Re: Ultimate debunking of Cantor's Theory

Related with this thread I enclose the following
article from my good friend Marcos Winnocur.

Fernando.
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A powerful knight is Sir Infinite

Introduction

"Evil is not the enemy, Infinity is", sentenced
Borges. When he didn't talk politics, the gentleman
demonstrated quite good aim. Of course, we can oppose
evil with good. But, what about the Infinite? Oppose
it with the Finite? The most evolved Finite we know,
that is, out selves? Nothingness? Waste of time. The
finite opposes the finite and the nothingness opposes
the being. And the Infinite devours them all. Written
with a capital I, nothing is external to it. It
encompasses all, it has no opponent. In these
conditions, the Enemy is invincible, escapes all
attempts at manipulation. However, this point of view
has caused crisis and it is interesting to remember
how human thought has conceived the infinite,
specially in mathematics.

It wasn't always this way. Two and a half millennial
back, Aristotle, the incomparable philosopher in
Ancient Greece described in Fisica the conclusion he
arrived at: "if it is divisible, any of it's parts
must be taken as infinite". Aristotle discarded this
alternative as absurd. Much later, in the XVII
century, another milestone in Thought, Galileo, an
Aristotelian critic in none other than the concept of
movement, is of similar opinion on infinite. We must
leave behind the Renaissance and the Industrial
Revolution and arrive at the second half of the XIX
century before we find a new thought attitude
confronting ?the enemy?

The whole and the part

We are accustomed to considering universal the
following proposal, born from common sense: "The
whole is bigger than the part". And so it was
recorded two millennial ago by Euclid in his
classical work Elements. However, the application
field of this proposal is reduced to the world of the
finite. It all changes when the wizard named Infinite
intervenes. There, in the XIX century, the
mathematician Dedekind said thepart is the same as
the whole. It is Aristotle?s ascertainment. With the
only difference being that Dedekind accepts what is
negated by the Greek. Thus the Aristotelian veto is
lifted. Despite how absurd it seems to our eyes, we
must accept infinite as it is: divisible and plural.
This way, the Infinite mentioned at the beginning,
omni-comprehensive, total absolute that admitted no
division, is placed aside, void of any functionality.
It may be conceived as necessary infinite in the act,
but it is not operable. Does it or does it not exist?
It doesn't matter. For the time being, it is not
useful.

An example. We take the series of "counting" numbers.
Half of them are odd, half of them are odd. When
compared, all three are infinite and in the same
degree.

There are no series which, having no beginning nor
end, are "bigger" than any other. But, following the
example, if we appoint a stretch of them leaving out
odd or even numbers, the stretch will be smaller in
content than the general series, which contains both
odd and even. Actually, we are comparing between
finite, and we can do nothing else. Imagine we set
out to find the beginning or end of infinite?

The reflection of mathematicians in the XIX and XX
century was versed upon these questions. From them
all, the main figure that stood out was Cantor. Who,
whilst he was alive, did not stand out and his
opponent, the mathematician Kronecker, was who
appeared to be the sane person as opposed to Cantor's
extravagancies, who spent the last twenty years of
his life in a deep depression.

Actual Infinite and potentially Infinite

Kronecker appeared vindicating Aristotle,a "finitist"
like himself. Going back to the ancient Greek, his is
the conception of infinite in act and potentially. It
was a time of deep reflection in the Greece of free
time and of the philosophers. The pitagorean school
had made a discovery: the relation between diameter
and circumference, the number "pi" which resisted
exactness, the count was never closed no matter how
many decimals were added after the ever more exact
measurements.

