A term's definition may require additional properties that are not listed in this table.
The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible.
Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set.
The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered.
If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.
Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set.
The position of each element within the ordered set is also given by an ordinal number.
Thus for finite n, the expression "n-th element" of a well-ordered set requires context to know whether this counts from zero or one.
In an expression "β-th element" where β can also be an infinite ordinal, it will typically count from zero.
For an infinite set, the order type determines the cardinality, but not conversely: sets of a particular infinite cardinality can have many different order types (see § Natural numbers, below, for an example).
The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds: This relation R can be visualized as follows: R is isomorphic to the ordinal number ω + ω.
Also Wacław Sierpiński proved that ZF + GCH (the generalized continuum hypothesis) imply the axiom of choice and hence a well order of the reals.
Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.
[3] However it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard ≤.
A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal to ω1 (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type.