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From: Archimedes Plutonium on 21 Jan 2010 02:25
I probably will have to cut this chapter out and do an entire book on
just correcting the Peano
Axioms.

And whether I do a solo book on Peano Axioms or include it in this big
book on overall
Correcting Math, I should start the discussion of Peano Axioms by
travelling through the
mind of someone who just learns the Peano Axioms and is thus
brainwashed and blinded
by not comprehending all the flaws and fakeries of what he/she just
thought they had
learned and understood. So let me travel through the mind of a typical
person having
learned the Peano Axioms and the Natural Numbers thereof.

Let me also remark about a poster I once saw. I think I saw it in the
University in the
1970s of a poster where the student goes to College and is met with by
a team of
surgeons that carefully saw into his cranium and so to speak "lift the
lid open" as seen
in the next frame. And then a frame in which a bunch of assistants
carrying some huge
pot of liquid and then pouring it into the student's cranium. Then the
student is sewn
up and graduates with a College degree. Now, many would be cynical and
adverse to this
poster but the poster rings with alot of truth to it. Especially in
the sciences, where more
often than not, that these educated people take their education so
inflexible that they
are never able to question what was poured into them as "education and
knowledge."
Especially mathematics in that if a student sees something in one book
and two or
three books then for the remainder of their lives, no matter how wrong
those tidbits are,
can never seem to examine or question those tidbits. I call it the
dogma of education
or the turning of science into a religion.

But now let me travel through the mind of some individual student who
has just learned
the Peano Axioms and the Natural Numbers they represent. So they learn
about a Successor axiom and an axiom that creates 0 (should create
both 0 and 1 together) but we will
not labor that point. So the student sees that we have a 0 and this
Successor that adds 1 to
0 and delivers a new number of 1. And continuing it delivers another
new number by adding
1 of the Successor function and arriving at 2. Now the student
realizes that this is a continuing
adding of 1, and endless adding of 1, delivering the Counting Numbers.
And those numbers never stop because if you stop, you can add another
1 to it and have a new number.

Now we reach a point in the mind of this student that the student
realizes since we never stop
adding 1 that these numbers go on endlessly and go to infinity. And
this is what the brain and
mind of every student sees in learning the Peano Axioms. That these
numbers are unstoppable since you endlessly keep adding 1 more to the
previous. And so this student
has a picture of the Peano Natural Numbers as numbers that start with
0 then 1 then 2
and go to infinity. Now most students never bother about asking
anything about whether each of those numbers is a finite-number,
because, well Peano never raises that issue of
finite-number and it is not written in any of his axioms. So basically
travelling through the mind
of a typical student who learns the Peano axioms of the Natural
Numbers comes away with a
picture that these are the counting numbers and that they occur by
adding 1 endlessly to produce a infinite set.

But later on, this student begins to learn that there are two types of
numbers, one called
finite-number and another called an infinite-number and where they
pick up this new found
knowledge is usually those inquisitive when they learn what a Real
Number is. They
learn that a Real Number has a unique decimal representation and that
consists of a
finite-string leftwards of the decimal point and a infinite-string
rightwards of the decimal
point. And they associate that finite-string as a Peano Natural-
Number. Usually the student
never examines this Real Number finite string with the Peano Natural
Number and asks
a relevant question. Since the Successor is an endless adding of 1,
well, will that endless
adding trespass into infinite-strings leftward and not just sticking
to finite-strings.

Usually, no student of mathematics ever gets to be that curious. Most
everyone
drop the issue of why does not the Successor produce infinite-numbers,
and not
just so called finite-numbers. Most students are not bright enough to
question
what the textbooks tell them.

Or when this student learns about the Hensel p-adics. And the
student learns that the Hensel p-adics have the same Successor
function only it
is in the form of a Series addition of endless adding 1. But the
bizarre thing is that
the student learns that the p-adics starting with 0 then 1 then 2,
goes to ....99999
which they are taught is the same as -1. But does the student begin to
be curious
as to why in the Peano axioms the Successor is supposed to create only
so called
finite-strings of numbers and is not supposed to create a infinite-
string. But in
Hensel p-adics that very same endless adding of 1 creates 2222....2222
or
9999....9998 or 9999....9999.

Now, let us say that Student A is super curious and smart and asks
questions as
he/she is learning the Peano Axioms, and says, well, since the
Successor is endless
adding 1, that the numbers that are spit out of the machine that
creates them from
the Successor that those numbers are going to be both finite and
infinite strings, because
the Successor cannot help but produce or create infinite-numbers since
it is an infinite
adding.

This smart student also knows that the Series 1 + 1 + 1 + . . . .+1
although you stop
it at intervals and can create 0 , 1, 2, 3 but since it is infinite
means it also creates
4444....44444 or 8888....812345 or 9999....99999.

Now if Peano did not want any infinite-numbers in his Natural Numbers,
he should have
defined where he would like for the Successor to stop, so that the
numbers like
9999....9999 were not part of his Natural Numbers.

And this is where it is extremely difficult for students to tackle or
wrestle with. There is no
stopping point or juncture where you can say from this point onwards
are only infinite-numbers
and below are only finite-numbers. The only way to do that is to
define Finite-number as
a specially selected number such as 10^500.

Now the above is only my first attempt of explaining what goes on in a
typical mind of a
student when he/she learns the Peano Axioms, and thinks he/she
understands them
and accepts them. The reason I want to perfect this post by editing
and re-editing in the
future is because it gets at the heart of the problem of lack of
comprehension on the part
of students of math and especially on the part of instructors of math
who have been
brainwashed worse than the students, and like the cutting open of the
cranium and pouring
in the broth of a pot and calling that a degree in mathematics from a
College.

I mean, in a sentence, how much more stupid can one be, to think that
if you have
an endless adding of 1, that all your numbers are going to be finite-
numbers, finite-strings,
when a reasoned person can easily grasp that the endless adding of 1,
whether Successor
or Series, will fetch infinite-strings or infinite-numbers. It is
fascinating, that virtually every
mathematician today, was brainwashed and blinded into not realizing
that a Successor
function must yield infinite-strings; must yield infinite-numbers.

So I have my work cut out for me to get this post edited and re-
edited, where I travel
down the brain and mind of a student learning the Peano axioms and
discussing in
detail, how easy it is to get brainwashed and blinded.

P.S. the next time any person tries to persuade you that the Successor
cannot create
an infinite-number. Then just turn the tables on him by saying " so
you think that the
series 1 + 1 + ....+ 1 does not go to infinity but is equal to a
finite-number.


Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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