Tuesday, December 23, 2014
It might seem that there is no connection between finding longitude, and finding a way to have just one connection, as opposed two. It might seem that these are separate problems requiring separate solutions. And for the 18th century this was the case, but then Georg Cantor drifted in. There were two pieces that he contributed. First, he found that infinity is different from finite, and then issued that there were at least two kinds of infinity: the natural number, and the, though we don't see this way in pure mathematics, real number. In this chapter we will deal with what this means, and why it is important.
There are some mathematicians who do work, and they do not attract any attention. Such as Leonard Euler, who made many contributions and, as far as the historical eye can see, did not raise any failures. This despite making important discoveries in different fields such as a infinitesimal calculus, among a host of others. Then there are mathematicians who generate controversy, this is the fate of Georg Cantor. Who became known for the important one-to-one correspondence between numbers. He came to mathematics not describing any infinite set theories, and left it by proving there are an unlimited number if infinite sets, and there are at least two separate infinite sets which are not countable between each other. What he did not prove, is if there are only two.
Now, numbers had been known to be irrational, that is not countable for a very long time, but what this did not take into account was there were numbers that were countable, and numbers which were not countable, and that both of these sets were infinite, and infinite in two separate ways. In other words, there was unlimited but countable, and unlimited and not countable, and these were different sets of numbers. Moreover, a small number of the countable numbers was exactly the same as a much larger share of them, and both of these were the same number.
Think about squares. It is obvious that there are more numbers that are not square then squares. It is also obvious there are the same number of squares, as numbers that are not square. It seems to be a contradiction, how can there be more not squares than squares, and at the same time an equal number of squares and not squares?
This is where Cantor fits in. Is more famous method is coming on the diagonals. What he does is show that numbers, such as 1, 2, 3, 4 and so on are the same as 1/2, 2/3, 3/4 and so on, that there is a 1 to 1 correspondence between two sets. There is a one-to-one correspondence between a finite number, but there is between an infinite number. First he shows that this could not be the case because there are an infinite number of 1/2, 2/3 and so for each 1, and so on. But he then counts not up-and-down, nor across, but along the diagonal. Then Cantor showed that there was an uncountable set, which had more members than number of countable but unlimited sets. These were sets which didn't end. Remember while there were unlimited number of numbers, the countable sets individually had a name – 1/2 for example, even if they did not have a name but could theoretically have a name, that made no difference. But numbers which could not be count, even by an infinitely patient man, were different. These are things such as pi, which my computer does not let me represent, our different. Thus there are two sets of infinite numbers, Countable and uncountable.
Realize immature amateur mathematicians still come up with a proof that they can “square the circle”, that is with a straight edge and compass built a square which is the same as a circle. And perhaps always will do so. I suppose it's one of the tests for an amateur mathematician.
What he does is divide the proof into two parts. The first part takes a random sequence of real numbers, x0,x1,..xn and a set of coefficients pairs [a,b] which has at least one pair is not part of the real numbers. He does by selecting pair of coefficients where a is increasing, and b is decreasing; and every member of a is smaller than every member of b. Then he breaks in to two cases: as said the finite case and the infinite case. If it is the finite case, then only one number can be inside an,bn . If it is of the infinite case then he doesn't need to counter, because, x0,x1,..xn is not in infinite order, remember that they are finite by definition. Actually, he goes one step further, one case where a=b, and the other one where a < b.
Let me take and example: 6x2+9x+5 it would become (1)(5)x0+(2)(3)x1+(3)(3)x2.
What Cantor does is the orders of the coefficient by their heigh and then orders, then orders the real roots by the same height, and shows this cannot be counted, either by infinite or finite means. This is not the normal way of doing things, but it is Cantor's way of doing things. And this is basically the first proof he ever publishes.
Cantor discussed the proof with Dedekind, and pointed out that it would be of interest if this proof could have produced a new theory of Liouville's that numbers are valid. He tries to do this in a proof, but it needs an additional proof, which he then supplies. He then announces just for Christmas 1873 a proof which not only answers whether the real numbers are counted, but that there are transcendental numbers. But Cantor realized that this was a bridge too far. So he omits this, and returns to prove with the first, that is the second proof, which he publishes.
That could be the end of Cantor's story.
But isn't yet, having proved there one time of countable infinity, he next proves that there is an uncountable kind of infinity, which is larger than the countable kind, but he doesn't know if the number of the continuum, which is what he calls it, is the next number in the sequence of uncountable figures. Thus he makes the first prove of zero, but he does not make the second proof of one, because he doesn't know if there is something beneath his second proof that his larger then the proof of zero, which he knows his the smallest kind of infinity, but he doesn't know if there is a larger infinity then the countable numbers, but smaller than his proof that uncountable numbers. In other words, he doesn't actually know, though he and we are pretty certain that it's true, that all of the numbers such as pi are really the next set of numbers. And this has been true for almost a century, we don't know that pi, and the other countless unending numbers are the next in the sequence. But as I said, most people are pretty sure. It's just that “pretty sure” it is not the same as “proven to be”.
And no one who is in their right mind is sure that “proven to be” really is true. Note that cantors proof doesn't mean that the numbers of x need to be countable numbers, they can be uncountable numbers, but each one needs to be a specific number, even though you don't know which member it is. This is strange that you may not know exactly which number it is, but you can say that it is a specific number.
So Cantor is known for the following ideas:
- There are a countable number of numbers.
- There are an uncountable number of numbers.
- The number of uncountable numbers is larger.
- There may be, though it is unlikely, that there may be a set between the first two which is larger than the countable numbers, but smaller than uncountable numbers.
And that will leave a mark that people will remember for at least 100 years.