Gödel 12 - Cohen II

For example:

Is a continued fraction, which represents 4/pi. But a slightly different continued fraction could not be represented as anything smaller. It would just go on like pi, but it would represent the ratio on some different Gaussian, one which we do not have any use for as yet. Which means there is no shorter version, unlike 4/pi.

And that is part of the problem, we do not know if we will ever have a use for it, or some time later some mathematician will find a use for it.

√3 means something, but other things do not – but the problem is, eventually someone will quite possibly find out that they do. But we cannot be certain, we know that there are God's Irrational Numbers because of the second proof by Cantor, but we do not know what they are. This is the difference between GIN, and those that we think are. We will never know - we can only know if they are proven not to be, but we can never know if they are truly GIN. But if we cannot know that they truly are, that will mean that the limit for aleph-0 will never actually be known. Which was the point, or Q. E. D.

Hence the rationales are Aleph 0, but so are a limited number of Irrational numbers, which we will investigate now. Clearly √3 and their ilk are Aleph 0, as are a division between them. But what about a radical which has an unlimited number – of such as ^pi√3? Clearly if we can write the character down, it is Aleph 0, but if we can only approximate the number, it might be not be. But here is the catch, if at some later point in time, we decide that it is useful to create a symbol - so that we can write it down - then it becomes Aleph 0. In other words, we cannot say that a number is not Aleph 0, we can only state that do not know. Again an answer which is neither to nor false, until it is known to be true. Hence we do not know it is God's irrational number, until we are dead. Not a very satisfactory reply from an attempt to make all answers true or false.

There is a small secret, this is not uncommon in mathematics. But if it is, then the entire framework of Boolean algebra is shot. There are not just one, but several places where GIN show up. This is why you can never create a Boolean algebra which is also infinite. Again: you can have a Boolean algebra which is finite; or you can have a Boolean algebra which has a finite number of axioms, but at least an entire subset which will either be false or not be able to know whether they are true or false; or a Boolean algebra with an unlimited number of axioms. That means that Paul J. Cohen proof is correct, but we have shown that by developing a proof which is far simpler. Of course we developed that proof by looking at Paul Cohan's proof. But that is why Cohen is a genius, and we are just fumblers trying to make it easier the line of proof which he so often set down in his book – our proofs work because of the ur- ideas behind the mathematics. In my case it is the ur- proof behind the concept of forcing. This is because forcing relies on the fact that forcing those only one way, and not the other.

Which, if you think about it, is one of the forcing proves – A → B; but not A ↔ B. in other words the very basis of our minds makes certain proofs easy, and others harder. So my proof works because of Cohen – a vignette upon this continuum. Just like i386, and its descendents, allow data to flow freely in one direction, but very hard in the reverse direction. If we find other creatures, they may not have the same quirks in their equivalent of a brain. Some things will be easier for them, and some things much harder.

So that means that however complex Boolean algebra is, it will have either one thing which is verifiably false, and will remain false even if you set up a symbol for it, or it will have one thing which is not known, again even if you set up a sign for it. Boolean algebra, for all of its great gifts is intrinsically false, but it is usefully false – just like the Newton mechanics is useful for problems within a defined box – even though we do not know what the boundaries are. This is why many physicists are also mystics. I have heard this from many people, and I convey that to you.

But for the moment, we are stuck with our own brain, and with computers who mimic – but nowhere near completely – our brains. And it is also the case that there are different configurations in our own brain, thus what was difficult for Cantor, Cohen, Gödel – was not difficult for others.

Thus we go on to Julia. And Mandelbrot.