What is the Axiom of Choice? | Maths and Musings - Medium

Here’s a fun 2-minute guide to the infamous Axiom of Choice, one of the stranger mathematical curiosities by Maths and Musings in Cantor’s Paradise.

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What is the Axiom of Choice?

The short answer, not at all rigorous answer, is that the axiom of choice allows mathematicians to extract elements from an infinite number of infinitely large sets at once. This turns out to be very important, because mathematicians are very fond indeed of making infinite-sized things, and even more fond of being able to formalise their manipulations of infinite-sized things.

But let’s dig a little deeper.

The axiom of choice allows us to pick elements from ‘indexed sets’. When dealing with ‘finite things’, this seems kinda obvious. For instance, if A={1,2,3}, B={3,4,5}, and C={5,6}, then it is easy to pick an element from each. Just pick, say, 1 from A, 3 from B, and 6 from C. In fact, if you only every deal with finite sets (e.g. dealing with finite numbers, finite graphs, a finite number of people, etc…), then you will never need the Axiom of Choice. That includes lots of interesting maths, from computer science to graph theory. And it probably includes everything in the real world.

But, Mathematicians don’t want to be constrained to the finite. How very dull it would be if we couldn’t make nearly as many weird and wonderful mathematical objects as our imagination allowed, while keeping coherency through the use of sensible axioms...

Is the Axiom of Choice even true?

… is the wrong way to look at it.

It has been proven that the Axiom of Choice is consistent with the other axioms of set theory, but so its negation. I.e. we don’t need to worry about it messing up our consistency with the apparatus we use for finite sets.

The Axiom of Choice extends what we are comfortable doing with finite sets, is consistent with the other axioms, and makes a vast amount of mathematics work, and much of this mathematics is extremely useful.

If you are looking for some kind of Platonist truth, then I think you are barking up the wrong tree. It is better to view mathematics as structure, much of which parallels the real world (such as Euclidean geometry giving real insights into objects which aren’t actually 2D and have perfectly smooth edges), rather than necessarily being some necessarily true statements.
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Source: Medium