Title is currently murdering what little is left of my brain, anyone here who can elaborate on it before gojibrain.exe terminates and the machine goes into sleep mode

But you're the best at math here

But that's wrong, the guy I'm turkey-signalling holds that spot

Quote from: GodspeedGojira! on November 06, 2014, 10:23:54 PM

Quote from: DeeJ on November 06, 2014, 10:21:16 PM

But you're the best at math here

But that's wrong, the guy I'm turkey-signalling holds that spot

Rc or Piranha?

HAHAHAHAHAHAHAHAH no im good at math, but nowhere near those guy's levels

Quote from: RC5908 on November 06, 2014, 10:30:12 PM

Quote from: DeeJ on November 06, 2014, 10:27:52 PM

Quote from: GodspeedGojira! on November 06, 2014, 10:23:54 PM

Quote from: DeeJ on November 06, 2014, 10:21:16 PM

But you're the best at math here

But that's wrong, the guy I'm turkey-signalling holds that spot

Rc or Piranha?

HAHAHAHAHAHAHAHAH no im good at math, but nowhere near those guy's levels

What did you get on the math portion of your SAT again? If a 630 puts me in the 85th percentile, then you're probably in the 98th.

like a 720 i think

OK goji, I've done a little research maybe I can help. Probably not, I'm shit at this stuff.

Essentially the diagonal method itself is just a way to come up with a number from a set of other numbers.

The argument?  Cantor wanted to prove that the set of all real numbers is not of the same cardinality as the set of all natural numbers. So he started by assuming the opposite.

Assuming they were, any number you could come up w/ should be able to fit into an infinite list of real numbers that corresponds w/ the infinite list of natural numbers. In fact you can just use the infinite set of real numbers between 0 and 1. Representing the real numbers symbolically generalizes the argument.
Ex.
 0.d_1,1d_1,2d_1,3. . .d_{n, n}. . .
(The first sub meaning the number corresponds to that real number and the second sub corresponding to the magnitude of the d).


Create a list based on this scheme and then apply the diagonal method to it. We will call the number we generate w/ Cantor method m.

m=0.(d₁₁+1)(d₂₂+1)(d₃₃+1). . .and so on.

If Cantor is wrong you should be able to shoehorn this number back into that list. You can't.

Cantor's argument tl;dr:

It's a proof by contradiction by showing that the set of of all real numbers {R} is not countable. Proof by contradiction is done by assuming the opposite and showing it couldn't be true. Because {R} is therefore uncountable, it must be larger than any countable set. There are different sizes of infinite sets, but any infinite set is larger [has a larger cardinality, to use the correct term] than any countable set. The set of all natural numbers has cardinality aleph-null (along with any other countable set).

So knowing this, Cantor set up his proof by making a hypothetical set of this form:


{.d11d12d12...,.d21d22d33...,...,.dn1, dn2,...}, Except he made it a chart like this:

 __n______|__number__
       1          |       .d11d12...
       2          |       .d21d22d23...
       3          |       .d31d32d33d34...

On the left is the order of the set that the number on the left is in. Left is natural and countable. Right is real and uncountable. You make the contradiction by trying to show that the numbers in this list have a one-to-one correspondence with real numbers (as that would show they have the same cardinality). Cantor found this number to be:

m = 0.(.d11+1)(d22+1)...(dnn+1)

Basically just the diagonal numbers from the previous list in a set, increased by 1. If they were of the same cardinality you'd be able to find m inside the previous set, but you can't. This might sound stupid, because obviously you could just add 1 to the old set like we did to the subset of diagonals, but doing so would just create a new set of diagonals and the contradiction remains true.

Set theory isn't really my forte despite being one of my favorite subjects. Mr. Sexy has a good summary, too.

This is awesome:

wuuuut

Fuck you and your faggot numbers.

Cantor's argument tl;dr:

It's a proof by contradiction by showing that the set of of all real numbers {R} is not countable. Proof by contradiction is done by assuming the opposite and showing it couldn't be true. Because {R} is therefore uncountable, it must be larger than any countable set. There are different sizes of infinite sets, but any infinite set is larger [has a larger cardinality, to use the correct term] than any countable set. The set of all natural numbers has cardinality aleph-null (along with any other countable set).

So knowing this, Cantor set up his proof by making a hypothetical set of this form:


{.d11d12d12...,.d21d22d33...,...,.dn1, dn2,...}, Except he made it a chart like this:

 __n______|__number__
       1          |       .d11d12...
       2          |       .d21d22d23...
       3          |       .d31d32d33d34...

On the left is the order of the set that the number on the left is in. Left is natural and countable. Right is real and uncountable. You make the contradiction by trying to show that the numbers in this list have a one-to-one correspondence with real numbers (as that would show they have the same cardinality). Cantor found this number to be:

m = 0.(.d11+1)(d22+1)...(dnn+1)

Basically just the diagonal numbers from the previous list in a set, increased by 1. If they were of the same cardinality you'd be able to find m inside the previous set, but you can't. This might sound stupid, because obviously you could just add 1 to the old set like we did to the subset of diagonals, but doing so would just create a new set of diagonals and the contradiction remains true.

Set theory isn't really my forte despite being one of my favorite subjects. Mr. Sexy has a good summary, too.

This is awesome:

Spoiler

I GET IT

I FINALLY ACTUALLY GET IT

I WAS THINKING ABOUT IT RANDOMLY ABOUT BINARY NUMBERS AND ALL OF A SUDDEN I REALIZED HOW TO GENERALIZE IT AND I WAS LIKE "OH DAMN THAT'S WHAT THEY WERE SAYING"

THANK YOUUUUUUUUUUUUU