how mathematics is changed with Natural Numbers = finite-integers + infinite-integers (p-adics)
Archimedes Plutonium
[ message deleted ]
You didn't answer my question. I can't make any sense of the claim "I
am confident that Natural Numbers = finite-integers +
infinite-integers (Adics)".
These strings .....00001, .....00002, .....000003, .....00004
which are called Natural Numbers = finite-integers become
......1111111, ......12121212., ......222222, ......333333..........999999999.
What is so difficult to understand about that.
There is nothing in the 5 axioms of Peano to stop or halt the finite-integers
from growing into infinite-integers (p-adics).
I can prove this because the 5 axioms of Peano have an infinite generator of
the Successor function which is endlessly adding 1. The p-adics are an Additive
infinite Series and thus the p-adics are also endlessly adding 1.
Hence, the burden is on you, Gal Binyamini to prove that the 5 Axioms of Peano
stop the finite-integers from spilling over into these infinite-integers such as
.......11111111
Prove that those 5 axioms of Peano are sufficient that every number produced
from those axioms has an infinite string of zeroes to the leftward side.
Just because you say so and because you say that some "definition" says so is
not sufficient. You must prove that the 5 axioms of Peano prevent them from
turning .....0000099999999999. into .......99999999999999. Show how the
Peano Axioms halt the zeroes from becoming endless nines.
Natural numbers is a name we give to some
set. It has no transcendent meaning beyong that.
Yea, well, the state of Wisconsin decided to "define" pi as 3.140000... to make
things easier for calculation.
Gal seems to think that once something is defined that it is true. That defining
for Gal is putting cloak of ultimate truth and reality on something. You cannot
even define something as basic as "life" and yet you want to make some
preposterous stance of stumping that defining the Natural Numbers as
finite-integers is etched in eternity.
Regardless of your
personal definitions, when most of us say "naturals" we mean a
specific set which has been rigourously defined. That's all we mean.
It is no personal definition. I have given 2 arguments that Naturals are the
finite-integers unioned the p-adics.
(1) endlessly adding 1 is contained in Peano Axioms and contained in p-adic
series
(2) Number Theory conjectures backlog hints that p-adics are a part of the
Naturals
The burden is now on you to prove that finite-integers do not become p-adics.
You may say "I don't think this is the right set for us to deal with",
or "I can suggest a more natural set" (natural in whatever way). But
the set N is what the set N is defined to be.
Prove that Peano Axioms halt ......0000009999999 from becoming .....9999
Prove it instead of writing another paragraph that the world is all neatly
defined.
And it is not defined to
contain the adics. If you can show some inherent problem in the
definition of N, in the form of a paradox or inconsistent axioms, go
ahead.
Number theory has 102,055 unsolved conjectures. The entire rest of
math put together has only 1,678 unsolved conjectures. I suppose to you that
such a reality is not inconsistent. I suppose to you that does not call for any
sort of alarm. That you cannot see the imbalance.
I suppose to you that when you make a change, of where
Natural Numbers = finite-integers union p-adics
and that Number theory then has only 118 unsolved problems and the rest of
mathematics put together has 1,678 unsolved problems. That such
PRACTICALITY is to be ignored by you.
That the cotton-gin should have never been invented because cotton picking was
always "defined" as human hands separating the seeds.
Just saying "both sets are constructed by adding 1" doesn't cut
it (and according to my understanding, this is not even true for the
Just saying that ......000009999 never spills into ......99999 does not cut it
You, Gal Binyamini, must prove from those 5 Peano Axioms that no numbers
with an endless leftward string of zeroes ever becomes an endless string of
nonzeroes.
You can spew out your mouth all day and night how the Naturals are defined as
finite, but in the end you must prove such.
adics: but I have a very superficial knowledge of adics, and please
correct me if I'm wrong).
The goldbach conjecture is stated for N as it is defined right now.
"Solving" the Goldbach conjecture means solving it for N, as N is
Not if N is ill-defined.
defined *right now*. What I get from your argument is that maybe
you're saying "Goldbach's conjecture is not interesting, because the
set N is not of any interest: we should be interested in my new N".
That may be true, but even it, you do not have a solution of the
Goldbach conjecture.
