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From: Archimedes Plutonium on 8 Jan 2010 14:49


Archimedes Plutonium wrote:
> Peano Axioms have been around for about 150 years now. They are at the
> heart of
> mathematics. But they are so seriously flawed, both in definition and
> conception
> that it is a wonder they lasted for 150 years without changes. This
> post is going to be
> a picture summary, showing the reader what the Peano Axioms produce
> and the
> major flaws of those axioms.
>
> I leave it to the reader to look up the Peano Axioms and Wikipedia
> gives a good
> summary.
>
> The Peano Axioms produces this set called the Natural Numbers:
>
> {0, 1, 2, 3, 4, 5, 6, .. 99, 100, 101, .. 9999, 10000, 10001, ... }
>
> Where the ellipsis of two-dots indicates a finite string of numbers
> and
> ellipsis three-dots indicates a infinite string of numbers.
>
> Every member or element of the set of Peano Natural Numbers is a
> "finite number".
>
> Peano never defined "finite number" nor have any of his followers ever
> given
> a precise definition of "finite".
>
> Peano and his followers, thus, assumed the definition of a "Finite
> number" to
> be a number in which its leftward string of digits is all zeroes to
> infinity. Example:
> the number 9999 is finite because it is 0000...009999.
>
> AP established that the endless adding of 1, or what Peano called the
> Successor
> Axiom actually delivers this set, instead of the set shown above:
>
> {0, 000...001, 000...002, 000...003, ... , 00999...999,
> 00999...998, ... , 999...997, 999...998,
> 999...999}
>
> And AP defines Finite as to the largest number known to Physics where
> there is no longer
> any physical measurement or experiment to conduct on numbers larger
> than that specified
> number. These are known as the Planck Units and the largest is about
> 10^500 which signifies
> the Coulomb Interactions in element 109.
>
> So to AP the Counting Numbers Set looks like this:
>
> {0, 000...001, 000...002, 000...003, . . 10^500, ... , 00999...999,
> 00999...998, ... , 999...997, 999...998, 999...999}
>
> Alright, I have everything established in a picture to explain, now,
> why Peano Axioms
> are seriously flawed and inconsistent and contradictory.
>
> Peano followers run with two hidden assumptions:
> (i) that every Peano Natural Number is a "finite number"
> (ii) assume the definition of finite-number as a number ending in an
> infinite string
> of zeroes leftward such as 81234 is finite because it is 000...0081234
>
> First Contradiction of Peano followers: Since they assumed the
> definition of
> finite-number as leftward string of zeroes. Then according to Peano
> Followers, the numbers like 00999...999 become ambiguous. The number
> 999...9999 becomes the largest integer and the number 1000...000 is
> about 10% of
> 999...999, and where the number 500...000 is about 50% of 999...999.
>
> So here we are faced with the first major contradiction of Peano
> Axioms. Peano assumed
> the definition of finite and what he assumed (along with his
> followers) is that some
> number between 0,1,2,3 and the number 100...000 (10% of all the
> Natural Numbers) lies
> the last finite number. It is not 0999...999 or 9% of all the Natural
> Numbers for that number
> is clearly an infinite-number. So where is Peano's last finite number
> given his "assumed
> definition of what finite is?"
>
> Second Contradiction of Peano and his followers: Peano and his
> followers state that their
> set of Natural Numbers forms an infinite set. They cite their axiom of
> endless adding of 1
> and thus unbounded and thus, to them an infinite set.
>
> That is alright, because 1+1+1+1 ad infinitum is an infinite number
> and here I trespass into
> the math subject of series and divergence and convergence.
>
> But the second-contradiction of Peano and followers is their
> insistence that all the Peano
> Natural Numbers be finite-numbers. They are not warranted with that
> conclusion. They have
> to prove that the endless adding of 1 to that of 0 gives this set:
>
> {0, 1, 2, 3, 4, 5, 6, .. 99, 100, 101, .. 9999, 10000, 10001, ... }
>
> and not this set:
>
> {0, 000...001, 000...002, 000...003, . . 10^500, ... , 00999...999,
> 00999...998, ... , 999...997, 999...998, 999...999}
>
> This is where the Second Contradiction is obvious: You cannot have an
> infinite set when
> all its members or elements are finite specimens. Every infinite set
> has to have at least
> one specimen, one member or one element which is infinite in
> characteristics.
>
> Peano and his followers assumed and believed they could have a set
> that is infinite,
> yet each and every one of its members is a finite number. That is
> impossible and is
> a contradiction.
>
> If Peano and his followers had said the Peano Axioms produces this set
> of numbers:
>
> {0, 1, 2, 3, ... , 999...999} then Peano and followers can claim their
> Natural Numbers
> are an infinite set because it contains an infinity of infinite-
> integers.
>
> So let me summarize the two major contradictions of Peano Axioms:
>
> First Contradiction is that Peano never defines "finite" and thus
> leaves ambiguous as
> to whether a number or a set is either finite or infinite, and yet,
> increasing the problem
> by stipulating that all the Peano Natural Numbers are finite-numbers.
> When a mathematician
> insists his numbers are all finite and yet never defines finite is a
> blatant contradiction and
> inconsistency.
>
> 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.
>
> Now I am going to use this post as a template for future posts because
> this post basically
> sums up my work on this subject for the past 20 years. It is a very
> difficult subject because
> a human mind is not really equipped to deal and handle the concept of
> infinity. And for that reason it is obvious that Peano and followers
> could get away with their travesty of logic
> for 150 years. And it is why it takes me the better of 20 years to
> finally come up with this
> template summary.
>

In this template post which I will repost and repost with
improvements, I need to
discuss a few concepts that Peano and his followers never had, never
recognized
in their 150 year lifespan of the Peano Axioms. If Peano and his
followers had had
these concepts some 150 years ago, then I doubt that the Peano Axioms
would
have been as flawed and mired as they were.

Here again is Peano's set of Natural Numbers according to his axioms:
{0, 1, 2, 3, 4, 5, 6, ... }

Notice that the ellipsis of three-dots masks more than it reveals. The
ellipsis
indicates that the numbers are unbounded and go to infinity.

But now look again at AP's set of Natural Numbers:

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

Notice that the set is infinite but it has a FrontView and a BackView
and that
the "infinity portion" is in between the FrontView and BackView. The
idea is that
we tuck infinity into the middle and can talk about both ends. We can
talk about the
Frontview of a Real Number and talk about the BackView of a Hensel p-
adic. So there
is no problem in combining both the FrontView with the BackView for
numbers or for
sets of numbers.

So these concepts were not available to Peano and his followers:
(a) Frontview and BackView
(b) infinity in the middle

Now if Peano and his followers had discovered FrontView and BackView,
would
they have made their mistakes of inconsistency? Would they have
realized that
they assumed what "finite" meant? Would they have realized that the
Successor
Axiom of endless adding of 1 entails a number such as 999...9999 had
to exist
in their set of all 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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