Exactness was negated in the house of mathematics! It
was deemed so incredible and subversive that two
resolves were decided: One, to keep "pi" a secret.
Two, to baptize it as irrational. Rationality negated
in the temple of mathematics. Well, Aristotle took
the bull by the horns. The Infinite is one, but its
manifestation is dual. Actual Infinite, that is, "the
whole" Infinite is a tribute to God. Potentially
Infinite, "incomplete", as it is offered to us
humans, to our intelligence which is a divine gift.
The relation between diameter and circumference is
potentially infinite as a testimony to the Actual
Infinite. Where the former couldn?t exist without the
latter. It is, to all intents and purposes, proof of
the existence of God. Thus, mathematics and mysticism
will go hand in hand. Cantor himself, a mathematics
heretic, will be a religious believer full of
fervour. Now then, it may me deemed more convincing
the secular formula designed by Peano for arithmetic
of cardinal numbers, about to be closed in the XIX
century: "The immediate successor to a number is a
number". Thus implies the renouncing of a final
number, bigger than the rest, any kind of
mathematical god, a renounce born from his own nature
and not actual infinite. At the same time as
independent entity, that the immediate successor
completes the personality of the immediate
predecessor. A necessary, absolutely necessary, sine
qua non, chain to the number is made.

Now then, if we take any two numbers in immediate
succession, would we say there is a lapse? It is a
well known fact that mathematics and temporality are
not easily put together. One could say the former has
escaped the latter. How old is a rectangle triangle?
Or a number? These are meaningless questions.
However, whether it is in the deployment of the
irrational number "pi", or in the series of numbers
"to count", there is an order to take into account.
If I write 0, 1, 2, 4 (...) I will have cheated and
will be more or less useless: that number 4 will
stand and satisfy the functions, of 3, that is, it
won't escape it's destiny: successor of 2. And I
cannot reach 4 without going through 3. So, what is
"going through 3"? In mathematical terms, it is to
summon an immortal entity, the number. In human
terms, to enunciate a symbol and give its place to
the successor. So, recapping on our original question
about whether there is a lapse between two numbers in
immediate succession, the affirmative answer will
lead us to potentially to infinite, whereas the
negative will lead us to actual infinity. But we
handle the first one, whereas the second one is
unreachable but we can assume as an eternal present.


There, where something is and its contrary also is;
where all is given to infinite and time is abolished
since all is actually happening with no need for past
or future, as both are happening at present. For
once, mathematics accept Mr. Time, for there is no
way to build a potential infinite not in
succession..

Geometry and a date with my girlfriend

Lets start with geometry. A point, special unit for
mathematics, "it is what has no parts". It was this
appointed from dawn by the father of geometry, the
Greek Euclid. It is a definition that ended up not
conforming, since the properties of the point
transcend the indivisibility. Thus, the mathematician
Hilbert, almost at the end of the XIX century,
developed the immaterial nature of the point without
actually defining it. It can be contrasted to the
infinite, since infinite contains an infinite number
of parts, and the point, however, contains none as
cited by Euclid. How can it be explained? It isn?t.
It just is.

This is so decreed, the Volunteerism legal standing
of the absurd. It nevertheless works fixing positions
in space whilst not actually taking up any space,
since it is void of the virtue of extension. Of
course, it works in a space that does not pretend to
be real but ideal; or, as it could be called now,
virtual. Actually, it is a worship to the goddess of
Perfection. The physical world prevents me to reach
the ultimate, divisibility cannot be discarded. It
was assumed that the atom was the physical unit,
which later resulted to be composed of the so called
elemental particles. In mathematics we can recreate
the universe. We can make true a unit without parts
nor extension, that is inexistent from the physical
point of view, that also has virtue of generation.
Capable of making a line of itself. From the line, a
plane. From the plane, a volume. That is, going from
the non-dimensional, to the uni-dimensional, to the
bi-dimensional, to the tri-dimensional, the volume
which finally restores us to materiality.

How do we achieve it? By means of magic. And the
magic is called infinite. An infinite number of
points results in a line. If the points go all in the
same direction, we call it a straight line. A second
stroke of magic and an infinite number of straight
lines placed ?side by side? gives us a plane. A third
stroke of magic and an infinite number of planes
?piled up one on top of the other? gives us a
volume.

