2 ms·

I understood the paradox but did not buy the explanation at all.

by hristov 17y ago

frig 17y ago
It's not a great explanation b/c it doesn't go into what "measure" and "volume" mean. Tough to do without going through a full course but a slightly-less-handwavy explanation would go like this: There's a mathematical tool (called measure) that formalizes-and-generalizes the notion of "volume". It generally behaves very much as you'd expect; your intuitions about how "volumes" combine and intersect will generally apply. There's a catch, though: the way the tool is constructed leaves open the possibility of a non-measurable set, meaning a set for which the definition of the tool leaves you unable to assign that set a well-defined "volume"; you can't just assume that the measure of a set exists. If you assume the Axiom of Choice then not only are such sets possible, but you can construct non-measurable sets. The core process in Banach-Tarksi looks like this: (1) take the sphere (a 'nice' set, which we'll say has "volume" V) (2) divide that sphere into some sub-sets that are non-measurable (are sets for which our tool cannot supply a measurement) (2.a) Effect on total volume: should have no impact, as the parts we have reassemble to an object of known volume (3) move the subsets around by sliding-and-rotating them (3.a) Effect on total volume: should have no impact, as neither sliding nor rotating changes volume (4) wind up with 2 spheres (both 'nice' sets, each with "volume" V)??? The paradox comes from getting double the volume through a sequence of operations that are apparently volume-conserving. You can go with this a couple different directions. I'm not convinced this should be an intuitive outcome. An intuitive, hand-wavy explanation for what's "really going on" would be something like non-measurable sets carry around infinite amounts of finely-detailed structure (too finely-detailed to perceive using our measuring tool); depending on how you position some sets relative to each other their finely-detailed structure might either cancel out (adding no volume) or reinforce each other (adding lots of volume). That said I think the real lesson here is that your intuition is trying to have its cake -- a non-measurable set -- and eat it too -- have the "volume" of a non-measurable set be preserved under volume-preserving operations.

I understood the paradox but did not buy the explanation at all.

by hristov 17y ago

I understood the paradox but did not buy the explanation at all.

frig 17y ago

It's not a great explanation b/c it doesn't go into what "measure" and "volume" mean. Tough to do without going through a full course but a slightly-less-handwavy explanation would go like this: There's a mathematical tool (called measure) that formalizes-and-generalizes the notion of "volume". It generally behaves very much as you'd expect; your intuitions about how "volumes" combine and intersect will generally apply. There's a catch, though: the way the tool is constructed leaves open the possibility of a non-measurable set, meaning a set for which the definition of the tool leaves you unable to assign that set a well-defined "volume"; you can't just assume that the measure of a set exists. If you assume the Axiom of Choice then not only are such sets possible, but you can construct non-measurable sets. The core process in Banach-Tarksi looks like this: (1) take the sphere (a 'nice' set, which we'll say has "volume" V) (2) divide that sphere into some sub-sets that are non-measurable (are sets for which our tool cannot supply a measurement) (2.a) Effect on total volume: should have no impact, as the parts we have reassemble to an object of known volume (3) move the subsets around by sliding-and-rotating them (3.a) Effect on total volume: should have no impact, as neither sliding nor rotating changes volume (4) wind up with 2 spheres (both 'nice' sets, each with "volume" V)??? The paradox comes from getting double the volume through a sequence of operations that are apparently volume-conserving. You can go with this a couple different directions. I'm not convinced this should be an intuitive outcome. An intuitive, hand-wavy explanation for what's "really going on" would be something like non-measurable sets carry around infinite amounts of finely-detailed structure (too finely-detailed to perceive using our measuring tool); depending on how you position some sets relative to each other their finely-detailed structure might either cancel out (adding no volume) or reinforce each other (adding lots of volume). That said I think the real lesson here is that your intuition is trying to have its cake -- a non-measurable set -- and eat it too -- have the "volume" of a non-measurable set be preserved under volume-preserving operations.