Friday, October 14, 2016

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Gödel 11 - Cohen

Cohen

You may have heard of Gödel, but you will probably not have heard of Cohen. This is because he is a great man, but does not have a great story – yet. A great man requires a great story, because without the great story he is not remembered. I say this because Cohen is a great man, but he does not have a story – and so is ignored, even though he came up with two of the most useful proofs in logic. What he did was monumental – he proved that the number of the continuum is neither true nor false, so to decide things in the number of the continuum, one has to have not one number line, but to which are opposed to each other. Just as the Zeta function of Riemann, probably, is.

Because after Gödel, he is the next person who finds problems to be solved in the nature of mathematical infinities. He takes trivial little things, and shows that they are important.
For example, he finds a proof to Cantor's problem of the number of the continuum. His proof rests upon what is called the Löwenhein-Skolem theorem, which extends what we have discovered with Gödel – only to a larger space. Remember the key thing about Gödel, is that his objects are points, but one needs to extend that to lines. What L-S does, is it takes the set of sets, and says that if the set is finite, then it is set of sets is also finite, but if it is infinite, then it is set of sets is also infinite. What Gödel did with points, Cohen did with lines.

But with Gödel, the points were true and false, with Cohen the case was somewhat different: not true or false, but finite or infinite – and what that does with proof. The reason that Cohen confuses people, is they do not have a road into his reasoning. We shall provide that road. His problem is not mathematical, but language. He did not notice that the language of math needed to change. So everything is in his paper – just as Cantor first published a proof of his theory, which was obtuse. The difference is that Cantor later realized there was a much simpler proof, which he then publicized. So Cohen got true math geeks going, but could not explain things to other than pure – and I do mean pure – math geeks. What he did not supply, I will add in two is astounding proof. There is nothing truly mathematical about what I am adding, just a simple occasion of forms. We first have to go over his proof as he left it, before supplying the simplification.

That everything comes from what happened before, and if we want to understand something new, we must first realize what it came from. With Cohen, the two parts of his theory are the incompatibility of the basic axioms, which he uses ZF, GHC and the axiom of choice – the three are needed for this. The second part of his process is by having “ forcing”, in a way similar to Einstein's general relativity. It is all right if you do not know how Einstein did it, because that is a class by itself.
Part of the problem is the kind of mind that Cohen is like is Gödel, as opposed to Turning and Nash. It is not that any of these thinkers are not geniuses, but some are finders of problems, while the other set is the solver by way of a solution. The field gravitates towards the finders of solutions, because there is always so much to do with tidying up or expanding. So thinkers like Gödel sit alone, attracting worship from some, but not having lesser minds wish to scale the heights of them. But then every so often another mind looks at the summit, and realizes there is another somewhat to climb. This is what beckoned to Cohen. So he first had to make Gödel work suited for his needs. One part was trivial, in that way that mathematicians use trivial. He needed to convert PM to a more suited form, and that was a variant on ZF. He admits in his paper that there are reasons why Gödel chose the route that he took – but he also explained why it was not for him, because he had a certain goal to attain.

What he wanted was to use Löwenhein-Skolem as a generator of Lemmas that would allow him to take the point nature of Gödel, and bridge it to the line nature that he was envisioning for proving the continuum theory. There is, naturally, a great deal of preparation – but it is not difficult if one can see beyond the trees and look to the forest.

With these two pieces, he sets up a one to one correspondence, in the manner of Cantor. While there is a great deal to explain, this is the basis of his proof. Essentially, ZF needs to pieces in addition – and that while it is not possible to explain everything, you can force something in two some particular shape and explain only that. One does not need to explain everything, just the proof which you are looking at. A great deal of the preparation can be handed to the student as a form of exercise – which I am going to do for you, the way it was done for me. Remember that LS is a very simple theory, though it has many contortions.

Now a great part of the lemmas that Cohen uses are really of the form: “ regular Boolen mechanics works here”, because he has two fit the proof around the jaws of Gödel. This for two reasons: first of all he does not want to prove that this is just a side light of Gödel's proof, and second of all the dozen which to have anything but true or false until he springs the continuum hypothesis.

First he must state the crooks of the matter, which is the Löwenheim-Skolem proof. Let us state the LS theory again: with a model for a collection of T constant and relational symbols there is an elementary submodel of M whose cardinality does not exceed that of T if T is infinite, and if finite will be countable T. in other words, there will be a closed countable T only if T is finite. Which again refers me back to a professor whose native language was German, and my grandfather said it would be all right if my grandfather stood in the first row to explain any translation features. The German professor accepted, and then proceeded to mix up in-fin-ite, and fixing this he made the faux pas of pronouncing fin-ite, as finit. The German professor seeing my grandfather pronounce the word correctly, excused himself and sat down for he made further blunders, or omissions.

