proof

[DrunkenArse]

a question of what mathematical proof is came up in my troll-bait thread that Xei took the bait out of yesterday. In answer I'm going to prove something that is of interest to almost everybody. Instead of putting this whole thing in that thread, I figured that it might be of interest to enough people to start a new thread. This is not a troll bait thread and I am willing to help anybody that wants to understand this proof but am not interested in arguing about it's legitimacy.That's the point of a proof. Any mathematician will back me up on this: It's true. This is one of the more profound low-hanging fruits of pure mathematics. It is entirely useless thus far but is very pretty. The result is that there are different magnitudes of infinity. It was proven by a mathematician named Cantor sometime in the 1870's i believe. We will prove this by proving that the set of real numbers between 0 and 1 is stricly larger then the set of natural numbers. We start by assuming that they have the same 'size'. that means that we can put them into a one to one correlation so that each natural number is associated with one and only one real number in the range from 0 to 1 and that each real number in the range occurs exactly once in this association. we don't really have a good way to write subscripts so read d1 as d sub 1. then we can draw a diagram like this: 1 : d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 .. .. .. 2 : d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 .. .. .. 3 : d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 .. .. .. 4 : d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 .. .. .. Interpret this as saying that dm in row n is the m'th digit of the real number associated with the number n. now what we do is construct a new real number. I'll use e for it's digits to keep it separate. We do this by starting at d1 in row 1. If d1 row 1 = 6 then we set e1 to 7. If d1 row 1 is not 6 then we then we set it to 6. We then get e2 from d2 row 2 in the same way. e3 comes from d3 row 3 and, in general, we get em from dm row m. so the number e looks like .67676667667776767676776777677677776767676767676.. ........ The point of this is that we know that e is different from every real number that appears in our association because it differs in at least one digit. That means that we have a contradiction on our hands because we assumed that every real number does appear in the association. Because the only thing we assumed was that such an association exists and making that assumption allows us to construct a contradiction, we know that that assumption must be false. So the two sets must not have the same size. We also know that the real numbers must be larger because we have at least one that does not appear in association. Thats it and that is mathematics.