is equal to the order type. Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). 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 a notation "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: well-ordered sets of a particular cardinality can have many different order types (see , below, for an example). For a countably infinite set, the set of possible order types is uncountable. Examples and counterexamples Natural numbers The standard ordering ≤ of the natural numbers is a well ordering and has the additional property that every non-zero natural number has a unique predecessor. Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds: 0 2 4 6 8 ... 1 3 5 7 9 ... This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1. Integers Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds: x = 0 x is positive, and y is negative x and y are both positive, and x ≤ y x and y are both negative, and |x| ≤ |y| This relation R can be visualized as follows: 0 1 2 3 4 ... −1 −2 −3 ... R is isomorphic to the ordinal number ω + ω. Another relation for well ordering the integers is the following definition: x ≤z y if and only if (|x| < |y| or (|x| =