Theorem of Zermelo
The theorem of Zermelo , also called theorem of the good order is a result of set theory, shown in 1904 by Ernst Zermelo which affirms:
Any unit can be provided with a structure of good order.
Thus, being given a unit X, there exists an order such as very nonempty part admits a smaller element.
This theorem is equivalent to the Lemme of Zorn, and thus to the Axiome of the choice.
Let us show for example that it implies the axiom of the choice. A nonempty unit, and P (E) are E the whole of its parts. Let us provide E with a good order. Then, for a nonempty part X of E, to choose an element of X, it is enough to take the smallest element of X.
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