#266 mathematics ends at about 10^500 Re: powerset
Archimedes Plutonium
Marshall wrote:
> On Dec 27, 12:24 pm, "plutonium.archime...@gmail.com"
> <plutonium.archime...@gmail.com> wrote:
> >
> > And if we were to multiply say 0.99... by 0.88... we end up with
> > 0.7..
>
> We get what now?
>
>
> Marshall
Marshall wrote:
> On Dec 27, 12:24 pm, "plutonium.archime...@gmail.com"
> <plutonium.archime...@gmail.com> wrote:
> >
> > And if we were to multiply say 0.99... by 0.88...
> we end up with
> > 0.7..
>
> We get what now?
>
>
> Marshall
Forget for a moment what you learned and memorized in school, which
maybe a difficult exercise.
Talking about infinity in mathematics is about the same as talking
about
philosophy in a science. Raising silly and absurd questions as to how
many
angels fit on the end of a needle.
There is no infinity in physics, is there? Have you ever entertained a
infinite item
in Physics?
What if we had something in physics that was infinite? Would it not
cover-up or hide or drown out everything else that was not infinite?
Let us say we had infinite photons in physics. Then would not the
Cosmos
be just a infinite ocean of photons that nothing else would be visible
or capable of existing.
Or let us say we had an infinitude of a virus, would it not drown out
the rest of the Universe by this infinite virus?
Lwalk wants to talk about ellipsis when he comes back in February,
which is a better idea then talking about ill-defined infinity and the
powerset which arises from an ill-definition.
You are talking about a powerset on an infinite set, for which you
never
defined-well enough what infinity is. Like talking about whether the
fire
breathing dragon has blue eyes or green eyes. This is not knowledge or
education but just imagination run amok.
What A Commonsense Person Would Do:
A person full of commonsense would say: (a) since physics has no
infinities
then let us only do Mathematics to the largest (smallest) numbers that
physics needs to use. Beyond that call it the Incognitum and far
beyond call
it the Infinite.
Planck Units are useful to physics to 10^500 is the maximum needed
number.
So all of Mathematics fits within the parameter of 10^500. If we go a
little
way beyond we are in the Incognitum and no longer mathematical for
there
is no more physics. Infinity is thus a mere abstraction but not
mathematics
since it is no longer has any Physics meaning.
So take the Powerset of a number like 10^80. Is it smaller than 10^500
and thus still have a Physics meaning? If so, then it is still
mathematics.
When we say the set of Primes is an infinite set, what we really are
saying is
nothing more than that prime numbers go beyond the number 10^500. That
Primes go into the Incognitum and beyond.
Much if not most of mathematics until now was just merely "philosophy"
and not a science. Philosophy is mostly chitter chatter, but not
science.
Can modern day mathematicians accept the idea that their subject ends
at a large finite number of 10^500? Probably not without a struggle.
So when Lwalk comes back in February delving into the ellipsis. What
is there to delve into when all of mathematics is confined to 10^500
as such:
Finite is 10^500 or below (ditto for inverse) and this is the realm of
mathematics
Incognitum is beyond 10^500 and where mathematics no longer exists but
it sort
of trespasses
Infinite is beyond incognitum.
The simple truth of the situation is that Physics is king, Physics
rules all the sciences, and that math is a tiny compartment of
physics. And since numbers have no meaning beyond the Planck Units,
then mathematics is
no longer existing beyond those Planck Units.
Another supporting edifice of what I say above is true is the edifice
of Quantum Mechanics
of duality. Duality creates strange phenomenon in Physics such as
Quantum Tunnelling or
Double Slit diffraction. So that if Physics is governed by duality, no-
one can know in mathematics anything about what "infinity" is and
whether a powerset of an infinite set
has any sort of meaning.
If Physics stops at 10^500 then mathematics is just a joke and a
preposterous joke beyond
10^500
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Marshall
<plutonium.archime...@gmail.com> wrote:
> Marshall wrote:
> > On Dec 27, 12:24 pm, "plutonium.archime...@gmail.com"
> > <plutonium.archime...@gmail.com> wrote:
>
> > > And if we were to multiply say 0.99... by 0.88...
> > > we end up with 0.7..
>
> > We get what now?
>
> maybe a difficult exercise.
That's bad advice; excuse me if I don't take it.
> Talking about infinity in mathematics is about the same as talking
> about philosophy in a science. Raising silly and absurd
> questions as to how many angels fit on the end of a needle.
The infinity of, say, the natural numbers is simple
and straightforward.
> There is no infinity in physics, is there? Have you ever entertained a
> infinite item in Physics?
Don't know and don't much care; whether there is or is not
infinities in the universe doesn't affect mathematics.
> What if we had something in physics that was infinite? Would it not
> cover-up or hide or drown out everything else that was not infinite?
Nope. Just like the infinitude of primes doesn't drown out all
the rational numbers.
> Lwalk wants to talk about ellipsis when he comes back in February,
> which is a better idea then talking about ill-defined infinity and the
> powerset which arises from an ill-definition.
It would make more sense to discuss the well-defined infinity
and powerset which arise from the usual definitions.
> You are talking about a powerset on an infinite set, for which you
> never defined-well enough what infinity is.
No, I was talking about your claim that "if we were to multiply
say 0.99... by 0.88... we end up with 0.7.." My post quoted
only a single sentence of yours; I am hard pressed to imagine
how you could possibly misunderstand what specifically I
was talking about.
> What A Commonsense Person Would Do:
>
> A person full of commonsense would say: (a) since physics has
What physics does or does not have is not relevant to
the question of what 1 * 8/9 is. Why are you claiming
that it's 7/9 when it's obviously 8/9?
Marshall
Archimedes Plutonium
Archimedes Plutonium wrote:
>
> Finite is 10^500 or below (ditto for inverse) and this is the realm of
> mathematics
> Incognitum is beyond 10^500 and where mathematics no longer exists but
> it sort
> of trespasses
> Infinite is beyond incognitum.
