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Tough Trolly Choices
(mander.xyz)
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this post was submitted on 21 Jun 2024
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The first lever I see in each group.
It might sound trivial but it is not! Imagine there is a lever at every point on the real number line; easy enough right? you might pick the lever at 0 as your “first” lever. Now imagine in another cluster I remove all the integer levers. You might say, pick the lever at 0.5. Now I remove all rational levers. You say, pick sqrt(2). Now I remove all algebraic numbers. On and on…
If we keep playing this game, can you keep coming up with which lever to pick indefinitely (as long as I haven’t removed all the levers)? If you think you can, that means you believe in the Axiom of Countable Choice.
Believing the axiom of countable choice is still not sufficient for this meme. Because now there are uncountably many clusters, meaning we can’t simply play the pick-a-lever game step-by-step; you have to pick levers continuously at every instant in time.
It seems to me that, since the set of real numbers has a total ordering, I could fairly trivially construct some choice function like "the element closest to 0" that will work no matter how many elements you remove, without needing any fancy axioms.
I don't know what to do if the set is unordered though.
If I give you the entire real line except the point at zero, what will you pick? Whatever you decide on, there will always be a number closer to zero then that.
I guess I can pick another number x to be closest to but it has the same problem unless I can guarantee it's in the set. And successfully picking a number in the set is the problem to begin with! Foiled again!