As for the last part, about Cantor's tranfinities, you completely lost
me there. Let's even assume what you say about the naturals is right
The p-adics are equinumerous with the Reals. There is an easy 1-1
correspondence with p-adics and Reals.
Proof: Take any Real number such as 3.343434.... and flip it over
3.3434..... when flipped over becomes a p-adic of ......4343.3
Hence, every Real when flipped over becomes a p-adic and thus a 1-1
correspondence.
Therefore, everything that Cantor did about the concept of Infinities is false.
(which I doubt). So what? Cantor has shown, in a proof less than a
page long, that the power set of set A has a bigger power than set A
Based upon Cantor's naive understanding of Natural Numbers. His Power Set
mechanism fails with the p-adic inclusion.
In fact, further, with Natural Numbers = finite-integers + p-adics. Most all of
Godel's work is dismissed as false. Godel's Incompleteness Theorem caves in
because Godel relies upon Cantor's non-equinumerous trait.
Natural Numbers = finite-integers + infinite-integers (adics)
is not a tiny revolution within mathematics but upturns the entire house of
mathematics with radical change.
itself. This means there is more than one "infinity", to put it in
informal terms. Have you found an error in this one-page proof that
has been reviewed by tens of thousands of people? If so, please point
it out. Otherwise, what's your point?
Gal
In the entire history of human thought there has been only one idea that has
continued unchanged and untampered with since it was first discovered. That
idea was the Atomic Theory of Democritus. Yes it has been improved by our
modern day Quantum Mechanics but essentially the same when Democritus
taught it some 2.5 millenium ago.
Every other idea in human history except the Atomic theory and the theory of
Natural Numbers has undergone radical change and reform.
So, do you not think that Natural Number Idea is also due for change, revision
and newer understanding? Or do you think that humanity in a million years from
now or 10 million years from now (should we survive that long) will be doing
mathematics with the Natural Numbers the same as year 2002?
Recently in the news a judge in California was under a cloud-storm controversy
when he declared "under God" in the National Anthem was unconstitutional.
Well, from the way Gal Binyamini and the rest of the mathematics community
treats the Natural Numbers by hiding behind the magic words "under Definition"
well, if the USA Supreme Court upholds that ruling and "under God" must go. I
think Gal Binyamini has a perfect replacement for those two words in the
National Anthem---- "under Definition". Of course, I am being sarcastic.
Tim Mellor
few comments:
As it stands you're definition from the reals it not quite well
defined. For example the series
\sum_{n=1}^{\omega} 9*\(10^{-n}) converges to 1, but your mapping
would take them to two different numbers in your system. This is not a
real problem though.
I am willing to believe that it is a ring (though there are some parts
worth checking), but there are some things one feels contradict our
intuition about "natural numbers" which are worth considering if we
are to postulate this as a better definition:
1. Zero dividers
2. Not well ordered, indeed not orderable as a ring.
3. Cardinality 2^{aleph_0}
4. Constructed as sets of sequences of legnth omega (which interprets
the standard naturals)
The system you have described may have other properties which make
ordinary number theory conjectures trivial when applied to it:
Can you show the following:
a) 2 (i.e. 1+1) is prime?
b) 2 is indivisible
c) Every indivisible is prime and every prime is indivisible
d) There is an indivisible or prime number
e) There are infinitely many indivisibles/primes
f) Every number is expressible uniquely as a product of primes
I suspect that unique factorisation will not hold and probably
property c) perhaps even d). Then it is not clear that number theory
conjectures about primes have much relevance.
Archimedes Plutonium
> I was interested to read your definition of natural numbers. I have a
> few comments:
Looks as though I should take you out of my killfile. But I just have a
new ISP
and will be having some posting discontinuities in this transition. One
of which
is that I cannot fully respond to the below, only fragmentary. I should
be on a
clear sailing course soon.
>
>
> As it stands you're definition from the reals it not quite well
> defined. For example the series
> \sum_{n=1}^{\omega} 9*\(10^{-n}) converges to 1, but your mapping
> would take them to two different numbers in your system. This is not a
> real problem though.
During 1993-1994 I did not spend much time in trying to get a better
handle on
these Infinite Integers. And left it as the union of p-adics and n-adics.
