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ARTWORK: Xuming Du
“to be chosen” ANDY YIN EDITED BY SAI CAMPBELL
Mathematics has the most demanding standards of truth of any field of study. For a fact to be considered mathematically true, it must follow from previouslyknown facts using precise rules of inference. In turn, those facts are justified by inference from previous facts, and so-on. But this process has to terminate at some point, or nothing could ever be justified. You need some facts which don’t need to be justified by something else. They’re called axioms — they are the sources of truth. Because axioms are assumed true without justification, choosing axioms to believe in is more subjective than maths usually is. Today, most mathematicians agree to work under the Zermelo-Fraenkel axioms. These axioms assert that there is only one kind of thing in existence: a set, i.e. a collection of things. Just as our understanding of people could be reduced to the movements of
atoms, many beautiful structures in mathematics – numbers, space, geometric objects — can be reduced to sets. The Zermelo-Fraenkel axioms include reasonable assumptions like “Two sets are the same set if they contain the same elements,” and “If you have two sets, then you can create a new set containing everything from both those sets.” Not especially controversial. But there was one axiom whose proposed inclusion in ZF caused a stir: The Axiom of Choice, also known simply as Choice. (You know a mathematical statement is famous when it has a mononym like Prince.) Choice can be written so innocuously as to believe its controversy: Given a collection of nonempty sets, it is possible to pick one element from each set.
