Earlier today, Archimedes Plutonium wrote:
>
> *** Proof set of all finite Peano Integers is a overall finite set
> itself ***
> Proof: Infinity and Finiteness are Quantum dualities as this book
> started off with
> defining Reals and AP-adics as dualities of particle versus wave.
> So to prove that Set of All Finite Integers is a Finite Set overall, I
> am going to use
> another Quantum duality characteristic of numbers and that is Even
> versus Odd.
>
> So in this proof I am going to switch finite for either odd or even
> and infinite for either
> odd or even.
>
> Alright, I am going to use the abbreviation of fin for finite integer
> and inf for infinite integer
>
> Given a set that contains every finite integer {fin, fin, fin,......
> ad infinitum} is this set
> overall finite or infinite? The answer is Finite since every element
> is finite. In other
> words, you cannot reach infinity if every element of a set is a mere
> finite element.
>
> Given a set that contains only Even integers {2,4,6,8,..... ad
> infinitum} is this set
> where we do pairwise addition or multiplication receive an answer that
> is always Even?
> Yes, and the total sum or total multiplication is also Even.
>
> Given a set that contains a mix of finite integers and infinite-
> integers then what is this set
> overall? Well it can be either finite or infinite overall. Now let us
> correspond that answer with
> even verus odd. Example {1,2,3,....., 3333....33333} has both finite
> and infinite so it is
> infinite but this set {11111.....11111,3333.....33333} contains two
> infinite integers but is
> overall finite.
>
> Given a set that contains a mix of even and odd integers { even, odd,
> odd, even,....}
> Well the sum or product pairwise or total overall can be either even
> or odd. So far we
> have total agreement between the concepts of finite versus infinite
> and even versus odd.
>
> Given a set that contains only infinite integers and what is it
> overall? Well we can have
> a finite set of only infinite integers and we can have an infinite set
> of only infinite integers
> such as the AP-adics and also the Hensel P-adics.
>
> Now does the even and odd match up to this last case? Well yes it does
> for given
> a set that is only odd numbers or only even numbers under
> multiplication its overall
> product is odd if all its members are odd and its overall product is
> even if all its members
> are even. As for addition, overall the sum can be either odd or even
> for the only-odd-set
> and must be even for the only-even-set.
>
> So yes, we have full agreement. So the concept of Finiteness versus
> Infinity match the concepts
> of Even versus Odd.
>
> So the Peano axioms of the modern day mathematician who believes they
> are a consistent
> set and that the concept of Finite Integer is a workable concept is
> fooling himself or herself.
> The above proof shows us that the Peano Axioms are inconsistent and
> flawed and that
> we lose the concept of Even versus Odd.
>
> *** end of proof***
>
I can refine the above and I do not need both addition and multiplication.
Let me tell the reader directly of the mechanism of the proof. What I
did above was Replace finite-integer with that of even-integer and
infinite-integer with that of odd-integer.
So what the proof yields is that the Peano notion of a set that is
composed strictly of nothing but finite-integers must be a finite
set itself. That you cannot have finite-integers yield an infinite set.
So the notion of the set of *all finite integers* is a finite set
overall, even if you tack on the words "out to infinity". So this
proof shows that the set of all finite integers is not an infinite set
itself. Now if you do not want to accept or believe or understand that
proof, then the proof forces you to accept that if you take a set that
is composed purely of even-integers and multiply them together that
you end up with a odd-integer.
So let me repeat what this proof forces upon a reader. If a reader
feels that the set of all finite integers is a infinite set overall,
well, then that reader must also believe and feel that when you take
a set composed purely of even-integers and multiply them together,
the product is an odd-integer.
So without any further ado, let me refine my above for I really need
only a few cases and can skip using addition and can use just
multiplication.
PROOF that set of all finite integers itself is a finite set overall:
We replace finite with even and we replace infinite with odd
(case 1) (a) Set of all even-integers question, is the product even or
odd?
Answer is the product is always even
(b) Set of all finite-integers and the question is whether the overall set
is finite or infinite and the Answer is that it must be finite since
very individual member was proscribed as finite
(case 2) (a) Mixed bag of a set that has both even and odd numbers so the
product is going to be a mixed bag of both even and odd
(b) Mixed set of containing both finite integers and infinite integers
and the end result is also a mixed result where some sets are finite
and some are infinite
(case 3) (a) Set of all odd-integers question, is whether the overall set
is odd or even under multiplication and the answer is that the product
is always odd-integer
(b) Set of all infinite-integers and the question is, what is that set?
Answer is quite obvious since every member is infinite that the overall
set is infinite.
So the proof is done except for the display of the ramifications. If we
insist that the Peano axioms produce *finite integers* and that these
are well-defined and go to infinity then we have to abandon the concept
that the multiplication of any set of even-integers is always an
even-number product. We have to abandon that, and thus we destroy
all of mathematics.
The proof above that I have laid out intertwines two concepts of
mathematics in that of finite versus infinite and the concepts of
even versus odd. This is quite acceptable as a proof for it is merely
"replacement of concepts". So that if we destroy the concept of
multiplication by even-integers, then, likewise we have destroyed
the concept of "set of all finite integers".
What this proof proves is that the Peano axioms are flawed and need
repair. They are flawed because they presume the reader believes a
set of all finite integers can go to infinity and still be well defined.
Finite Integers are merely an abbreviation for a few infinite integers.
And the only infinite sets of Counting Numbers are sets that contain at
least one member which is infinite itself.
But to be absolutely clear, all sets of Counting Numbers, whether
they be finite or infinite, have only or contain only
infinite-integers where every member is an infinite-integer.
The reason the subject of mathematics had a mountain of unsolved
Number theory problems and conjectures, is not because those were
hard problems, in fact they are all simple problems, but because the
old mathematics community never had a true understanding of what
the Counting Numbers were in the first place, well, then almost every
problem in Number theory becomes another Fermat's Last Theorem or
a Riemann Hypothesis. When you truly do not understand what you start
out with, then it is rare that you are ever going to solve problems.
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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