Basically, Physics is king and math is a tiny subset, albeit a very
important subset.
And we have to take the very essence of Physics-Logic and put it into
Mathematics.
This chore or task has never been done before.
The very essence of Physics-Logic is Quantum Duality. Duality is a
existence or
existor function. In order for anything to exist, it must be a
"duality" For a picture
to exist, it must have the entire rest of the Universe as a background
or frame.
For "finite" to exist, it must have "infinity" as a background and
vice versa. For
a Wave to exist, there must be a background "Particle" that exists and
vice versa.
So mathematics never started off with their subject as that of a
subset of Physics and
to utilize the essence of Physics-- duality. Math made only a fleeting
attempt to
incorporate duality such as in the regular-polyhedra of a duality
relationship of sides with
angles with other properties. And in Projective Geometry there was a
smattering of duality
of point to line et al. But Math never incorporated Duality for all of
Math, where geometry is a
duality to algebra, or where finite is a duality of infinity.
Mathematics until these writings was a unidirectional simplistic and
linear train of thought,
which could not even well-define what "finite" was. All of the math
books written to date,
assume and presume the reader to think that "finite" means "ending in
zeroes to the left
of a number such as ....0000087 which we call 87. But that is ill
defined because a number
like 0088888....8888 is then a finite number which we obviously see as
infinite.
So math never told the truth and always hedged about its definition of
"finite".
So we go to the King of science-- Physics to ask what physics would
define as finite.
Here, Physics would say that there is no more physics beyond the
numbers of the Planck
Units. A Planck Unit such as all the Coulomb Interactions inside a
atom of element 109
which is about 10^500. So there is no more physics beyond 10^500 and
its inverse.
So if Physics stops at 10^500 and its inverse, and since math is but a
tiny subset of
physics, well, then certainly, it is preposterous of mathematics or
mathematicians to
claim to having knowledge of numbers or concepts beyond 10^500.
Sure, mathematics has a proof that the prime numbers form an infinite
set. So how do we define, a well-defined concept of infinity? We use
physics.
(i) Finite is up to 10^500 and inverse
(ii) Incognitum is a little beyond 10^500
(iii) Infinity is beyond Incognitum
The above is what would be a Physics outline of a definition of Finite
and Infinity
with Duality playing a role of Logic.
We build all of mathematics around these largest numbers of Planck
Units. And if
our calculations go beyond 10^500 we are in a sort of Quantum zone,
where mathematics
is no longer precise. We have discovered prime numbers that are far
larger than
10^500, and those would be primes in the Incognitum. But since Physics
logic is based
on Duality, we cannot know "infinite prime numbers" for we cannot be
sure that the
concept-of-prime exists beyond finite.
Just as we know that the speed of light is a finite number, we do not
know that anything
can have a speed of infinity.
Much of the 19th and 20th century mathematics that dealt with infinity
and transfinite
were all just bogus chitter chatter.
The Reals of the 20th century was a set that is infinite. But to claim
that between
any two Reals on a number line exists a third Real are statements that
mathematics
simply could never make, since Physics could never vouch for such. In
Physics,
Quantum Mechanics does not permit of "infinite betweenness". Duality
does not permit
it.
So, much of mathematics, is a vast wasteland of garbage. Much of
mathematics was
never well-defined, but left vast stretches of mathematics for the
reader to assume
and presume what is meant by "Finite". And the fallout of this ill-
definitions of
mathematics is the reason that there never were proofs of Fermat's
Last Theorem,
Riemann Hypothesis, Goldbach conjecture, Perfect Numbers Conjecture
and a
long string of unsolved problems.
Mathematics needs to come home. And its home is inside Physics. Much
of
19th and 20th century mathematics is not mathematics but rather is
what can be
called philosophy-math of just endless dribble arguing.
Archimedes Plutonium
byproduct of
physics of the Atom Totality. Because atoms are numerous have have
arithematic. Because
atoms have sizes and shapes we have geometry. Because the Atom
Totality has a special
waist band of the Cosmic atom of 22 subshells inside of 7 shells in
rational-approximation
means that "pi" will have that value and because only 19 are occupied
subshells delivers
a value on what "e" must be in our universe.
So we see that mathematics is a tiny subset of physics, since atoms
account for everything.
And we go to Physics to define what "finite" is compared to
"infinite." And since physics
has no more meaning if it cannot be measurable then the limit of
physics is the largest
numbers of the Planck Units which is about 10^500 for Coulomb
Interactions in element
109. If there is no more physics beyond 10^500, then math a subset of
physics would
be meaningless beyond 10^500. I say meaningless because physics is
governed by
Quantum Logic which has duality. And finite versus infinite is one of
those dualities.
The mathematics enterprize from Pythagoras to modern times is a
Aristotlean Logic
which is not dualistic, but a simple-minded linear logic. Dualism is
nonlinear. And that
is the reason so much of Quantum Mechanics is strange as to how a
particle can be both
a particle and a wave at the same time.
So math should adopt what Physics does with treating infinity. In
physics there are no
known infinite things or items or properties. About the only known
"infinity" in physics
is the "electricmagnetic potential" goes to infinity, but that is only
a "potential", not a
reality.
In short, physics has no infinity. And whenever infinities creep into
physics, the physicist
"renormalizes" the equations to get rid of the infinities.
So Math should have a similar scheme that treats and defines infinite
and finite as such:
(i) Finite is up to 10^500 (and inverse in the microworld)
(ii) Incognitum is a little beyond 10^500
(iii) Infinity is beyond Incognitum
So when doing mathematics one may discover a prime number that is
larger than the
finite boundary of 10^500 and that prime would thus exist in the
Incognitum zone. Where
it has no meaning other than you found a prime that is in the
Incognitum.