Abain later
tried to help me with some other constructions. The worse thing about the
adics is
there base-dependency which leaves questions as to what numbers truly
exist in
p-adics and n-adics and whether they are independent. Is a 1 in 2-adics
different
from a 1 in 10-adics?
So I am glad to be back with this old nemesis and trying to make progress
to clear
things up.
>
>
> I am willing to believe that it is a ring (though there are some parts
> worth checking), but there are some things one feels contradict our
> intuition about "natural numbers" which are worth considering if we
> are to postulate this as a better definition:
> 1. Zero dividers
> 2. Not well ordered, indeed not orderable as a ring.
> 3. Cardinality 2^{aleph_0}
> 4. Constructed as sets of sequences of legnth omega (which interprets
> the standard naturals)
>
Funny how physicists never tried to force Nature to behave to some rules
that the
physicists dream up. Instead physicists try to discover how Nature
operates. But
in mathematics there is not this sense that mathematics is out there with
independent
existence of human mind waiting to be discovered. It is a pitiful shame
that so many
students of mathematics have this view that (1) write down some rules (2)
see what
follows from those rules and whether a "ring" is produced. As if every
quest has to
have a pretty lady as an endgoal. Mathematics proper is more like new
physics,
wherein there are no preset and preconceived goals to achieve along the
way.
>
> The system you have described may have other properties which make
> ordinary number theory conjectures trivial when applied to it:
>
Many of them trivial such as Goldbach but many of them
false such as FLT, Beal, Catalan.
>
> Can you show the following:
> a) 2 (i.e. 1+1) is prime?
> b) 2 is indivisible
> c) Every indivisible is prime and every prime is indivisible
> d) There is an indivisible or prime number
> e) There are infinitely many indivisibles/primes
> f) Every number is expressible uniquely as a product of primes
>
> I suspect that unique factorisation will not hold and probably
> property c) perhaps even d). Then it is not clear that number theory
> conjectures about primes have much relevance.
You are not getting the gist of this.
I call them the Infinite Integers. They are the Reals only flipped over
with or without a tiny
finite component and the rest of the string infinite. For Reals are
finite to the left and infinite to the right. Whereas Infinite Integers
are finite right and infinite left.
Interesting Properties:
(a) no need for negative numbers since these Infinite Numbers contain
both positive
and negative without any need for a negative sign.
(b) there is a largest number and so there is no need for a symbol of
infinity. The number
is ....99999999. It is -1 under a different guise and we step on 0 and
then go to 1 and start
all over again.
(c) you do need unique factorization because you need to start to build
these Infinite Integers and we start with 0 and 1 and the Peano Axioms
which lead into adics.
By looking at the Reals flipped over, I can snapshot freeze all of the
Infinite Integers without any base-dependency.
Remember the 5-adic that is sqrt(-1). It is also a Real when flipped back
over.
We use the p-adics and the n-adics to help us to understand any one of
these flipped over Reals.
Archimedes Plutonium
sets. All of Mathematical Numbers are either of the set Reals or Adics.
The entire Universe of Mathematical Numbers consists of _ 2 and only 2 sets
_
Reals = infinite rightward strings with a finite leftward portion
Reals = rationals union irrationals union complex (i)
Adics = infinite leftward strings with a finite rightward portion
Adics = finite-integers union infinite-integers
Nothing really new for physics since All of Physics is really one of two
sets
Particle or Wave.
So, that leaves the question of Doubly Infinites, strings that are infinite
both rightward and leftward. They are nonsense leaving us in mathematics
with 2 and only 2 types of numbers Reals and Adics. Leaving us in Physics
with only 2 entities-- Particle and Wave.
What is the reality or substance of the Reals. It is easy to see that the
Reals are like a tape-measure or ruler of Euclidean Geometry. A MICROSCOPE
of the
world, because between each *whole Real* are infinite number of tinier marks
such as on a ruler. So the Reals are a microscope of differences between
whole
Reals.
What are the Adics? If the Reals are microscope such as in biology going
from
large animal to tiny to microscopic bacteria. The Adics are TELESCOPIC. They
are the Faraday lines of force (circles and ellipses) going out throughout
space.
Reals want to get in close between two marks. Adics spread out throughout
infinite space as Faraday lines of force.
