Timeline for answer to Axiom of choice, Banach-Tarski and reality by Timothy Chow
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Aug 8, 2025 at 20:49 | comment | added | Hypnosifl | Apart from those who see Banach-Tarksi as paradoxical bc of desire to model physical world, may be others who have philosophical predisposition that notion of randomly choosing a real number from a given interval like [0,1] makes sense, and if you make an infinite series of such random selections, there should be a truth about the fraction that are in any particular subset (so all subsets measurable). van Lambalgen's suggestion of alternative axiom to AC at jstor.org/stable/2275368 was based on taking randomness as primitive, page 1277 talks abt giving measure to all sets of reals | |
| Dec 1, 2022 at 5:48 | comment | added | Timothy Chow | @user112009 It sounds like you think that the problem with the axiom of choice is that it claims that certain "undefinable" operations are possible. But for example, in $L$, there is a definable well ordering of the universe, and everything you do with the axiom of choice is going to be definable. So this type of criticism of the axiom of choice is unconvincing. Joel David Hamkins explains this point well in another MO answer. | |
| Feb 7, 2018 at 21:14 | comment | added | Vladimir Reshetnikov | @user112009 Following your reasoning, do you have to distinguish and name all (uncountably many) real numbers to put them into their usual linear order? Or do you have to distinguish and name all (uncountably many) countable ordinals to put them into their usual well-order? | |
| Aug 5, 2017 at 15:16 | comment | added | user112009 | The axiom of choice is in contradiction with logic. In order to well-order a set you have to distinguish all its elements. That means you have to give a finite name of its own to each one. That is impossible for an uncountable set. The reason for maintaining the axiom is simply that otherwise great parts of set theory would break down. | |
| Jan 20, 2017 at 23:46 | comment | added | Andreas Blass | My answer at mathoverflow.net/q/34863 is relevant to the last sentence of your answer here. | |
| Jan 20, 2017 at 23:08 | history | edited | Timothy Chow | CC BY-SA 3.0 |
fixed typos
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| Jan 20, 2017 at 20:07 | history | answered | Timothy Chow | CC BY-SA 3.0 |