Since physics can no longer be carried out beyond 10^500, then we can
never really be
assured of knowing what is beyond 10^500 or the quality of knowledge
of something
beyond 10^500 since so much of Quantum Mechanics is strange with
tunnelling and other
strange phenomenon.
Likewise in math, talking about primes in infinity is very much a
waste of time because
we cannot know if the concept of "prime" extends into infinity region.
As in physics if we
say infinite mass, well, would the entire rest of the cosmos be
crowded out, or if we said that
the H1N1 virus was infinite would it not imply the crowding out of all
other atoms in the cosmos because an infinite supply of viri would
need the room and space occupied by
other things.
So the reader can sense the strangeness or the failure of concepts to
think that infinity
can reside next to finiteness. And since physics is nothing but a
gaggle of nonsense if
there are infinite quantities or parameters, why should the
mathematician be so naively
arrogant to think that he/she has any sort of handle on infinity and
the properties or
characteristics of infinity? Pretty naive and arrogant.
But physics does need infinity as a counterweight or dualism to
finite. So it does exist,
except we can have little to no knowledge of the properties or
characteristics of infinity.
And math, a subset of physics should take that humble repose. That all
of mathematics
be confined to the "Finite level".
Ostap S. B. M. Bender Jr.
<plutonium.archime...@gmail.com> wrote:
>
> Math should view "infinity" in the same way that Physics views infinity
>
If it were up to you, genius, math would view "infinity" in the same
way as your own tribe Piraha:
http://www.newyorker.com/reporting/2007/04/16/070416fa_fact_colapinto#ixzz0bcct6IS3
Everett told him about the Pirahã’s limited “one,” “two,” and “many”
counting system. Other tribes, in Australia, the South Sea Islands,
Africa, and the Amazon, have a “one-two-many” numerical system, but
with an important difference: they are able to learn to count in
another language. The Pirahã have never been able to do this, despite
concerted efforts by the Everetts to teach them to count to ten in
Portuguese
Transfer Principle
> On Dec 28, 12:12 pm, Archimedes Plutonium
> > > > we end up with 0.7..
> the question of what 1 * 8/9 is. Why are you claiming
> that it's 7/9 when it's obviously 8/9?
0.9 * 0.8 = 0.72
0.99 * 0.88 = 0.8712
0.999 * 0.888 = 0.887112
0.9999 * 0.8888 = 0.88871112
0.99999 * 0.88888 = 0.8888711112
Therefore, we ought to have:
0.99...99 * 0.88...88 = 0.88...88711...112
and not 0.77...77 -- pending, of course, a more rigorous
definition of ellipsis.
This is, of course, the rigorous definition of ellipsis
that I hope to work on and unveil next month. Note that
the ellipsis definition I plan on working on is intended
to satisfy some of the ideas of _several_ so-called
"cranks," not just AP. Thus, the answers to certain
problems in my notation will not necessarily be the same
as the answers obtain using AP's notation. But I do
promise that 0.99...99 will be strictly less than unity
(by an infinitesimal, of course), since it is virtually
unanimous among ellipsis "cranks" that 0.99...99 ought
to be strictly less than unity. This will be in direct
contrast to standard analysis, where 0.99... = 1 exactly.
Incidentally, this also answer tommy1729's question from
earlier in this thread:
"[T]he so-called "cranks" will be able to use their
ellipsis notation without objections from those who
prefer to use standard set theory and analysis."
tommy1729:
"what *ellipsis* ? 1 + 2 + 3 + *...* 100 = 5050 ?"
No, I mean the ellipsis as used by AP and several other
"cranks" when using notation such as 0.99...99. Those who
prefer the use of standard analysis often object to AP
and others who wish to use ellipsis notation without a
rigorous definition. Therefore, in February I want to
come up with a rigorous definition of the ellipsis in
order to help the "cranks" answer those who ask them for
a rigorous definition of ellipsis.
MoeBlee
> in February I want to
> come up with a rigorous definition of the ellipsis in
> order to help the "cranks" answer those who ask them for
> a rigorous definition of ellipsis.
Keep up the good works. It's a sure bet that one day you will be made
the Patron Saint of Cranks.
MoeBlee
Ostap S. B. M. Bender Jr.
<plutonium.archime...@gmail.com> wrote:
> Marshall wrote:
> > On Dec 27, 12:24 pm, "plutonium.archime...@gmail.com"
> > <plutonium.archime...@gmail.com> wrote:
>
> > > And if we were to multiply say 0.99... by 0.88... we end up with
> > > 0.7..
>
> > We get what now?
>
> maybe a difficult exercise.
>
Marshall, he is right: if we forget all our knowledge and assume that
0.99... x 0.88... = 0.7... , then we can indeed deduce anything he
wants.
Archimedes Plutonium
Transfer Principle wrote:
> On Dec 28 2009, 12:57 pm, Marshall <marshall.spi...@gmail.com> wrote:
> > On Dec 28, 12:12 pm, Archimedes Plutonium
> > > > > And if we were to multiply say 0.99... by 0.88...
> > > > > we end up with 0.7..
I was only speaking off the cough in broad sweeping generalities. And
to people who
strive not to learn anything but shovel hatred, they usually pick on
something like this
to dwell on their hatred.
I honestly do not know why you, LWalk continues to make conversations
with these
hate-mongers whose goal is not to learn or push math forward. You are
too much of
a "too kind of a soul."
I do not care about hatemongers replies, for they just, usually waste
my time.
What I meant by 0.99.. x 0.88.. = 0.7... in the context of my writings
of 10^500
as the definition of Finite is that of
0.99..-9 x 0.88..-8 = 0.7.. -?
where the -9 and -8 and -? are the 10^500 decimal place value. I think
some modern
day algorithm of math, maybe no computer can work it out but some
algorithm could
probably tell us exactly what the -? exact digit is.
> > What physics does or does not have is not relevant to
> > the question of what 1 * 8/9 is. Why are you claiming
> > that it's 7/9 when it's obviously 8/9?