The Adics are so telescopic that they even give us infinity as a number
which is
.....9999999. No need to ever write infinity as a sideways 8. Write Infinity
as what it actually and really is, which is .....9999999. Some of you know
that this number can be represented as -1. For the Adics have no need of
negative numbers.
But in physics, is that not really what infinity is all about if we start at
one spot and traverse the entire universe and end up back at the same spot
with nothing changed. Is that not infinity.
I said previously that if we take any Real number and flip-it-over that we
obtain a Adic. But we need only flipp over all the Reals between 0 and 1 of
the Whole-Reals. Because, all the Reals between 1 and 2 are identical once
we delete the 1 and 2 finite portion. So, we can focus only on the Reals
between 0 and 1 and flipp-them-over and we should have a good understanding
of the Adics that is not base-dependent.
What do we see immediately after flipping-over all the Reals between 0 and
1? We see that, unlike the Reals which form a straight-line-segment. The
flipped over Reals have a curvature like a circle and that what was
.9999999..... Real becomes the Adic ......99999999. which is for Real as 1.0
and for Adic which is
-1.
The Reals always keep their Euclidean flatness and the Reals are a measure
of microscopic physical reality. The Adics on the other hand have a natural
curvature such as a circle or ellipses. If we were to envision all the Reals
they would be on a ruler that is a straight line and infinite precision to
small distances.
The Adics vision is much different. The Adics are Faradays lines of force in
Space and if you can picture the Universe as endless dots of electric-charge
with entangling lines of force then you have a picture of the Adics.
The Reals are to mathematics what the Wave nature of light is to physics.
The Adics are to mathematics what the Particle nature of light is to
physics. There is no 3rd nature to physics just as Doubly Infinites is
nonsense in mathematics.
Archimedes Plutonium
The history of mathematics will follow in a somewhat similar path that the
history of physics followed concerning the debate and finally enlightenment
and understanding of Duality. That Nature is dual to particle and wave.
Existence of anything must have competing dualisms. Nothing exists unless it is
dualistic.
The history of physics first started with particles. The Ancient Greeks and the
Atomic theory was particle. They never saw the importance of Wave nature
until about the 19th and 20th century.
The history of mathematics, one can assess has been a retarded science for
approximately 2,000 years. Mathematics at present still does not realize that
Dualism runs through the entire fabric of mathematics. Especially Numbers.
There are 2 and only 2 numbers. They are either Real or Adic (either wave or
particle).
Unlike physics history that dwelt with Particle Reality from Ancient Greek time
until about the 19th century. The history of mathematics dwelt with only the
Reals from Ancienct Greek time right up to year 2002. Modern mathematics is
ignorant of the Particle nature of mathematics which is the Adics. In other
words, Natural Numbers = finite integers is as primitive and comparable to the
Thales notion that all reality is water or 4 elements notion.
Ironic that physics history dwelt mostly with particle reality until 19th
century and that mathematics history dwelt mostly with wave-reality as
represented by Real numbers. And then Quantum Mechanics merged particle with
wave reality by 1920s. Yet the history of mathematics is so very primitive that
the mathematics community is not even aware of a Duality and has no clue as to
the full realization of the particle-nature of mathematics. They believed that
Natural Numbers = finite integers was the particle nature of reality, but their
understanding of Natural Numbers was as "primitive" as if Thales water
hypothesis in modern times.
Archimedes Plutonium
> I'm sure this has been tried before, but...
>
I am sure that every new student to Calculus upon seeing the number
1.99999.......
>
> Let N be a 'finite integer' whose successor, N+1, is a 'non-finite
> integer'.
>
Will be fighting 1.99999....... every step of the way and dream that there
is a 0 and more zeroes out there.
>
> Look at the last digit of N+1.
Even the Calculus teacher shows the dismayed student of the asymptotic
graph
where 1.9999............. becomes 2
Or, that, 1.99999...... was always 2
>
>
> If this digit were anything other than 0, then N would differ from N+1
> only in the last digit. Since N+1 does not have an infinite string of
> zeroes on the left (else it would be a 'finite integer'), neither would
> N.
>
And so now, we have some new pupil in the form of Jacob Haller who
says ............000000000009999999999999999999999999999
those zeroes will never be filled with all 9s.