A stroke of magic also transforms a regular polygon
into a circle. How? By giving it infinite sides, its
perimeter is transformed into a circumference. It is
another stroke of magic. The circle appears thus as
the ultimate possible figure which is no longer a
polygon. Or results in a polygon with infinite sides,
which is the same. Allow me. I can be five minutes
late to the date with my girlfriend, half an hour
late, an hour late, etc. But, just how late can I be?
I have no alternative. The latest I can be is?
forever. But forever no longer belongs to the late
category. My girlfriend has coupled up with someone
else, is a grandmother and I'm driven from home with
my feet in front? to my date fifty years ago I will
be as late as possible, that is, forever.

But there is more. The geometrical definition of the
infinite was recognized two millennia ago by Sexto
Empirico, an ancient Greek, and confirmed by modern
thinkers such as Nicolas de Cusa. What is de
geometrical definition? The infinite is a circle
whose centre is everywhere and whose circumference is
none. In a word, the a circle negated by itself. The
centre is everywhere seems a ?comfortable? option but
it is actually a trap: it is resolved in any given
point that is then stripped of the circumference. In
the widest sense: it is not found anywhere since it
would limit the circle to which it belongs. But then
it is actually a circumference that is confused by
the diameter, come about from an infinite straight
line. More clearly: the circumference is inflated as
a two-dimensional balloon, an asintotic arc ever
closer to a straight line which it will never achieve
since it needs the stroke of magic of the infinite.
Thus, the circle has vanished and its centre has lost
all meaning, resulting in any given point. The circle
has succumbed, it has too many centres and it lacks a
circumference.

Conclusions

We are back in materiality, having started from
nothing, since that is a point: a nothing. Who beat
the Nothing? The Infinite. He is capable of absorbing
the biggest absurd, transmitting his virtue: the
number of points is infinite, as is the straight line
by definition, as is the plane. The volume, however,
is finite. Of course, for its full materialism we
will have to provide it with movement, thus adding
the fourth dimension, time. But the step from the
virtual to the real is taken. From the mathematical
fantasy to the physical world. To the spoon with
which you fed the mouth with soup. For me, with
pasta, please. To the airplane you took to fly over
great distances. I asked for a window seat, look it
is written here and now someone else is sitting in my
seat. Thusm the spoon and the soop, aisplane and the
distances flown to leave me in Honolulu, the planet,
the galaxy, the cluster of galaxies, however
excessive they seem, are all objects of the finite.

Ours is the world of the finite. But, curiously, it
is due to the infinite. Spoon, airplane, planet,
galaxies? and then? There has to be something else,
and after that, a second something else, and after
that, another, and another, and so on and so forth
indefinitely as it happens with numbers "for
counting". Even if after that there is nothing, that
nothing, to these effects, is something. We cannot
conceive that after "the last star", the universe
ends or that numbers run out, or that there is a
final one, bigger than the rest. No, Russel himself,
who assumed it sometime, later recognized his
mistake. We lack the perception of the infinite but
we intuitively know it is necessary. And mathematics
give us news from the universe that physics don?t
dare to: if finite to suit our senses, infinite to
our minds that refuses to conceive "the last". We
cannot imagine the infinite unless as a continuity.
At the same time, that continuity is necessary to
indefinitely renew itself since we cannot imagine the
infinite to end. Potentially and actually, that grand
wizard that sometimes is called God, pushes us to
agnosticism: his anthropomorphic presence is not as
conceivable as his absence.

So you see, as powerful as no one else, he sometimes
tempts us to name him in capitals and appoint him a
knight. Sir Infinite, a.k.a. the Agnostic and a.k.a.
the Pantheist, disciple of the Philosopher Spinoza.
Sir Infinite, powerful knight, whose will in the
universe can be read in the inexorable physical laws
and whose imaginary are mathematics.
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you are a child writing poetry

not an adult mathematician.

tommy1729
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