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This isn't necessarily true; you can also limit which sets are valid sets in a way that guarantees that choosing an element from these sets is well defined. For

This isn't necessarily true; you can also limit which sets are valid sets in a way that guarantees that choosing an element from these sets is well defined. For example, the Axiom of Constructibility basically limits which sets exist, in a way that allows you to always use the Axiom of Choice because your sets have a specific fine grained structure you can use to well-order them. For example, if I were to require "all sets are finite", then the Axiom of Choice is trivial. The Axiom of Constructibility still allows infinite sets (and even uncountable sets), but it's still limited enough that the Axiom of Choice holds. You can almost think of it as requiring all sets to have source code, and then just alphabetizing the source code in order to pick a set.

jerf 5y ago
Yes, it's true that you can use the Axiom of Choice in ways that don't invoke infinite information, and yes, those would be OK with me from an aesthetic point of view. (Again let me emphasize my complaint here is not that it is "invalid" but that I find it very unaesthetic, contra many conventional mathematicians.) But things like the Banach-Tarski paradox definitely require infinite amounts of information (uncountable infinite in this case, I believe) with, as far as I know, no mechanism for producing it. As far as I know, it's just an existence proof. When discussing hypothetical FTL technologies, I often like to say that it's no great surprise that if you allow one impossibility (negative mass) it's no surprise that you get another (FTL). Similarly, if you allow an uncountably infinite amount of information to be magicked into your proof, it's no surprising that you may get a confusing result like the Banach-Tarski paradox. From my perspective, the confusing step isn't when you have two spheres where you used to have one, the confusing step is when you made uncountably-infinitely-precise cuts with no ability to produce the cuts in question. I'm not confused by the end result, I'm confused at that step. So to speak. I'm not literally confused, obviously, only my sense of aesthetics is.

jerf 5y ago

Yes, it's true that you can use the Axiom of Choice in ways that don't invoke infinite information, and yes, those would be OK with me from an aesthetic point of view. (Again let me emphasize my complaint here is not that it is "invalid" but that I find it very unaesthetic, contra many conventional mathematicians.) But things like the Banach-Tarski paradox definitely require infinite amounts of information (uncountable infinite in this case, I believe) with, as far as I know, no mechanism for producing it. As far as I know, it's just an existence proof. When discussing hypothetical FTL technologies, I often like to say that it's no great surprise that if you allow one impossibility (negative mass) it's no surprise that you get another (FTL). Similarly, if you allow an uncountably infinite amount of information to be magicked into your proof, it's no surprising that you may get a confusing result like the Banach-Tarski paradox. From my perspective, the confusing step isn't when you have two spheres where you used to have one, the confusing step is when you made uncountably-infinitely-precise cuts with no ability to produce the cuts in question. I'm not confused by the end result, I'm confused at that step. So to speak. I'm not literally confused, obviously, only my sense of aesthetics is.