>
> 0.9 * 0.8 = 0.72
> 0.99 * 0.88 = 0.8712
> 0.999 * 0.888 = 0.887112
> 0.9999 * 0.8888 = 0.88871112
> 0.99999 * 0.88888 = 0.8888711112
>
> Therefore, we ought to have:
>
> 0.99...99 * 0.88...88 = 0.88...88711...112
Yes, thanks, LWalk, true to my invention of the AP-adics = Infinite
Integers.
I invented those because I despised the base-dependency of P-adics. I
wanted
a number system that carried out addition and multiplication not tied
to the base
involved.
Other reasons I invented AP-adics is to find the natural numbers
native to the
NonEuclidean Geometries, where the Reals are native to Euclidean
Geometry.
The result of the AP-adics is that mathematics has only one type of
infinity. There
are no hierarchies of different types of infinities.
But I am on another math topic when I am doing the definition of
Finite. That
mathematics is not able to tread beyond the Finite Range because
Physics
has no "infinities". And since Physics has no infinities, the logic of
Physics
beyond Finite is not a logic that Mathematics can ever handle.
Since Physics has no numbers beyond about 10^500, we simply have a
breakdown
in any mathematics beyond 10^500. We do not know if a number such as
the
Infinite Integer 9999....999997 is prime or not prime. There is
nothing in mathematics
that can decide that question and the reason is that we cannot know if
the concept of
prime breaksdown once we pass 10^500, and if it breaks down, where
does it breakdown.
The Logic of Physics is no longer the Aristotlean linear logic of
either yes or no. The
logic of Physics becomes a duality or nonlinear.
Aristotlean Logic is alright for Physics of its range of 10^500 to
10^-500 because we
can get direct experimentation to tell us if things are correct but
once we go beyond
those Planck Units, we can have no idea of when a duality logic plays
tricks on us
and destroys the concept you are working with.
Well, that is good that you will clarify the "ellipsis", but nothing
you
do Lwalk will sway any of those hatemongers that pursue you. They
are not here for mathematics but their psychological chips on their
shoulders, which is far cheaper than paying for psychiatric sessions.
Feel free to exploit the AP-adics notation. I came up with it because
I
just dreaded the P-adics base dependency where I could not visualize
the numbers nor the operations.
But please, keep in mind, LWalk, that I do not believe a number like
9999....99997 or 999...131211109876543210 have any "reality" , but
that
numbers such as 9999..99-7 where the -7 is the 10^500 decimal place
value
has reality and is where mathematics as a subject has to end, a
meaningful
end since beyond that number there is no "reliable physics" since that
is
beyond the Physics Plank Units.
So where the upshot of the invention of the AP-adics = Infinite
Integers
is that there is one and only one type or kind of infinity. The upshot
of defining
a well-defined Finite in mathematics is that infinity is not a
mathematics subject
and that mathematics cannot go beyond the boundaries of physics.
Sure, I admit that infinity exists, for example the infinitude of
primes exists, but
once I go beyond the number 10^500, I can no longer use Aristotlean
Logic that
mathematics is based upon and must use Physics logic which is
nonlinear.
So that I know 2, 3, 5, 7 are primes and I know there is a prime
number near the
boundary of 10^500 and there is a prime beyond 10^500, and that there
are
an infinitude of primes. But once I go beyond 10^500, no logic but
Physics logic
can sort out what prime means beyond that boundary.
Your clarification of "ellipsis" is a help to me, and others and the
math community
but it will not phase any of that gaggle of hatemongers that chase
after you, because
their interests are to fill their psychiatric holes, not mathematical.
Archimedes Plutonium
Archimedes Plutonium wrote:
> Transfer Principle wrote:
> > On Dec 28 2009, 12:57 pm, Marshall <marshall.spi...@gmail.com> wrote:
> > > On Dec 28, 12:12 pm, Archimedes Plutonium
> > > > > > And if we were to multiply say 0.99... by 0.88...
> > > > > > we end up with 0.7..
>
> I was only speaking off the cough in broad sweeping generalities. And
> to people who
> strive not to learn anything but shovel hatred, they usually pick on
> something like this
> to dwell on their hatred.
>
> I honestly do not know why you, LWalk continues to make conversations
> with these
> hate-mongers whose goal is not to learn or push math forward. You are
> too much of
> a "too kind of a soul."
>
> I do not care about hatemongers replies, for they just, usually waste
> my time.
>
> What I meant by 0.99.. x 0.88.. = 0.7... in the context of my writings
> of 10^500
> as the definition of Finite is that of
>
> 0.99..-9 x 0.88..-8 = 0.7.. -?
>
> where the -9 and -8 and -? are the 10^500 decimal place value. I think
> some modern
Here and elsewhere I made a sloppy mistake where that should read
10^-500
and I corrected it on the original with a (sic) sign.
Ostap S. B. M. Bender Jr.
Run algorithms in short time without a computer? How? Have you
invented a new computing device? A turbo-finite-state-machine?
Well, I can't tell off the cuff whether 9999....999997 is prime or not
prime, but I know that 9999....99999721 is not prime. So, there are a
lot of things that we know about very large numbers.
I fail to see what's so magic about 10^500 and not, say, 10^300 or
10^7000.
Mathematics is an abstract mental exercise and doesn't need to limit
itself only to numbers and objects that humans can find in Nature.
If you want to study only bounded mathematics - go ahead. Maybe you'll
derive some interesting facts. That's the beauty of math - there are
probably 1000000 different axiom structures that have been
investigated. Your research may be 1000001'st.
But don't expect mathematicians to give up work on the existing
infinite structures.
Archimedes Plutonium
Ostap S. B. M. Bender Jr. wrote:
(snipped)
> >
> > 0.99..-9 x 0.88..-8 = 0.7.. -?
> >
Alright, according to LWalk the first digits are "0.888" and not 0.7..