So, this pupil Jacob Haller is going to say the ENDLESS adding of 1.
The SUCCESSOR Axiom is not going to fill up all of those 0s and that
there will always be a zero to the left.
But then, is pupil Haller denying Successor. Is Haller saying, oh,
Successor
you can stop now. Oh, 1.99999..... is not 2. Oh, Successor is not
infinite.
>
> So N+1's last digit must be 0.
Hey, pupil Haller. What is the reason for that? Thin air?
And Haller never believed nor understood that 1.999.... was 2
>
>
> How about the second-to-last digit?
Someone should write a song titled the Birdbrains of Mathematics
dedicated of Jacob Haller.
>
>
> If the second-to-last digit is not zero, then N+1 would end in "90",
> "80", ..., or "10", and N would end it "89", "79," ..., or "09"
> respectively, differing from N+1 in only the last two decimal places.
> Again, in these cases, if N+1 is non-finite then so is N.
>
> So N+1's second-to-last digit must be 0 also.
>
> Similar reasoning shows that all of N+1 digits must be 0.
>
> But then N+1 is ....00000. But that's a finite integer! Specifically,
> it's 0.
>
> This is a contradiction.
>
> Therefore, my assumption that there was a 'finite integer' whose
> successor is a 'non-finite integer' must be false.
>
> What do you think?
>
> -jwgh
What do I think? I think you failed to ever understand and will always
fail to understand that Successor is Infinity and being infinity means
that this number
............9999999999999999999 exists as a Natural Number of the Peano
Axioms.
Mathematics has no *time* element or time function to impose.
And so, ........000000000999999999 becomes ........9999999999
instantaneously
when you invoke the Successor Axiom.
When you inject a Successor Axiom of Peano you instantly have infinite
integers. You create 0 and 1 and then you create the Successor Axiom. At
that
instant of time, you have created .........99999999999999.
And the number .........000000009999999999 is for the weak birdbrain
people
who cannot even get over the fact that 1.9999...... is 2.
What Jacob Haller and his birdbrain argument needs is a halting axiom. An
axiom that stops the endless adding of 1.
Peano never gave a Halting Axiom.
The Reals themselves also have this Disease, just as the Natural Numbers.
What is the Real Number .......9999999999_._999999999....... It is not a
Real Number for it is a Doubly Infinite.
So, Mathematics needs a halting axiom. I give it as saying that Doubly
INfinites are Nonsense and do not exist. In that way, I halt the Natural
Numbers by saying they are infinite in only the leftward string. And that
Reals are halted by saying they are infinite only in the rightward string.
All of this has a parallel in physics. Existence occurs with duality.
Particle Wave duality. The Reals are wave-like and the Natural Numbers are
particle like.
Is there Triality in Physics? No. Triality is nonsense. Likewise for
mathematics in that Duality gives birth to existence and so you have this
Duality
Reals = infinite strings rightward, finite portion leftward
Naturals = infinite strings leftward, finite portion rightward
Archimedes Plutonium
>
>
> So, Mathematics needs a halting axiom. I give it as saying that Doubly
> INfinites are Nonsense and do not exist. In that way, I halt the Natural
> Numbers by saying they are infinite in only the leftward string. And that
> Reals are halted by saying they are infinite only in the rightward string.
>
> All of this has a parallel in physics. Existence occurs with duality.
> Particle Wave duality. The Reals are wave-like and the Natural Numbers are
> particle like.
> Is there Triality in Physics? No. Triality is nonsense. Likewise for
> mathematics in that Duality gives birth to existence and so you have this
> Duality
>
> Reals = infinite strings rightward, finite portion leftward
> Naturals = infinite strings leftward, finite portion rightward
I cannot resist on commenting about the above implication of duality of
physics that pervades mathematics.
Physics has Complimentarity and so does mathematics.
We know that the Reals have imaginary i number sqrt(-1) and the Reals have pi
and e readily available. But the Natural Numbers = P-adics does not have pi
or e available. Instead, the Adics have imaginary i readily available.
So, where the Reals are lacking of numbers such as i, then the Adics have the
number. And where the Adics are lacking of pi and e, the Reals have them.
