skine wrote: take the pieces and construct two spheres with the same radius as the first.
Two completely solid spheres.
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skine wrote: take the pieces and construct two spheres with the same radius as the first.
majikthise wrote:On the other hand, it could lead to people devoting entire lifetimes to proving something without even knowing whether it's possible.
BlackSails wrote:If people make decisions not based upon physical data and not predictable from the physical data, then so do electrons
Suffusion of Yellow wrote:Question: I read that Continuum Hypothesis was proved unprovable, but since you can always disprove it by just pointing to a set with cardinality between rationals and reals, doesn't its unprovability imply no such sets exist, proving the hypothesis and contradicting it being unprovable?
TNorthover wrote:Suffusion of Yellow wrote:Question: I read that Continuum Hypothesis was proved unprovable, but since you can always disprove it by just pointing to a set with cardinality between rationals and reals, doesn't its unprovability imply no such sets exist, proving the hypothesis and contradicting it being unprovable?
Because of (or perhaps as part of) the undecidability result there's no set whose cardinality you can prove is between the integers and reals. That doesn't stop you guessing one -- in fact I think that if you drop the axiom of choice then it's consistent to posit that a countable union of countable sets suffices, but there'll be no proof that it's between the two in ZF.
Imagine the warehouse at the end of "Raiders of the Lost Arc," and assume every crate is not empty. Obviously, you can take something out of each crate. But if there are an infinite number of crates, you can't prove that "taking something out of each crate" quite makes sense (as a function). Even though this makes sense, it has been proven that , given this, you can take a sphere apart (not by individual points), take the pieces and construct two spheres with the same radius as the first.
Something Awesome wrote:A countable union of countable sets is countable, with or without Choice. Since countable sets have a bijection with the naturals by definition, you can easily put them each into an [imath]\omega[/imath]-sequence, and then use something like the proof of the countability of the rationals (which also doesn't depend on Choice).
Tirian wrote:A more mathematical example is the parallel postulate in plane geometry. People spent two thousand years trying to prove it from the other four axioms, until eventually it was discovered that if you assumed that you could draw more than one line through a given point parallel to a given line, you'd come up with a consistent system (called elliptical geometry), and if you assumed that you could never draw a line through a given point parallel to a given line, you'd come up with another one (called hyperbolic geometry). None of these systems are "better" than the others, just more or less applicable to whatever problem you're trying to solve. To give an example of that, people studying relativity from a mathematical perspective are greatly aided by considering that space-time conforms to the rules of hyperbolic geometry, but people who are building a house prefer Euclid's model.
GyRo567 wrote:Tirian wrote:A more mathematical example is the parallel postulate in plane geometry. Blah blah blah
I think you may have this backward.
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