And I reckon that supercomputers can easily reveal what all the digits
are
for the answer. And obviously the last digits will be what LWalk
pointed out,
and for which I was sleeping and not thinking as "1112"
But can a supercomputer fill in the entire rest of the number as the
exact product?
> > where the -9 and -8 and -? are the 10^500 decimal place value. I think
> > some modern
> > day algorithm of math, maybe no computer can work it out but some
> > algorithm could
> > probably tell us exactly what the -? exact digit is.
> >
>
> Run algorithms in short time without a computer? How? Have you
> invented a new computing device? A turbo-finite-state-machine?
>
I think simple operations such as divide or multiply or add are easily
done
for a 10^500 place value. I do not know what the computer time is for
such
a feat.
> >
> > > > What physics does or does not have is not relevant to
> > > > the question of what 1 * 8/9 is. Why are you claiming
> > > > that it's 7/9 when it's obviously 8/9?
> >
> > > 0.9 * 0.8 = 0.72
> > > 0.99 * 0.88 = 0.8712
> > > 0.999 * 0.888 = 0.887112
> > > 0.9999 * 0.8888 = 0.88871112
> > > 0.99999 * 0.88888 = 0.8888711112
> >
Yes, it looks like alot of "8" digits in the front. And it will be
"1112" at the end.
But can the computer fill in the rest of the number to its exact
product?
And can someone tell me if the numbers of the two multipliers were
messy, at about
what number as Finite, would our modern day computer not be able to
answer the
product with exact digits? Is it 10^500 or can our computers handle a
finite definition
as say 10^1000?
Maybe the Planck Units are somehow related to the efficiency in time
of rendering
exact products of numbers, though I doubt there is such a link. A link
of where a
intelligent life can nucleosynthesize 231Pu and thus is limited by
calculations of
its best computers to 10^500? I doubt it.
>
> Well, I can't tell off the cuff whether 9999....999997 is prime or not
> prime, but I know that 9999....99999721 is not prime. So, there are a
> lot of things that we know about very large numbers.
Yes, I do not know why I wrote "cough" when I meant "cuff".
The thing about knowing for sure that 9999....99999 is not prime, yet
a number
such as 9999....99997 is never decidable (perhaps the first know case
of undecidable
in mathematics) as to whether it is prime or not prime and I have had
many arguments
by various persons-- Dik Winter, LWalk, et al as to whether
9999....99997 is prime or
not prime, because it depends on the place value of the "7", even
though it is infinite
place value.
So this is what I mean by saying that Physics gives out at 10^500 and
that Quantum
Logic of duality (circular logic) comes in. We are fine and dandy with
all the numbers
with a boundary at 10^500 as all obeying Aristotliean Logic and the
Physics Experiments
are fine and dandy in this range also. But when we talk of infinite
sets where 999...9997
is a member, we lose Aristotliean Logic and where the Quantum Logic of
duality enters.
So Physics no longer is meaningful beyond Planck Units and thus
mathematics is also
no longer meaningful. So all of mathematics must be confined to a
range of 10^500 or
thereabouts. It could be 10^600.
>
> I fail to see what's so magic about 10^500 and not, say, 10^300 or
> 10^7000.
>
Obviously you have no education in Physics, Quantum Mechanics and
Planck Units.
If you knew what Planck Units are, you would not ask such a question.
> Mathematics is an abstract mental exercise and doesn't need to limit
> itself only to numbers and objects that humans can find in Nature.
>
And, from your above sentence, you are a novice of mathematics
Physics also is axiomatic, only they are called "principles". Duality
is one
such principle. The Maxwell Equations are axioms of physics. The
Uncertainty,
the Pauli and the Complementarity principles are axioms.
All mathematics written to date, have an assumed axiom-- Aristotliean
Logic
of linear logic where everything is either yes or no, true or false.
In Physics,
for Finite we have Aristotliean Logic but for beyond finite we have
Quantum Logic.
> If you want to study only bounded mathematics - go ahead. Maybe you'll
> derive some interesting facts. That's the beauty of math - there are
> probably 1000000 different axiom structures that have been
> investigated. Your research may be 1000001'st.
You have a big fault here. But do not feel bad since noone else in
mathematics
ever defined Finite with a "well defined definition."
So your mathematics is all contradictory since you never defined what
you mean
precisely by finite and infinite. Your mathematics is a gaggle of
contradiction. You
say the Natural Numbers are a set that is only finite numbers yet you
say this set
is infinite. And worse, you cannot perceive your mistake, but only
assured by the
mistaken mass of other mathematicians.
When asked to define "finite precisely", you resort to namecalling
such as "crank".
You hide in amoungst the thousands of others who assume what finite
means
and unable to give a precision definition.
>
> But don't expect mathematicians to give up work on the existing
> infinite structures.
>
I expect in the future that all mathematicians will give up on
"infinite structures".
You should give up on mathematics because you do not have a good
handle on the
subject and should study some physics. Take any college physics
textbook and look
up how much time is devoted to "infinite structure". You will find
very little and about
the only lengthy discussion on "infinite structure" is in quantum
electrodynamics
where they try to get rid of the nuisance infinites with
renormalization techniques.
In summary:
Math only exists because physics exists
Math is a tiny subset of physics and it arises because atoms are
numerous
giving us arithematic and atoms have shape and size giving us
geometry.
Since Physics ends at the Planck Units since no experiments can be
conducted beyond those Units, then Mathematics gives out also at
about 10^500 and that is the precise definition of "finite"
Archimedes Plutonium
Ostap S. B. M. Bender Jr.
<plutonium.archime...@gmail.com> wrote:
> Ostap S. B. M. Bender Jr. wrote:
> (snipped)
>
>
>
> > > 0.99..-9 x 0.88..-8 = 0.7.. -?
>
> Alright, according to LWalk the first digits are "0.888" and not 0.7..
