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.
