|y| and x ≤ y)). This well order can be visualized as follows: 0 −1 1 −2 2 −3 3 −4 4 ... This has the order type ω. Reals The standard ordering ≤ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals. 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. 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. An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well order: Suppose X is a subset of R well ordered by ≤. For each x in X, let s(x) be the successor of x in ≤ ordering on X (unless x is the last element of X). Let A = { (x, s(x)) | x ∈ X } whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an injective function from A to Q. There is an injection from X to A (except possibly for a last element of X which could be mapped to zero later). And it is well known that there is an injection from Q to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to the natural numbers which means that X is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard "≤". For example, The natural numbers are a well order under the standard ordering ≤. The set {1/n : n =1,2,3,...} has no least element and is therefore not a well order under standard ordering ≤. Examples of well orders: The set of numbers { − 2−n