Wednesday, October 19, 2016

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Gödel 13



Wrapping up the Chapter

If you ask a man or a woman in English what “Julia” means, they will think of it as a first name, for girls. But if you ask a mathematician they will think of it as a last name, and they will reply “Gaston Julia discovered one of the first sets...” such as the difference between asking anyone what Julia means, and asking someone specifically. Right now, hurricane Gaston is leaping out over the Atlantic with the intent of slamming into the Azores – and Julia is on the way. Someone at NOAA has a wicked streak; and every four years this will recur in the Atlantic as his or her joke. The connections that we make when we are not making connections, because it was this moment where the whole concept of infinity's did not have a tome, fictional or not, for people to see the scope of.
When one looks at the Gödel incompleteness theorem – and remember he did great deal after starting his career with a bang – you see it is of the nature of aleph null. But aleph null is only the fundamental generator of infinities. There must be a problem which has as its solution the number of the continuum. And this is what we are going to talk about.

The problem exists in the area of game theory. We have talked about Gödel, Turning, Nash, and Cohen – and these all sit very well in game theory – though Cohen is still a particularly hard subject. These all set with the idea that each individual proof boils down to a number, and since every symbol can be converted to a number, this means that a great deal of human knowledge – not just mathematics, but economics - and things that are based on economics – fit within it quite nicely. But now we are going to show that points are not the only thing which has truth in it, there are lines as well. And the nature of proofs is different for lines.

The story begins with a young man called Gaston Julia, who discovered that there was a line which denoted a series of complex points on a 2-D number line, so that points outside of this line would go to infinity, and points inside the number line would go to 0 – but there was indeed a line which would neither go to infinity or go to 0. Eventually this would be discovered by Mandelbrot, and be called the Mandelbrot set. With Julia discovering the set, and Mandelbrot showing how it was quite different from the ordinary sets, it came in to the public consciousness. But that consciousness did not understand the deep significance.

If you take a point, and either it goes to infinity, or goes to 0 – then it will eventually behave like points in game theory. But it may take a very very very very long time to do that. But if it never goes to infinity or 0, it will never behave like them. For example, take 1. it will change between -1 and 0. this remember is part of the proof of Gödel, but it branches off from this point. So if you look at it in one way, it starts out as a proof of Gödel, but it does not continue down with a proof of prime numbers – instead it looks at everything as a line, which is to say, it delves into the region of Cohen's proof of the continuum hypothesis. But we have not squared this circle yet.

Before going on to this problem, let us first go over the odd behavior that the Julia and Mandelbrot set have.

The first thing that we notice, is that all of the Julia sets that we encounter are not concave, which in many parts of mathematics is an important start. Nor do the sets have a finite number of non-concave points - for example, 1/x has only one place where it is not concave: 0, where the negative side of the equation heads towards negative infinity, and the positive side heads towards infinity. (Remember on a machine that will mean that, stealthily, they will head to the same place – because on a machine it is not a plane but a circle. ) instead be Julia sets are infinitely convex, and there is as many non-concave fracture points ahead of us As there were before, in that one to one infinity matchup. So, which infinity is it?

The first guess is aleph-0. But clearly some of the points are not on the rational part of the number line. For example take √3/√5 Clearly it is not one of the rationals. But because by making it accessible through a number plus a symbol, it can be Gödeliazable, as we have just done. What the really mean by unfixed irrational, is a number which cannot be easily accessed by this means. But any place that we can look for a number, we can create a symbol and a simple rule which makes it accessible by this means.

The only problem is that a physical mind has limited space – and even if we took all of the power of many brains – there would still be numbers for which Cantor's definition would still apply. That is, it could not be related on the diagonal. So the problem is that we know what will happen, but any time we look, we will find out that it matches up 1 to 1. we blink, and it is gone – we know it would happen, but when we look it does not. In other words the number of the continuum is neither true nor false. At this is why the number of the continuum is neither true nor false, no matter what Woodin will argue. It is like the blind spot in our vision, when we look directly on it, it vanishes.

Irrational numbers which not only do not repeat, but they cannot be fixed – we put “...” in the definition. So there are at least two kinds of irrational numbers, those which can be fixed and those that cannot be fixed. And we know that if we limit our attention to the fixed numbers, that they are part of aleph zero.That means that √3/√5 is part of aleph 0, but that a continued fraction, that cannot be expression in a finite form cannot. The problem with this is we do not have a way of showing that a continued fraction can be re-expressed in this way – for example pi can be re-expressed as a host of continued fractions – it is only that we know that it can also be re-expressed as pi. This also means that all physical numbers are in the first category, where as mathematical numbers are not – because the Plank number has a physical limit, and physically all numbers can be reduced to the Plank number.

There are an unlimited number of fractions do not have such proclivities as pi – indeed just change one number out of pi, and it loses any meaning – just a string of infinite digits, which means very little except as pi in some Gaussian form. In other words, there are physical infinities, probable physical infinities, and “god's infinities” - is just that we cannot tell the second class from the third class.

But there are “god's infinities”? Yes, because we have Cantor's proof to this already.What this does tell us is that there is an infinity that is smaller than the number of the continuum, which was what Cohen hoped to prove, and what Woodin may have done. This is why Julia and Mandelbrot are important, because they attack the line completely differently. Or rather, they are interested in the line, and go about proving its fractal nature at a diagonal. This means that they use Cantor's ideas in a different way – they use recursion on the number line itself – the generator of the Mandelbrot set is recursive. So instead of using a proof that is recursive, they use a set which is recursive.

That is the beauty of Julia and Mandelbrot sets – because remember, a proof in Gödel form is also a “construction”. Thus the proof of Gödel is also a construction of primes.

It is here that we notice different styles of proof among the great mathematicians of infinity. Cantor tries to establish the simple proof – partially because he knows that no one else believes him. He first publishes a difficult versions of his proofs, but then found easier ones. Which is why he est. the counting by diagonals proof, which showed that things that seemed different were the same; and he est. the number of the continuum, which showed that it was different from the lower order of infinities. In fact I have used these proofs to show that some irrational numbers are the same as aleph-0. Gödel was different, he est. a single proof that was flawless, and showed that he was thinking about proofs that he could not describe. Cohen however, these many proofs as an exercise for the student, if student could describe the workings of a mind that can chew through his set theory. Turing uses a different sort of proof, and it will often take several times through – or even with help from a trusted source. But he comes up with some astonishing proofs. Nash tends to write simple proofs, that astonish with their complexity, which often he did not think of.

We should touch on the subject of madness. Nash was from time to time imagining characters that he thought were real. Thus he spent a good deal of his life not knowing if someone were real, and he talked with them even when he knew they were not real. Sometimes I wonder would I take the trade of having imaginary friends, and lots of stupendous genius. Gödel was thought mad, but there is extremely little proof of that. What there is is a sign that he did not want to be bothered by any but few strange cases of brilliance; for example he corresponded with Cohen, and there was no sign that he was anything but polite.



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