So, the Reals compliment the Adics and the Adics compliment the Reals. This
observation does not constitute a proof of anything but it does support the
notion that Naturals = finite-integers-only is a falsehood.
DavCrav
$\sigma$ is a set of setences having a model of infinite cardinality
$\kappa$ then $\sigma$ has a model of cardinality $\geq \lambda$ for
any $\lambda \geq \kappa$.
(I'm not all that good with logic, so maybe we can't).
Presumably N is a infinite model of the Peano axioms (wno-one's saying
that there are only finitely many integers). Then this theorem can be
used to produce non-standard models of Peano arithmetic.
Archimedes Plutonium
(snipped)
> Can't we use the Upward Löwenheim-Skolem theorem from logic: If
>
Part of the massive flaws of mathematics is the whole genesis of Number
systems. The present day community feels that the origins of Numbers is
a good origin. Let us review it:
1) It starts by having the Natural Numbers = finite-integers only as a
given
from Peano Axioms.
2) Given the Natural Numbers math community then constructs Rationals
and then the Irrationals and then clumps them into one set and calls them
the Reals.
But does this genesis of Number Systems makes sense given other sciences
such as Physics? Does Physics have any genesis approaching that of
Numbers
in Mathematics?
Here is my Genesis of Numbers which is heavily influenced by Physics.
I don't begin with taking the Finite Integers as a given (what one
mathematician
said that God gave us the integers and all the rest was a man-made
creation)
My genesis is this:
1) you have infinite strings for numbers. Either infinite righwards or
leftwards with a finite portion.
2) call the infinite rightwards the Reals. Call the infinite leftwards
the Natural numbers. Peano Axioms are superfluous and not needed.
3) if the Reals are given, we can produce the Naturals. If the Naturals
are given we produce the Reals for one type is the compliment of the
other type.
In my genesis the two number systems are duals and compliments of each
other. They are like particle and wave is to physics.
The old way was to say God gives us the Finite Integers and we
manufacture everything else including the Reals. So the Reals are
dependent on the Finite Integers.
In my view the Reals are independent of the Naturals because they are
fundamentally different because they are rightward infinite string.
In my view, Peano Axiom system is basically a commentary. A poor
commentary that cannot capture the Naturals. And in my view no axiom
system is needed to make either the Reals or Naturals. Both exist a
priori. And humans learn new characteristics and features of Reals and
Naturals as time goes on.
The Peano Axioms of mathematics is a straightjacket, not a foundation of
mathematics. It would be like the ancient Atomic theory of Democritus set
as a axiom system and denying all of 20th century physics because so much
of the atomic view had changed.
My view rejects the idea that any mathematics is axiomatic. And to impose
axiomatics in mathematics or any other science is akin to imposing dogma
by
a religion.
Kris Marshall
Your arrogance is pathetic and it seems you did not understand and
will always fail to understand that Successor and Predecessor of
genius is Infinity and not you, nor your predecessors or successors
reign on the world of numbers. Take some ethic course, the one they
give in kindergarden.
Kris
Archimedes Plutonium
> Archimedes Plutonium <a_plu...@dtgnet.com> wrote in message news:<3D2388E0...@dtgnet.com>...
>
> > Wrong. The moment you utter the Successor Axiom you have infinite
> > integers. There is no "eventual". There is no waiting. The moment you
> > put wet ink to a Successor Axiom, you instantaneously have infinite integers.
> > Instantly you have .....0000099. but also ......9999999.
> >
> I think I have a sense of what you are trying to assert here. Let N be
> the set of all natural numbers {0,1,2,...}. It follows by the
> "Successor Axiom" that
>
> (For all finite n in N)(There exists m in N) m > n.
>
> What you are asserting is that the above also implies
>
> (There exists m in N) (For all finite n in N) m > n.
No. What I am asserting is that
Successor Axiom yields a number ....999999 in base-10
In base-123456790 the infinite-integer ......123456789123456789.
is produced.
The Peano Axioms are *base independent*.
Meaning that when you collect all of the bases together into one pile.
That it has all of the finite-integers and all of the infinite-integers.
And thus,
Natural Numbers = finite-integers + infinite-integers
How to get back to the old days of bliss and ignorance? Answer:
drop the Successor Axiom. But if you do that, well, you lost all the Naturals.
