It is used to prove a theorem stating that no first-order theory can control the cardinality of their infinite models, which is the Lowenstein-Skolem theorem.
The Goodman-Myhill theorem shows that this result is enough to prove the law of the excluded middle. The proof of the Cantor-Bernstein-Schroeder theorem notably does not rely on this result. It is used to prove a theorem stating that no first-order theory can control the cardinality of their infinite models, which is the Lowenstein-Skolem theorem. Equivalent to the statement that every vector space has a basis, this result plus ZF was shown by Paul Cohen to be independent of the continuum hypothesis. Often stated as the idea that the Cartesian product of a collection of non-empty sets is non-empty, this statement is equivalent to the idea that every non- empty set of positive integers has a least element, which is called the well-ordering principle. For 10 points, name this statement from mathematics, which allows any set to have some function to pick elements of the set.
The Goodman-Myhill theorem shows that this result is enough to prove the law of the excluded middle.
axiom of choice [or Zorn's lemma; or well-ordering principle until it is read]