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Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set of natural numbers has an infimum, say .
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set or woset. [1]
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are ...
The positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b. The set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if s is a proper substring of t. The set N × N of pairs of natural numbers, ordered by (n 1, n 2) < (m 1, m 2) if and only if n 1 < m 1 ...
The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets.
Three well-orderings on the set of natural numbers with distinct order types (top to bottom): , +, and +. Every well-ordered set is order-equivalent to exactly one ordinal number , by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified ...
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The Cartesian product, S × T, of two well-ordered sets S and T can be well-ordered by a variant of lexicographical order that puts the least significant position first. Effectively, each element of T is replaced by a disjoint copy of S. The order-type of the Cartesian product is the ordinal that results from multiplying the order-types of S and T.