[HN Gopher] The Point of the Banach-Tarski Theorem
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       The Point of the Banach-Tarski Theorem
        
       Author : ColinWright
       Score  : 30 points
       Date   : 2023-01-22 21:05 UTC (1 hours ago)
        
 (HTM) web link (www.solipsys.co.uk)
 (TXT) w3m dump (www.solipsys.co.uk)
        
       | CJefferson wrote:
       | I always feel part of the confusion with Banach-Tarski is that
       | lots of words don't use their "natural definitions", which makes
       | the proof more surprising. People (not this article) often
       | talking about "cutting" a sphere, which is really misleading.
       | 
       | This result is, in many ways, quite similar to the idea I can
       | "cut" the integers into the odd integers and even integers (but
       | with many more fine details).
       | 
       | This is still a nice article, which explains the actual result
       | well.
        
       | azeemba wrote:
       | As the author points out that Banach-Tarski theorem is an example
       | of hard-to-accept result that comes out of the easy-to-accept
       | axiom of choice.
       | 
       | There is a popular quote that related to this:
       | 
       | > The axiom of choice is obviously true, the well-ordering
       | principle obviously false, and who can tell about Zorn's lemma?
       | 
       | From https://en.wikipedia.org/wiki/Axiom_of_choice
       | 
       | Axiom of choice, the well-ordering principle and Zorn's lemma are
       | equivalent statements (any one proves the other two). But each
       | has a very different "believability" feel to it.
        
       | whatshisface wrote:
       | Here is a potentially daft question that I nonetheless would
       | appreciate if someone could answer. Is it possible to deny the
       | axiom of choice for the purposes of measures while accepting it
       | for vector spaces? I am wondering if you could say, "there are
       | two kinds of sets, ones equipped with a choice function and ones
       | without it, and measurable sets are of the latter kind."
        
         | joerichey wrote:
         | If you accept that _all_ vector spaces have a (Hamel) basis,
         | you can then prove the Axiom of Choice:
         | http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf
         | 
         | This means if you want to deny the Axiom in some cases, you
         | will also have to allow for the existence of vector spaces
         | without a basis.
        
         | contravariant wrote:
         | Not in a meaningful way I think. I mean you could weaken it to
         | 'all finite vector spaces have a basis', but I think regular
         | induction is enough to prove that, you don't need the axiom of
         | choice.
        
       | moloch-hai wrote:
       | I asked a mathematician about what a wacky conclusion it is. He
       | said that whenever you allow infinity, you get results like that.
       | It relies on uncountably-infinite division of an object, which
       | corresponds to no real-world experience anywhere in the universe.
       | Real objects have, you know, atoms.
       | 
       | We use real numbers a lot, but we are careful never to rely on
       | their more extreme properties anywhere it would matter. In
       | practice, in fact, we use floating-point numbers, not reals, when
       | doing actual calculations, and use numerical analysis to stay
       | well clear of nonsensical results. If you tried to rely on BT in
       | a real calculation, you would find a lot of NaNs and Infs.
        
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