>
>
> Such an m would be an infinite integer. The above interchange of
> quantifiers is not permitted in classical logic. By your assertion,
> you are rejecting classical logic and concluding that infinitely many
No, the Successor Axiom is an infinite process, so it should not alarm or
disturb you in that it has gone beyond producing finite-integers.
Besides, where would 20th century physics be if Niels Bohr had not
gone beyond the Classical Physics of Maxwell in that the electron orbits in
only discrete radii. Sometimes Truth forces you to break with Classical
notions.
>
> finite positive natural numbers cannot exist. Another way of looking
> at your assertion is that you are applying the following induction:
>
> "every finite n in N is not an upper bound for N" --> "all finite
> natural numbers are not upper bounds for N --> "there must exist an
> infinite natural number in N."
>
By this point you have entered irrelevancies and tried to cobble together
those irrelevancies. You would make a good philosopher.
>
> Of course, classical logic denies the validity of the above induction
> (which in my opinion, will lead to the conclusion that N belongs to N,
No doubt this is indubitable knowledge.
>
> and N is an absolutely infinite integer). My own personal opinion is
> that it is simply not possible to reason consistently with infinite
> sets and one must give up the notion of N as a "completed infinity". N
In your above you are being contradictory for you hold faith in the Successor
Axiom which is an "infinite process" and here you disparage infinite sets
with the inability to grapple completed-infinity with open-ended-infinity.
If you like the Successor and the infinity it brings, why disparage infinity.
If you like the Successor Axiom, then, don't be afraid of what that
axiom leads to.
>
> is not a set, but a proper class containing all (and only) finite
> natural numbers. A proper class by definition cannot belong to any set
> and would correspond to the notion of "potential" rather than
> "completed" infinity. This would limit mathematics, but would make it
> (possibly) consistent.
> Sincerely,
> R. Srinivasan srad...@in.ibm.com
Mathematics, like Physics is independent of human thought. We can dream up these axiom systems like
Peano in hopes of capturing the essence of something. But as time goes on we find those axioms
inadequate. Just as we find axiom-systems in physics inadequate (only they are called theories). The
Ptolemy axiom system of astronomy lasted for 2,000 years. I have lead the charge into breaking down the
Peano Axiom system of mathematics. It's greatest
flaw was that it ignored and never recognized the Infinite-integers that existed alongside the
finite-integers.
Archimedes Plutonium, 5JUL02, a_plu...@hotmail.com
Archimedes Plutonium
(snipped)
>
> OK. Starting at the right end of the line of checkers you see
>
> ...000
> ...001
> ...002
> ...003
> [etc]
>
I don't remember if the poster of number representation was in this
thread. Some poster spoke of removing all number representation and calling
for sticks to represent numbers. One stick is 1. Two sticks is 2. In such a
representation of sticks we dispense with fancy digit representation do we
not? We no longer write .....000059 for instead we have a pile of 59 sticks.
But the tricky thing comes when you represent Peano's Successor Axiom as
sticks. It would be the endless adding of 1 more stick. Current mathematics
has a symbol for it and a name. The name is infinity and the symbol is a
sideways 8. But a better symbol and name is ....999999. The stick-man would
not want to write 59 or .....00000059 but have a pile of 59 sticks.
Jacob writes:
...000
...001
...002
...003
[etc]
But, Jacob, that can be also written as such. Keeping in mind that
.......9999999 in base-10 is -1.0000...... Now Jacob, what if I add 1 to
.....999999. We would have 1 + .....9999999 and we would get
.....0000000.
So, now then, Jacob, let us rewrite your above. Most people do not like stick
representation for numbers because they are cumbersome and unpractical.
Most like to write 0 instead of ....00000. Most like to write infinity as
sideways
8 instead of .......99999999
So, where Jacob writes:
...000
...001
...002
...003
[etc]
Can also be written as
1 + .....999999 is ....000
[1 + .....999999]1 is ....0001
[1 + .....999999]2 is ....0002
[1 + .....999999]3 is ....0003
So, now, the issue that burns on Jacob's mind as to where do these finite
integers
end and where do the infinite-integers begin, that question just evaporates into
nothing. The entire concept of finite without any componentry of infinity was
ill-begot and ill-defined.
