Is God a Mathematician_ - Mario Livio [90]
The Non-Euclidean Crisis All Over Again?
In 1908, the German mathematician Ernst Zermelo (1871–1953) followed a path very similar to that originally paved by Euclid around 300 BC. Euclid formulated a few unproved but supposedly self-evident postulates about points and lines and then constructed geometry on the basis of those axioms. Zermelo—who discovered Russell’s paradox independently as early as 1900—proposed a way to build set theory on a corresponding axiomatic foundation. Russell’s paradox was bypassed in this theory by a careful choice of construction principles that eliminated contradictory ideas such as “the set of all sets.” Zermelo’s scheme was further augmented in 1922 by the Israeli mathematician Abraham Fraenkel (1891–1965) to form what has become known as the Zermelo-Fraenkel set theory (other important changes were added by John von Neumann in 1925). Things would have been nearly perfect (consistency was yet to be demonstrated) were it not for some nagging suspicions. There was one axiom—the axiom of choice—that just like Euclid’s famous “fifth” was causing mathematicians serious heartburn. Put simply, the axiom of choice states: If X is a collection (set) of nonempty sets, then we can choose a single member from each and every set in X to form a new set Y. You can easily check that this statement is true if the collection X is not infinite. For instance, if we have one hundred boxes, each one containing at least one marble, we can easily choose one marble from each box to form a new set Y that contains one hundred marbles. In such a case, we do not need a special axiom; we can actually prove that a choice is possible. The statement is true even for infinite collections X, as long as we can precisely specify how we make the choice. Imagine, for instance, an infinite collection of nonempty sets of natural numbers. The members of this collection might be sets such as {2, 6, 7}, {1, 0}, {346, 5, 11, 1257}, {all the natural numbers between 381 and 10,457}, and so on. In every set of natural numbers, there is always one member that is the smallest. Our choice could therefore be uniquely described this way: “From each set we choose the smallest element.” In this case again the need for the axiom of choice can be dodged. The problem arises for infinite collections in those instances in which we cannot define the choice. Under such circumstances the choice process never ends, and the existence of a set consisting of precisely one element from each of the members of the collection X becomes a matter of faith.
From its inception, the axiom of choice generated considerable controversy among mathematicians. The fact that the axiom asserts the existence of certain mathematical objects (e.g., choices), without actually providing any tangible example of one, has drawn fire, especially from adherents to the school of thought known as constructivism (which was philosophically related to intuitionism). The constructivists argued that anything that exists should also be explicitly constructible. Other mathematicians also tended to avoid the axiom of choice and only used the other axioms in the Zermelo-Fraenkel set theory.
Due to the perceived drawbacks of the axiom of choice, mathematicians started to wonder whether the axiom could either be proved using the other axioms or refuted by them. The history of Euclid’s fifth axiom was literally repeating itself. A partial answer was finally given in the late 1930s. Kurt Gödel (1906–78), one of the most influential logicians of all time, proved that the axiom of choice and another famous conjecture due to the founder of set theory, Georg Cantor, known as the continuum hypothesis, were both consistent with the other Zermelo-Fraenkel axioms. That is, neither of the two hypotheses could be refuted using the other standard set theory axioms. Additional proofs in 1963 by the American mathematician Paul Cohen (1934–2007, who sadly passed away during the time I was writing this book) established the complete independence of the axiom of choice and the continuum hypothesis.