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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.
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, 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 usual order of rational numbers (Cannot be explicitly written as an ordered list!) In both examples of well orders here, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element. This is the key definition that determines whether a total order is also a well order.
That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice , every set is well-orderable , so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers.
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.
This is a well-ordering and hence gives an ordinal number. The definition of exponentiation can also be given by transfinite recursion on the exponent β. When the exponent β = 0, ordinary exponentiation gives α 0 = 1 for any α. For β > 0, the value of α β is the smallest ordinal greater than or equal to α δ · α for all δ < β ...
For example, being well-ordered is a hereditary property, and so it being finite. [ 12 ] If we instantiate in the above schema Φ ( x ) {\displaystyle \Phi (x)} with " x has cardinality less than κ", we obtain the more general notion of a set being hereditarily of cardinality less than κ, usually denoted by H κ {\displaystyle H_{\kappa ...