Axiom of Choice, 13 Essence of Set Theory

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The Axiom of Choice and two of its slightly paradoxical applications -- infinite prisoners with hats, and a non-measurable set.

[wilderuhl]

Underrated channel. Thanks for the great videos.

[lior_shiboli]

This deserves so many more views
It's really interesting

[hugoballroom]

Thank you for this series. I have been looking for a video that explicates the motivation this Axiom. Yours is the only one i can find. The animation and narration are so clear and concise. 

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[StratosFair]

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[dsagman]

this video should have many thousands more views

[AntonParamonov-ed]

Thank you for the wonderful playlist!

By the way, the problem about hats and prisoners can be solved more “optimally”, namely with only one prisoner having a chance to guess his hat wrong. The strategy is the first prisoner encoding the parity of a distance from a sequence he is seeing to the class representative (that is correct since the distance is finite). Knowing this information the second prisoner can guess his hat for sure, knowing the parity and the second prisoners guess the third prisoner can guess his color and so on.

[GiovannaIwishyou]

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[ostihpem]

Too complicated. Here is an easy example. We have a set x with non-empty sets, disjunctive to each other, but we do not know anything about the elements of these sets of x. So we cannot use the axiom of separation or replacement or whatnot because we couldn‘t even formulate the rules to apply. But AC guarantees us a set with exactly one element of the sets of x as its elements.