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From: Archimedes Plutonium on 9 Jan 2010 01:49
AP wrote:
>
> Second Contradiction is that Peano and followers then insist the set
> formed by all the Natural
> Numbers is an Infinite Set of numbers. But how can they be an infinite
> set when Peano
> and followers claim every Peano Natural Number is a "finite-number."
> No infinite set is possible in which every element of the set is a
> finite specimen.

This process of correcting the Peano Axioms has taken me about 20
years,
so this is not a superficial jaunt for me.

What Peano Axioms produce is this set:

{0, 1, 2, 3, 4, 5, 6, 7, ...} where each element or member is a finite
number
and where the three-dot ellipsis indicates -- out to infinity.

So the contradiction is that how can you have a infinite set when
every one of its
elements is a finite-number. And Peano and his followers would say the
endless
adding of 1, means the set must be infinite, even though each element
is finite.

So here is what AP says happens when you endlessly add 1 as a
Successor axiom
that the set produced is not the above set, but this set:

{0, 000...001, 000...002, 000...003, ... , 999...997, 999...998,
999...999 }

Now Peano and followers never defined finite nor infinite and expected
everyone
to come into mathematics brandishing their own notions about what is
finite and
what is infinite. And they are told that every Peano Natural Number is
a finite number.

So what I did was precisely define Finite as that of 10^500 and beyond
is the Infinite.
That is clear and precise of a definition. Peano could have done the
very same, by
picking a huge number and declaring it as finite and the rest as
infinite. In fact, if you
give it some thought, that is the only way to precisely define Finite
versus Infinite. Only
I did not pick 10^500 arbitrarily. I picked that number as the largest
Planck Unit in Physics.
Because there is no more physics going on beyond that number since we
can never do
any experiments beyond that number. So where Physics gives out, math,
being a subset
of physics gives out.

But back to the discussion.

You see, my set of AP-Natural Numbers is identical to Peano Natural
Numbers, except the fact that I included the numbers that go to
infinity. I included the Infinite Integers since the
endless adding of 1 yields or creates or produces infinite-integers.

Peano and his followers never seemed to grasp the idea that a Success
axiom is going to
go beyond "finite" and yield numbers that are themselves infinite-
numbers.

So that Peano and his followers who never defined Finite nor Infinite,
have imposed upon everyone to dictate that his numbers are all
"finite" yet Peano never defines finite
and then dictate that his set must be infinite whilst containing only
finite numbers.

You see, my set is Infinite because obviously it contains infinite
numbers as members of the set. But examining Peano's Natural Numbers
set all of its members are dictated as being
finite-numbers.

Now we all know that in mathematics when you add two finite numbers
together the answer is never an infinite number. The answer to adding
two finite numbers is always another finite number. So in the Peano
Natural Numbers set as illustrated above, we can add pairwise every
two of those numbers until we end up with just one gigantic number. Is
it finite or is it infinite?
Well, according to Peano and his followers the theorem that addition
of every two finite numbers ends up as a finite number, means that the
Peano Natural Numbers when added up
is a gigantic Finite Number. And thus the Peano Natural Numbers does
not form an infinite set,
but is a finite set, since every one of its members is alleged or
dictated to be finite.

So there is one contradiction exposed of the Peano Axioms. It is
caused by the fact that Peano and followers never defined with
precision what they meant by a number as being finite
nor did they bother with well-defining infinite. This lack of
definition of finite versus infinite,
means the Peano Axioms are a gaggle of inconsistency.

Any Axiom system that has a Successor Function of endless adding of 1
has a infinite set
produced immediately. For the life of me, I cannot understand why
Peano and his followers
needed, demanded and dictated that their numbers had to be all "finite
numbers", like some
Freudian complex that they wanted and insisted each Natural Number to
be finite.

So they needed the Successor Axiom, but they wanted also the dictation
that every number
is finite. Well you just cannot have those two conditions:
(i) every Natural Number is finite number
(ii) have a Successor Axiom

So any axiom system that has (i) and (ii) is immediately a
contradictory and inconsistent
system. The two just simply tear each other apart.

So although Peano and his followers were all fooled and lulled into
foolishness, the foolishness is easy to see, in that you focus on the
endless adding of 1 so that 678 goes to 679 and on and on, we can
easily be fooled and lulled into thinking that every one of those
numbers is
a finite-number yet the entire set formed is infinite. So Peano and
his followers were fooled
for 150 years.

And the Correction is so simple. That the Successor does not give the
Peano set shown above but yields that set of this:

{0, 1, 2, 3, ..., 999...998, 999...999}

That set is an infinite set and it is this type of format that yields
divergence to infinity
or convergence to a finite number in series or sequence mathematics.

In math, if I were to ask you what does 1+1+1+1+... yield? You would
say it diverges to
infinity for it is like the Successor axiom only it is a endless
adding of 1. In fact that
series is equal to 999...999.

So all that Peano and followers needed to do was to define Finite
precisely and they could
have chosen 10^60000000000. And they would have realized that their
Successor Axiom
would go beyond their Finite definition. And thus, they would have
correctly stated that their
Natural Numbers formed an infinite set. But they would have also
admitted that some of
the members of the Natural Numbers are finite whereas the rest are
infinite-numbers.

Also, some of the blame for why Peano and followers went so far astray
of mathematics
can be blamed on the time period when Cantor was santering about with
his phony mathematics of different types of infinity. The world has
only one type of infinity and most
of Cantor's arguments are severely flawed arguments such as the
alleged uncountability
of the Reals when in fact the cardinality of the Reals matches the
cardinality of the Natural
Numbers.

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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