Ordered well together

A together ordered ( E , ≤) is ordered well and the relation ≤ is a good order if the following condition is satisfied:

Toute nonempty part of E has a Plus small element.

It is shown that any well ordered unit is completely ordered. Indeed, that is to say $(E, \ prec)$ a well ordered unit, and $(X, there) \ in E^2$ . According to the property of good order of $E$ , the unit $\ {X, \}$ admits a smaller element there. In other words: $\ forall (X, there) \ in E^2, \ quad X \ prec there \ text {or} there \ prec x$ .

So moreover the Axiome of the dependant choice (a weak version of the axiom of the choice) is checked, this property (to be well ordered) is equivalent to say that the order is total and that the associated strict relation is quite founded (there does not exist strictly decreasing infinite continuation). According to the Theorem of Zermelo, the Axiome of the choice in all its force is equivalent to the fact that any unit can be well ordered, and thus can be made isomorphous with an ordinal .

Examples

the Empty set, provided with the only order which is possible there: (Ø, Ø) (it is smallest ordinal)
the whole of the natural whole , provided with the usual order of the entireties: ( $\ mathbb NR, \ le_ {\, _ \ mathbb NR} \,$ ), often noted ω in this context (it is smallest ordinal infinite)
In a general way, all ordinal, by definition, is well ordered
In a certain direction, very good order on R is monstrous.

Subtleties

That is to say ( E , ≤) is a nonempty well ordered unit. It has a smaller element, and it has or does not have a larger element: the whole of the entireties ω, which has 0 for smaller element, does not have any larger but nothing prevents of him to add one of them - it is the whole beginning of a naive construction of ordinal transfinite. That is to say α∈ E : if α is not the largest element of E , there exists smaller β∈ E strictly higher than α, called successor of α and often noted α+1, whose α is the predecessor . An element of E has with more the one predecessor; the smallest element does not have any obviously and it is the only case for E =ω, but in general there are many elements of E which do not have any - it is what makes the charm of ordinal transfinite. Think, to have an little idea, with the dictionary of all the finished words built starting from alphabet a finished or infinite good ordered. An element of E having a predecessor is known as of first species , and second species if not. This distinction is often useful to reason by transfinite Récurrence.

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