What is important is not the little features, but the large program which Cohen is trying to reach for, because several of the proofs needed can be streamlined, if they were important. This is why going to LS in slightly different ways is far from the best proof – but it is a side instance, which can be cleaned up later if it is important. But it is not, what is really important are the disconnectability of GHC and AC from ZH – that is you cannot prove either GHC or AC from ZH, and the idea of forcing. Thus such proofs as our needed to connect LS can be left up to students of the cause. On page 83 he notes that this implies that Gödel was right.

This brings us then to forcing, and within it the proof that Gödel's numbers reside within. What he wants to get to, is the list that ranks by induction of P forcing A, and the results upon A that P forces. He is again quick to note that very simple rules of propositional calculus do not apply to forcing – such as ~~A does not imply A. this is because we do not know what mathematics P forces. In a sense, we are talking about a mathematics which we do not know the order of. For example, while in numbers A + B = B + A, in strings, abc + def does not equal def + abc. The reason is that “+” does not mean the same thing. In numbers the abc becomes a single number, where as in strings it does not. There is an ur-lesson to this: our brains require that numbers join more easily than letters.
This means that some introductory lemmas are in order, proving that either A or ~A will be held true, and not another case. Remember we are in the world of Gödel, where such things need to be approved, and not merely in some Boolean way dismissed. He also needs to prove that a complete sequence exists, but again, this can be homework for that class that you will be taking.
But step back and look at what forcing is doing for you, Gödel proved that numbers could be not true and not false, and Cohen is proving that cardinality's of numbers can also be not true and not false. This is why he has to prove that it is not part of the number sequence, but an original idea. In other words he wants to prove something different from Gödel.

That is why he “forces” two sets of numbers: a finite one and an infinite one, we will take the infinite one first, rather than as Cohen takes the finite one first. He does this because limited is easier to handle, but since then we have found a way to treat unlimited first.

Unlimited:


1. P forces ∃xB(x) if for some c ∊ S, P forces B(c) The Element case
2. P forces ∀ x B(x) if for all c ∊ S Q⊇P, Q does not force ~B(c) The Every Case
3. P forces ~B, if for all Q⊇P, Q does not force B The Nand Case
4. P forces B&C if P forces B , P forces C The And Case
5. P forces B v C If either P forces B or P forces C The Xor Case
6. P forces A → B if either B, or P forces ~A The Or Case
7. P forces A ↔ B if either P forces A → B and P forces B → A The If and Only If case
8. P forces c1∊c2 or c1=c2 if it forces them as limit statements.


Limited:
1. P forces ∃xB(x) if for some c ∊ SB B>A P forces B(c)
2. P forces ∀ x B(x) if for all Q⊇P and c∊SB ,B>A Q does not force ~B(c)
3. P forces ~B if for all Q⊇P, Q does not force B
4. P forces B&C if P forces B , P forces C
5. P forces B v C If either P forces B or P forces C
6. P forces A → B If either P forces B or P forces ~A
7. P forces A ↔ B if either P forces A → B and P forces B → A
8. P forces c1=c2 where c1∈Sa, c2∈SB, r = max(A,B) If either r = 0 and c1=c2 as elements of S0 or r>0 and P forces ∀rx(x∈c1 ↔ x∈c2)
9. P forces c1∈c2 where c1∈Sa, c2∈SB, A<B if P forces A(c1) where A(x) is the formula defining c2
10. 9. P forces c1∈c2 where c1∈Sa, c2∈SB, A>=B (where A=B=0) If for some c3∈Sr,r<B if B>0,r=0, P forces ∀Ax(x∈c1 ↔ x∈c3) & (c3∈c2) (with the caveat that there might be a reduction in rank.)
11. P forces c1∈c2 where c1,c2 ∈S0 if c1,c2 ∈w and c1∈c2 or c2 = a and the statement c1∈a is in P

The seven of the forcing algorithms are a direct result of the axioms for Z1on page 21 of Cohen's famous Set Theory and the Continuum Hypothesis.

Remember that there are only five correspondences to P: ~,&,v, and ↔.
You will note that they are very similar, and indeed the darken parts are analogs – a change from unlimited to limited. So in reality cases 1 through 7 are the same, and it is only the case if one examines 8 through 11. so there are really 6 cases: the unitary cases, the case where the unlimited drops down to a limited, and the four cases of the limited – which remember the unlimited case marked 8 will drop down to.