>
Not only to him. Anybody with at least 3 years of school can figure
this out, because even 0.99*0.88 > 0.8
>
> And I reckon that supercomputers can easily reveal what all the digits
> are
> for the answer.
>
So can LWalk and I. See below.
>
> And obviously the last digits will be what LWalk
> pointed out,
> and for which I was sleeping and not thinking as "1112"
>
> But can a supercomputer fill in the entire rest of the number as the
> exact product?
>
Of course not. It is practically impossible to write down 2*10^500
digits. But why do you need to write down all these digit? You have
been already told what this number is exactly:
0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
71111...1(K-1 times)2
Just substitute K = 10^500 there, or for that matter, K = 10^50000.
Mathematicians can figure out a lot of things about enormously large
number, even if these numbers are too large to occur in the real
world, as long as these numbers have some compactly describable
pattern. And if they don't have compactly describable pattern - then
you can't even describe them.
>
> > > where the -9 and -8 and -? are the 10^500 decimal place value. I think
> > > some modern
> > > day algorithm of math, maybe no computer can work it out but some
> > > algorithm could
> > > probably tell us exactly what the -? exact digit is.
>
> > Run algorithms in short time without a computer? How? Have you
> > invented a new computing device? A turbo-finite-state-machine?
>
> I think simple operations such as divide or multiply or add are easily
> done
> for a 10^500 place value. I do not know what the computer time is for
> such
> a feat.
>
Oh, maybe 10^480 years or so, give or take a few gazillions, if you
need all 2*10^500 digits.
>
>
> > > > > What physics does or does not have is not relevant to
> > > > > the question of what 1 * 8/9 is. Why are you claiming
> > > > > that it's 7/9 when it's obviously 8/9?
>
> > > > 0.9 * 0.8 = 0.72
> > > > 0.99 * 0.88 = 0.8712
> > > > 0.999 * 0.888 = 0.887112
> > > > 0.9999 * 0.8888 = 0.88871112
> > > > 0.99999 * 0.88888 = 0.8888711112
>
> Yes, it looks like alot of "8" digits in the front. And it will be
> "1112" at the end.
>
Let me repeat:
0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
71111...1(K-1 times)2
Just substitute K = 10^500 there.
>
> But can the computer fill in the rest of the number to its exact
> product?
>
Isn't the above result exact? Surely, a computer can do the same
thing, if you program it right.
>
> And can someone tell me if the numbers of the two multipliers were
> messy, at about
> what number as Finite, would our modern day computer not be able to
> answer the
> product with exact digits? Is it 10^500 or can our computers handle a
> finite definition
> as say 10^1000?
>
I doubt if there exist more than 10^400 molecules in the entire
Universe. So, you cannot even input your 10^500-digit numbers into a
computer. So, how can you expect any output if you can't give it the
exact input? Garbage in - garbage out.
However, as I said, there is a compact way of representing and
multiplying periodic-looking decimal numbers even if they are 10^5000
digits long. Such as:
0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
71111...1(K-1 times)2
>
> > Well, I can't tell off the cuff whether 9999....999997 is prime or not
> > prime, but I know that 9999....99999721 is not prime. So, there are a
> > lot of things that we know about very large numbers.
>
> Yes, I do not know why I wrote "cough" when I meant "cuff".
>
You are missing my point here.
>
> The thing about knowing for sure that 9999....99999 is not prime, yet
> a number
> such as 9999....99997 is never decidable (perhaps the first know case
> of undecidable
> in mathematics)
>
Never? A number such as 9999....999951 is certainly decidable.
Robert
there be dragons.
Jesse F. Hughes
> Of course not. It is practically impossible to write down 2*10^500
> digits. But why do you need to write down all these digit? You have
> been already told what this number is exactly:
>
> 0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
> 71111...1(K-1 times)2
>
> Just substitute K = 10^500 there, or for that matter, K = 10^50000.
Surely, AP *can't* substitute K=10^500 there, because the result would
be too long to be a number (according to AP).
At least, if I understand his latest wackiness right. He's come to a
weird mixture of ultrafinitism and the claim that there are infinite
integers, and the result is pretty unclear to me.
--
"But remember, as long as one human being follows the rules of
mathematics, then mathematics as a human discipline survives.
Right now I'm that one human being, so mathematics survives."
-- James S. Harris
Archimedes Plutonium
Ostap S. B. M. Bender Jr. wrote:
> On Jan 5, 10:43 pm, Archimedes Plutonium
> <plutonium.archime...@gmail.com> wrote:
> > Ostap S. B. M. Bender Jr. wrote:
> > (snipped)
> >
> >
> >
> > > > 0.99..-9 x 0.88..-8 = 0.7.. -?
> >
> > Alright, according to LWalk the first digits are "0.888" and not 0.7..
> >
>
> Not only to him. Anybody with at least 3 years of school can figure
> this out, because even 0.99*0.88 > 0.8
>
> >
> > And I reckon that supercomputers can easily reveal what all the digits
> > are
> > for the answer.
> >
>
> So can LWalk and I. See below.
>
Well, then, write out the complete and exact product of 0.999..-9 X
0.888..-8
where the -9 and -8 are in the 10^-500 place value and the final
answer
looks like this in basic form 0.88..-? where the -? is in the 10^-500
place value.
So your final answer is fully exact and has 501 digits.
> >
> > And obviously the last digits will be what LWalk
> > pointed out,
> > and for which I was sleeping and not thinking as "1112"
> >
> > But can a supercomputer fill in the entire rest of the number as the
> > exact product?
> >
>
> Of course not. It is practically impossible to write down 2*10^500
> digits. But why do you need to write down all these digit? You have
> been already told what this number is exactly:
>
> 0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
> 71111...1(K-1 times)2
>
> Just substitute K = 10^500 there, or for that matter, K = 10^50000.
>
> Mathematicians can figure out a lot of things about enormously large
> number, even if these numbers are too large to occur in the real
> world, as long as these numbers have some compactly describable
> pattern. And if they don't have compactly describable pattern - then
> you can't even describe them.
