mathbb {R} }$ (and the subset relation ${\displaystyle \subseteq }$ is transitive), then also every set ${\displaystyle A\subseteq \mathbb {Z} _{\geq 0}}$ has an infimum ${\displaystyle a}$, which implies that there exists an integer ${\displaystyle n}$ such that ${\displaystyle a}$ lies in the half-open interval ${\displaystyle (n-1,n]}$, which implies that ${\displaystyle a=n}$ and ${\displaystyle n\in A}$.