You will also note that the entire thing is written in Boolean algebra, even though it is results do not offer things as true or false until one decides whether they force each other. So now there are two questions which do not work out to be true or false. Which means that either there must be further forms – which would break the Copernican unity, because Boolean logic has a flaw as does Aristotelian logic – or we half to live with the fact that true or false do not answer all questions, but that is anathema to the very core of Boolean logic.

What is missing from the proof is what is not, not what is. There is no mention of digital recording of numbers – and remember this would not be amiss to the Greeks and the Romans, because they did not have the concept. In no small part because they reacted with horror at the addition of the number “0”. the addition of digital writing of numbers was a great stride, so to cast it away for some reason has to be a monumental leap. What we are doing here is saying that there is a difference between finite numbering – which is the same as before – and infinite numbering, which is the way we have presented this. This small change is what makes Cohen see much farther than other people had. This alone would be enough for mathematicians to remember Cohen by. It would be a significant there in its own right. But what he does with it is show that cardinalities have an intimate relationship with points – so that while Gödel proved that one could prove the existence of the number of the continuum, Cohen proved that nor can one not prove it either. So that meant that the theory was always going to be a theory in the larger sense, because there is a third option, which Gödel had already proved. Once again the Boolean true or false test is broken.

The infinite one lays out a set of proofs, which are the Boolean way of cardinality's. So, 1, is for every B, there must be B(c); and 2 is for any B there is not some B(c) that is forced – and so on. But with proof 8, he moves back to the limited proofs, because he has run out of the unlimited proves scope – which he shows that if one is an element of the second, or they are equal to each other, then you must go to the limited proofs.

When he takes up the limited proofs, one can see the same method, first every, then any, then ~, then “and”, and then “or”. He then goes on to show that if one proves the other, and then if they both prove one another. Then we are with the last four proofs, which are slightly more vicious in their construction. 8 says that if they are equal to each other then any element x, will be equal to a third thing that will be equal to each other. This is a classic Euclid prove, updated for modern times.
9 is saying that if 1 is an element of 2, then there will be an A(x) that will be an element of A. It is also an updated version of an old proof, if the 2 things are not not equal to zero, then there will be an x(a) which will be equal to the number which slides down – 17(Gen(1)). In other words, this is the point where Gödel slips in like a ghost. Do not think that will get this, most people do not. But think on it and it may come to you.

The last proof says that if element one is part of element two, then either element 1 is equal to element to 2 or A(1) is an element of A(2).

Now this may seem complicated, but that is because it is complicated until you stare at it for a long time. Gödel said it was like reading a play. The point that Cohen makes here, is that there will be a slot for B and/or/nor/nand C – and then P will force the cardinality into place. Remember that Gödel proved numbers, and Cohen is trying to prove cardinality in the same way. Thus it is complicated in the same way that Gödel was complicated at the beginning, and gradually other people made it smoother. The key step in this forcing, is to arrange the Boolean proofs that will carved out of a relationship between two numbers. So what he is doing is grinding out all of the relationships between two numbers, and shows that there is a relationship. Because it covers all of mathematics, neither Cohen, or anyone else, will know the solution to any given set of numbers. He can just say that given a certain set of relationships between numbers, he can tell you what cardinality's will result.

Again there are certain Lemmas that have to be proven in the wake of all of this, but again there, in a mathematical sense, trivial. Which means hard. The most important is that a A will be true if for some part of N, Pn forces A. What we have done here is taken and unknown set which describes an unknown mathematics, and says that once we describe the set, then it is logic will also be forced.
But what good is this? Surely there had to be a point, and you would be right in thinking this – because without a punchline, there would not be much of a point. So what if we can do mathematics on some set, even if we do not know the mathematics before hand?

Let us get back to the fundamental axiom, that GHC and AC are separate, and cannot be proven from ZH, or from each other – in other words there is a triangle, which all three need to be there. But what happens if the axiom of choice fails, does this mean that ZH will also fail? Or that GHC will fail?
It might seem like the answer is “yes” - for example Cantor thought so. But Cohen shows that if one arranges the axiom of choice after the continuum hypothesis, the answer to this question is “no”, the axiom of choice does not need to be there, and life will go happily along. What it does mean is that GHC is not proven. In other words as with numbers in Gödel, it is neither proven nor disproven. Which is the goal which Cohen sets for himself. How he does that, leads him to the number set of the infinite: aleph. Remember that Cantor proved that there was more than one infinity, but he did not prove that the number of the continuum was the next link, he proved that it was larger, but he did not prove that there was not some aleph-1 which was smaller than the continuum, but larger than the x/y – that is to say aleph-0. What Cohen wanted to prove is that the number of the continuum could be aleph-1 or could not be, in short, the number would have to be decided under the existing rules of mathematics. In other words, first you have to decide if you want to do mathematics on the set of the continuum, before you do any mathematics on it, and you have two decide in a mathematical, not linguistic, format.

