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  2. Well-order - Wikipedia

    en.wikipedia.org/wiki/Well-order

    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.

  3. Well-ordering principle - Wikipedia

    en.wikipedia.org/wiki/Well-ordering_principle

    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 .

  4. Order statistic - Wikipedia

    en.wikipedia.org/wiki/Order_statistic

    Order statistics have a lot of applications in areas as reliability theory, financial mathematics, survival analysis, epidemiology, sports, quality control, actuarial risk, etc. There is an extensive literature devoted to studies on applications of order statistics in these fields.

  5. Order type - Wikipedia

    en.wikipedia.org/wiki/Order_type

    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 with the corresponding ordinal. Order types thus often take the form of arithmetic expressions of ordinals.

  6. Von Neumann cardinal assignment - Wikipedia

    en.wikipedia.org/wiki/Von_Neumann_cardinal...

    Also, is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and is the order type of that set), is the smallest ordinal whose cardinality is greater than , and so on, and is the limit of for natural ...

  7. List of types of numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_types_of_numbers

    Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers : Finite and infinite numbers used to describe the cardinalities of sets . Infinitesimals : These are smaller than any positive real number, but are nonetheless greater than zero.

  8. Order theory - Wikipedia

    en.wikipedia.org/wiki/Order_theory

    A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these properties, one immediately sees that the well-known orders on natural numbers, integers, rational numbers and reals are all orders in the above sense.

  9. Well-ordering theorem - Wikipedia

    en.wikipedia.org/wiki/Well-ordering_theorem

    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 ...