Tarski's theorem about choice
In mathematics, Tarski's theorem, proved by , states that in ZF the theorem "For every infinite set formula_1, there is a bijective map between the sets formula_1 and formula_3" implies the axiom of choice. The opposite direction was already known, thus the theorem and axiom of choice are equivalent. Tarski told that when he tried to publish the theorem in Comptes Rendus de l'Académie des Sciences Paris, Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between  two false propositions is of no interest. Proof. The goal is to prove that the axiom of choice is implied by the statement "for every infinite set formula_1: formula_5". It is known that the well-ordering theorem is equivalent to the axiom of choice; thus it is enough to show that the statement implies that for every set formula_6 there exists a well-order. Since the collection of all ordinals such that there exists a surjective function from formula_6 to the ordinal is a set, there exists an infinite ordinal, formula_8, such that there is no surjective function from formula_6  to formula_8. We assume without loss of generality that the sets formula_6 and formula_8 are disjoint. By the initial assumption, formula_13, thus there exists a bijection formula_14. For every formula_15, it is impossible that formula_16, because otherwise we could define a surjective function from formula_6 to formula_8. Therefore, there exists at least one ordinal formula_19 such that formula_20, so the set formula_21 is not empty. We can define a new function: formula_22. This function is well defined since formula_23 is a non-empty set of ordinals, and so has a minimum. For every formula_24 the sets formula_23 and formula_26 are  disjoint. Therefore, we can define a well order on formula_6, for every formula_28 we define formula_29, since the image of formula_30, i.e. formula_31, is a set of ordinals and therefore well ordered.