So the maze of Lemma's in Cohen's proof hinges on the same idea as Gödel's proof: how do you define things that are not definable in mathematics? Gödel proved that this led to a contradiction with numbers, and in a later paper showed the first have of a proof on cardinality. What Cohen did was to provide the last half of the proof on cardnality, and showed that by “forcing” a large number of other proves could be ascertained.

The heart of his proof is then that the axiom of choice fails for pairs of elements P(P(w)).
But there is an easier way to show this, one that does not go through so many Lemmas. What it does do is show that there are many kinds of irrational numbers, not just one.

-

So now let us supply the simplification. It is that to words in the old proof, need to be different. This is the same way that numbers and letters lead to different forms of arithmetic: numbers can be mashed together but letters need to be a part.

The words that need a different meaning are rational and real. Because in the old definition, rational means can be divided in two digits: 10, 2, 8 – or any other form. So 18 is a rational number because we can place it in some buckets that each contain one digit. So for example, 1234 is actually 1 * (thousand) + 2 * (hundred) + 3 * (tens) + 4 * (units). That is the whole definition, and real numbers are completely different in that holder definition.

But in the world of infinities, some real numbers are actually closer to rational, than others. This is because they can be written down as a clear concise number. If you ask me what the one millionth digit of √3 is I can tell you almost as quick as you can ask it. But a number which goes on and on, I will have two work harder. Because I have to compute the digits and then do the arithmetic, as opposed to just doing the arithmetic.

So in the new definition, rationals and reals that one can write out, are the same; but rationales which cannot be written out our, perhaps, different. I say perhaps because we do not have the brainpower to sort out – yet. But there will come a genius, or series of geniuses which will figure them out. But as to Cohen, the rationales and the discrete reals can be gathered together under the same heading – Aleph 0. He supplies the facts,  I just supply a detail.

This is the simplification that makes his proof easy to understand: if it can be written down, it is Aleph 0, but if it cannot we do not know whether it is some failure of our mathematical abilities, or it belongs in the category of God's irrational numbers. Which we will never know in a single script.
So we can lay out the following:

Rationals - Aleph 0
Reals that we can write out exactly - Aleph 0
Reals That we can not write out exactly - Not aleph 0 yet
Reals that we can never write out - Aleph 1, at least

The problem is we cannot know if the number is real and we just do not know if it will be forever, or eventually it will be decided to invent a symbol for it. That is, whether it will be God's irrational number or not. The phrase “God's Irrational Number (GIN)” is an homage to the rubrics cube, where “God's Algorithm” is the best way of solving the cube, even if we mortals do not know how it is done – because we know the number of moves, even if we do not know what they are. This is to say that a great deal of Cube solvers do not believe in the God in the Bible.

Three not two.

The basis of Cohen is that every number has a square root, with the exception of those numbers who have a whole number, is irrational, in the old meaning of the word. Headed to this are irrational numbers which have some meaning. Thus there are more irrational numbers which have meaning, which are a subset of irrational numbers. This means that the rational numbers with some meaning are less than the whole set of irrational numbers, but larger than the set of whole numbers. But what that meeting is, can only be defined by words, and those linguistic words often do not have a solubility, in the Gödel context. Thus, they can be defined only by reaching out from mathematics. But linguistically also means computers, and aliens, and human beings who are wired differently, and anything else that we can imagine uses linguistics.

But what if there are different kinds of irrational numbers? For example √3 is a specific number. We cannot write it down in decimal, or any kind of whole number, but we knew exactly where it is on the number line. But not every number is like that, many of them go on and on never stopping at a specific number – we can only get approximately their. This is the basis of what is called in mathematics a continued fraction. The reason these things are different from √3, or any other specific number, is that with a continued fraction we only know approximately where it is. So there are at least three times of irrational numbers. Those that we know where they are, but cannot express them in digital terms. Those which we think we cannot place on the number line, but we are not sure. And those that cannot be placed on the number line. The problem is that we cannot say for sure which of the two classes it is between “ were not sure that it can be placed on the number line” and “ we are sure that it cannot be placed on the number line.” This distinction, between “not being sure” and “there is no possible way” means that we know the lower limit to aleph-0, but we do not know how far it goes. We know that if we take all of the numbers of the continuum, that is certainly more than aleph-0. So can no that something is true, such as a rational number is aleph 0, and we can know that it is false, such as the number of the continuum is not aleph 0, but there are things which we neither know nor do not know, unless we set our parameters.

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