>
Okay, yes, I made another mistake because once the multiplication is
finished I am going to have to lopp off all the digits that go beyond
the
10^-500 mark of place value, and so the 1112 string is going to be
lopped
off.
What I want is the exact digits of the product to the 10^-500 place
value.
Now a line here on sci.math Usenet contains about 50 characters or 50
digits
and so, you can give the exact answer of the product on half a page.
And just as 9999...9999 is decidable and just as all even numbers are
decidable.
But the "jinx" is that some numbers that are odd can never be decided
as to whether
they are prime or composite because of the place-value. Numbers such
as
999....99991 or 9999....9997 or 9999...99989. These three numbers are
sometimes
prime depending on where you "call your place value at infinity".
So this is an example of Duality Logic in Mathematics. It is common in
Physics
since duality is physics. And the above is probable the heart of what
Godel was
thinking about with undecidability. Only he never really needed to do
a proof. All he
needed to do, like I have done, is simply acknowledge that mathematics
has no
precision definition of finite versus infinite. So once you define
Finite precisely-- such
as 10^500 and below. Then you can come to grips with the fact that the
concept of
"prime number" is no longer valid beyond 10^500 since you can never
"decide"
whether a certain number beyond 10^500 is truly prime or composite.
This is what
a physicist would say that you no longer have the ability to measure,
or test or
experiment beyond the Planck Unit of 10^500. All of the Godel work on
undecidability
is nothing more than the fact that once you define Finite precisely,
all of mathematics
is confined to that finite region.
Archimedes Plutonium
Robert wrote:
> what is 10^500 + 1?
>
> there be dragons.
Pray tell, is it prime? Is this number
10000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000001
A prime number?
And with a definition, a precise definition of Finite as the Planck
Unit
of 10^500
Then 10^500 +1 is no longer a finite number but is in the Incognitum
where numbers have "no meaning" because there is no longer a reliable
physics going on to test and experiment.
So the structure of mathematics becomes this:
Finite -- 10^500 and below and where all of math is carried out
Incognitum -- little beyond Finite and the answers that lie here
are not mathematical because they are not trustworthy
Infinite -- beyond the Incognitum such numbers as 9999....9997
where they are undecidable as to whether they are prime or
composite and most other concepts of mathematics.
Physics logic is duality and only in the Finite region can we
have Aristotliean logic of linear logic and so mathematics
is a subject that can only be precise within the confines
of Aristotlean logic of 10^500 or less.
Alan Smaill
> Robert wrote:
> > what is 10^500 + 1?
> >
> > there be dragons.
>
>
> Pray tell, is it prime? Is this number
>
> 10000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000001
>
> A prime number?
It is divisble by 73, according to my feeble small computer.
> Archimedes Plutonium
> www.iw.net/~a_plutonium
> whole entire Universe is just one big atom
> where dots of the electron-dot-cloud are galaxies
--
Alan Smaill
Owen Jacobson
<plutonium.archime...@gmail.com> wrote:
> Robert wrote:
> > what is 10^500 + 1?
>
> > there be dragons.
>
> Pray tell, is it prime? Is this number
>
> 10000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000001
That is not 10^500 + 1 (it's 10^499 + 1 -- remember, 10^n has n+1
digits, not n digits). This is 10^500 + 1:
10000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000001
And, the quotient when divided by 73:
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
136986301369863013698630136986301369863013698630
1369863013698630137
> A prime number?
No. You can verify this with a short Python snippet:
>>> import decimal
>>> (decimal.Decimal('1E500') + 1) / 73
(The decimal module is an arbitrary-precision math library.)
-o
Ostap S. B. M. Bender Jr.
<plutonium.archime...@gmail.com> wrote:
> Ostap S. B. M. Bender Jr. wrote:
>
>
>
> > On Jan 5, 10:43 pm, Archimedes Plutonium
> > <plutonium.archime...@gmail.com> wrote:
> > > Ostap S. B. M. Bender Jr. wrote:
> > > (snipped)
>
> > > > > 0.99..-9 x 0.88..-8 = 0.7.. -?
>
> > > Alright, according to LWalk the first digits are "0.888" and not 0.7..
>
> > Not only to him. Anybody with at least 3 years of school can figure
> > this out, because even 0.99*0.88 > 0.8
>
> > > And I reckon that supercomputers can easily reveal what all the digits
> > > are
> > > for the answer.
>
> > So can LWalk and I. See below.
>
> Well, then, write out the complete and exact product of 0.999..-9 X
> 0.888..-8
>
First, write out the complete and exact decimal representation of
'0.999..-9'.
First of all, if you don't write out the complete and exact decimal
representation of your inputs, how do you expect to get the complete
and exact decimal representation of the output? Garbage in - garbage
out.
Second - why are you so hung up on complete and exact decimal
representations? Decimal representation is just one myriads of other
representations of rational/real numbers.
You yourself use your weird notation:
'0.999..-9'
That's not traditional decimal representation, is it?
So, here is my answer, which is perfectly exact from the mathematical
point of view:
0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
71111...1(K-1 times)2
Just substitute K = 10^500 there, or for that matter, K = 10^50000.
Can't you understand?
>
> where the -9 and -8 are in the 10^-500 place value and the final
> answer
> looks like this in basic form 0.88..-? where the -? is in the 10^-500
> place value.
>
> So your final answer is fully exact and has 501 digits.
>
What? My answer doesn't have 501 digits. If viewed as a decimal
fraction, it has 2*10^500 digits. Count them:
10^500 - 1 digits '8'
One digit '7'
10^500 - 1 digits '1'
One digit '2'
Of course, in my notation above, it has a very short and compact
representation:
0.99...9(K times) * 0.8888...8(K times) = 0.88...8(K-1 times)711...1
(K-1 times)2
>
> > > And obviously the last digits will be what LWalk
> > > pointed out,
> > > and for which I was sleeping and not thinking as "1112"
>
> > > But can a supercomputer fill in the entire rest of the number as the
> > > exact product?
