/sci/ - Science & Math

>>11342761
Assuming that every subset of R has to be measurable is a much sillier assumption than the axiom of choice is.

>>11342823
Is there a Peano arithmetic like system for set theory?

>>11342761
That's a stupid interpretation, the set is simply not measurable.

>>11342761
why allow an abstract definition of (newtonian) physical measurement and try to generalize it over something non-physical in an intuitive way and expect it not to cause a "paradox"? I'm pretty sure we could prove a theorem that states "if your definition of measurement is not identically distance formula, then it will either be paradoxical or not completely robust for its purposes."

>>11343237
Explain? How can you just assert that it's not measurable.

>>11342761
>tfw no cute gf to explain me trivial set theory concepts

>>11342761
why should every set be measurable? why is it so intuitive to you that every set should have a length?
measurable sets are just as confusing and strange as unmeasurable ones. just look at the fat cantor set. it contains no intervals, it's just a bunch of disconnected points. and it's fucking closed! so it's nowhere dense. but it's got positive lebesgue measure. what the fuck??? where do you squeeze measure out of it?

measure theory is NOT intuitive at all, and you should not act like it is. just look at the definitions for the lebesgue measure on a set that isn't just an interval, it's a limiting process which should absolutely not exist for every set. in fact, the fact that it is so hard to construct unmeasurable sets is astounding.

you need to struggle through a course on measure theory before you're allowed to have opinions about things like unmeasurable sets or the banach tarski paradox.

>>11343360
because if you have infinitely many of the same thing, which when you add them together make something finite, then you've got a problem. but obviously you can do this if your finite thing is made up of an infinite number of points.

>>11344436
>why is it so intuitive to you that every set should have a length?
I guess it's because (talking as a layman) when mathematicians talk about things like making a copy of a sphere by breaking up the sphere and putting it back together, we assume they are talking about a physical sphere, or at least some abstract model of a physical sphere. If you're defining sets, or spheres, or quantities to have no size, or infinite size, then you should make it very explicit that these are ideas that have nothing to do with our universe or reality.

I'm not saying that these theories are wrong, just that what they are modelling is a purely mathematical universe.

>>11344438
If you add up an infinite number of points (i.e. zeroes) you still get a point (zero). The set that is the result of the sum must have a finite but non-zero size, as the video shows.