Someone in these newsgroups said it better than I. He said our question is often
vague and full of misunderstanding and full of illogic.
Jacob seems to think that Naturals can be such a list as this:
....0000
....00001
....00002
....00003
.
.
.
.....99999
and that all of those integers which have a string of 0s to the left are neatly
classified as "finite" and those integers without a string of 0s to the left are
infinite-integers.
But the sad truth of the matter is that all integers are infinite integers and
that
this finite-integer business was all a delusion.
The stickman cannot show infinity with a pile of infinite sticks. But I can show
infinity with this largest integer in base-10. It is ....999999. So very large
is this
infinity number that like a circle it comes all the way back around and is just
1
shy of becoming the starting point. For on a circle, is not infinity the ability
to
come back to the starting point.
So, now, when Jacob in this post wonders where does the finite-integers stop
being finite and with the addition of 1, this so called last finite-integer
becomes
the first infinite-integer. I doubt that Jacob, nor 99% of the rest of the
people
reading this post can understand that every integer is an infinite-integer. And
it is only their stupendous bias and prejudice that they want to fantasize that
....00001 is 100% finite, when in fact, this number 1 carries infinity along
with it.
For, 1 is not only ....00001, but also [1+...9999]1. And to the stickman with
his
one stick to represent 1, he forgets that in order for 1 to exist means there is
a background universe to accomodate an infinity of sticks. He deludes himself by
ignoring this infinite background universe to hold not just one stick but
infinity
of sticks.
So, I hope the above will finally quell and quench the curiosity of Jacob. For
if he can read this and understand it, he would then realize that every integer
is
infinite and so the question as to when and where a finite goes into infinite is
a flawed-question because every integer is infinite integer.
>
> > >
> > >
> > > If you looked at a different place on the line you'd see
> > >
> > > 5,932,255 5,932,256 5,932,257
> >
> > Keeping with convention it should be 5,932,257 5,932,256 5,932,255
>
> Further to the left you see
>
> ...0005,932,255
> ...0005,932,256
> ...0005,932,257
>
> > > Are any infinite integers to be found in this line of checkers?
> >
> > Yes, the infinite integer .....9999999999999999. at the end of the line
> > at the extreme left. In fact this number is infinity itself and there is no
> > need for a sideways 8 to represent infinity for .......99999 is infinity.
>
> OK, fine. Would you agree that since the line of checkers contains both
> finite and infinite integers then at some point a checker containing a
> finite integer must be next to a checker containing an infinite integer?
>
> Thanks,
> jwgh
>
As I said above, the concept of a "finite integer" is a marred concept. A flawed
concept. Because any string of 0s such as ......00000047 is also a string
of 1+....9999 such that the 47 portion is also the number [1+.....9999]47.
But, let us work backwards, just out of curiosity and fun.
Let us suppose we are Mr. Peano himself and a backwards Peano. We don't
like endless adding of 1. Instead we like endless subtracting of 1. And so
when we sit down and begin crafting the famous Peano Axiom system that
will systematize the Natural Numbers for the 20th century. We do not
write the Successor Axiom as endless adding of 1. We write it as endless
subtracting of 1 and we begin not at 0 and 1 but instead at infinity of
.....9999999. So, Jacob is all excited about where in the endless subtraction
that
the infinite integer of .....99999999 will begin to start looking
like this.....00000000xyz
Okay we start subtracting 1 from ....99999 and we have .....999998. We
subtract another 1 and we have .....999997. We do alot more subtracting of
1s and we come to the point where our number looks like this ....99990000.
But the zeroes are to the right of the infinite string of 9s. What Jacob
would like is for something like this instead ......00009999. But alas again,
we are saved. because ....999999 is -1 and if we subtract 1 we get ....999998
which is -2. Peano Axioms backwards, and instead of Induction we have
Infinite Descent. We subtract another 1 and we have .....99997 which is
-3.
And now we do the same technique. We know that 1+ ....9999 is ....0000
So we have this progression of ....., ...., .....999997, ....999998, ......9999
and we apply 1+ .....9999 = ....00000
and we end up with ...., ....0000-3 , ....0000-2, ....0000-1.
Again, the question of where does a finite-integer stop and a infinite-integer
begin is again a ill-conceived question.