A surprisingly simple concept shook the foundations of mathematics
Baking is not my strong point. So when I have houseguests, I dash to a bakery to grab dessert and am often spoiled for choice. Given the huge selection of appetizing cakes and tarts, I find it difficult to decide. My strategy is to say, “Oh, why don’t you pack me one piece of each?”
For that reason, some experts specifically indicate when they use the axiom of choice in a proof—and there are mathematicians who have wanted to overhaul the subject without this axiom. Yet a world without the axiom of choice is even stranger.To understand the debate, we first need to consider what distinguishes mathematics from the other natural sciences.
Mathematician Ernst Zermelo quickly realized, however, that these eight fundamental truths were not enough. In 1904 he therefore introduced the axiom of choice. And thus began the conflict.The axiom of choice allows you to select one element at a time from a series of nonempty sets—much as I can have a slice of several cakes at the bakery. At first, it seems only natural that this is possible.
Then, later in 1904, Zermelo, who had introduced the axiom of choice, found an extremely counterintuitive result that he could only prove with the help of the axiom of choice. He showed that any quantity can be well ordered. From this so-called well-ordering theorem, it follows, among other things, that every set has a smallest element according to that ordering.
Without the axiom of choice, for example, it is not possible to ensure that every vector space has a basis. This property is implied by another statement calledand sounds abstract, but physicists and mathematicians refer to it time and again. You can visualize this with the help of a sheet of paper . If you draw two arrows on this sheet, one pointing horizontally and one pointing vertically, you can reach any point on the sheet from these arrows. For example, you can move 0.
But the reverse is also true. You can add the negation of the axiom of choice to the eight Zermelo-Fraenkel axioms without encountering any contradictions. This means that the axiom of choice can be regarded as either true or false, and mathematicians are free to choose one of the two possibilities.“One commonly hears the axiom of choice described as useful for certain mathematical arguments, but problematic in light of the Banach-Tarski paradox and other counterintuitive consequences.
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