Assertion: set | /r/HasProperty | well-ordered
Facets:
value: partially
type: DEGREE
Source sentences:
text: The Axiom of Choice is equivalent, modulo ZF, to the Well-ordering Principle, which asserts that every set can be well-ordered, i.e., it can be linearly ordered so that every non-empty subset has a minimal element.
text: A set paired with a total order is called a chain, a totally ordered set, a simply ordered set, a linearly ordered set, or a loset.
text: The axiom of choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order.
text: In 1904 Ernst Zermelo proved by means of the axiom of choice (which was introduced for this reason) that every set can be well-ordered.
text: Since every infinite well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infinite set is Dedekind-infinite.
text: A set with a partial order is called a partially ordered set (also called a poset).