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Good Math by Mark C. Chu-Carroll

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The Insanity of Choice

The only one of the axioms that’s really difficult to understand is the axiom of choice. And that itself makes sense, because even among mathematicians, the axiom of choice is downright controversial in some circles. There’s a bunch of variations of it, or different ways of ultimately saying the same thing. But they all end up with the same problems. And those problems are whoppers. But dumping choice also has some serious problems.

To understand just what makes choice such a problem, it’s easiest to use a variation of it. To be clear about what I mean by a variation: it’s not just a restatement of the axiom of choice in different language, but a statement that ends up proving the same things. For example, we’re going ...

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