>
> > Of course not. It is practically impossible to write down 2*10^500
> > digits. But why do you need to write down all these digit? You have
> > been already told what this number is exactly:
>
> > 0.99999...9(K times) * 0.88888...8(K times) = 0.8888...8(K-1 times)
> > 71111...1(K-1 times)2
>
> > Just substitute K = 10^500 there, or for that matter, K = 10^50000.
>
> > Mathematicians can figure out a lot of things about enormously large
> > number, even if these numbers are too large to occur in the real
> > world, as long as these numbers have some compactly describable
> > pattern. And if they don't have compactly describable pattern - then
> > you can't even describe them.
>
> Okay, yes, I made another mistake because once the multiplication is
> finished I am going to have to lopp off all the digits that go beyond
> the
> 10^-500 mark of place value, and so the 1112 string is going to be
> lopped
> off.
>
> What I want is the exact digits of the product to the 10^-500 place
> value.
>
I am sorry, but decimal representations of rational numbers are
useless when talking about rationals that are more than, say, 10^2
digits long. For such numbers, people use some other representation
instead, as did you when you wrote:
"0.99..-9 x 0.88..-8 = 0.7.. -?"
>
> Now a line here on sci.math Usenet contains about 50 characters or 50
> digits
> and so, you can give the exact answer of the product on half a page.
>
Can you give an "exact answer" to your definition of '0.7.. -?' or of
'0.99..-9'?
mike3
Hmm. So beyond some magic threshold, rules like those of arithmetic
are no longer "trustworthy"?! So I guess you can't trust that
10000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000001 + 1 =
10000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000002
then, right? It could equal
32975092734097329040239749023704323927408237402374
89235892369560976509236406027931632646203140622347
32720314702340732407347803427340783873528705084380
48743874387043784387042387047283874238794238742398
74327434872437802734878042748374237804278478043870
43078234807432807423087423807408727402780423807423
87480487042080428394830965904619756209164790327659
08621379564378567093265873467659734985623652736837
82365943675097289562459276237590837248905346343243
43248963246346394327983427324934293131247932575623
79156793254732547013527048523784783254780352078406
23863824768321774032750325407532078463207846732014
67082316470231674021678450872347802136407823647062
03140236409430006660006660006660006660006660006660
00666000666066606660666066606660666066606660666066
606660666
or something, for all we know. It could equal 42. It could equal
something
else altogether. It might be a non-number. Yeah, right, I don't THEENK
so,
matey.
Archimedes Plutonium
Alan Smaill wrote:
> Archimedes Plutonium <plutonium....@gmail.com> writes:
>
> > Robert wrote:
> > > what is 10^500 + 1?
> > >
> > > there be dragons.
> >
> >
> > Pray tell, is it prime? Is this number
> >
> > 10000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000001
> >
> > A prime number?
>
> It is divisble by 73, according to my feeble small computer.
>
Yes, thanks Alan. I probably forgot to include a 0 in the above
representation for there. I often do not check for errors, feeling
that
the message clarity if more important and there will always be typo
errors.
The reason I wanted to represent 10^500 +1 as in the above is in hopes
that someone will use a computer, like LWalk and post what is
0.999..-9 X 0.888..-8 with the answer in full representation to
10^-500 place
value. And to additionally post how long it required that computer to
spit out
the full answer.
I suspect that in the future, that representation above will be
commonplace
because it is the boundary line where mathematics beyond does not
matter.
But that is no big deal, or big news today since most people only use
three
place value to the right of the decimal point as anything significant,
let alone
10^-500 place value.
Alan, it is obvious that every odd number that ends in a "1" (ditto
for 3, 7, 9)
is a candidate for being a prime number. So now, we know the primes
are infinite,
means that this infinity has to be culled out of the odd numbers who
end in
1, 3, 7, 9.
So that if the Precision definition of Finite is 10^500 and a short
way beyond is the Incognitum
and beyond the Incognitum is Infinity means that somewhere along this
path of
10^500 +1
10^501 +1
10^502 +1
etc. etc
Somewhere along that path will come upon a prime number.
This means that we must have a representation for an infinte-number
such as this 9999....99999 or this 10000...0000 where the ellipsis
means
an infinity of digits between the frontview and backview.
If all numbers were finite numbers, then the set of all primes is not
infinite, because
every Finite number has a successor.
So here comes the friction, or what Godel calls undecidable. If we
define Finite as
10^500 then far beyond are infinity and infinite numbers such as
999...999 and we
can see it is composite but we cannot see whether 9999....9997 is
prime or composite
nor whether 9999...9999 +2 is prime or composite. There is nothing in
present day
mathematics that could ever sort out whether 999...9997 is prime or
composite because
it is no longer Aristotliean logic but is a circular logic of duality,
of Quantum logic.
Modern mathematics never addressed the problem of defining Finite with
precision.
From Pythagoras to Von Neumann, mathematics has avoided defining what
it means
to be finite versus infinite. And only with the Atom Totality theory
can we even venture
to give "finite versus infinite" a precision definition.
Noone has ever shown the math community a example of a Godel
undecidable statement.
Some have suggested the Riemann Hypothesis or the Goldbach Conjecture.
Well, both are
Godel undecidable. But mine above of whether 999...9997 is prime or
composite is probably
the most simple mathematical statement of undecidable.
It is impossible to make any mathematical system of axioms with
Aristotliean logic, in which
you precisely define Finite versus Infinite and can tell you that
999...9997 is prime or is composite. It is impossible. And the reason
is very simple, because Physics is king and
math is a subset. And in Physics, infinity is a duality of finite. So
when you leave the world
of Planck Units, you enter into the incognitum and Infinity where
Aristotliean Logic no longer
exists and where nonlinear-logic comes into play, or duality logic.
The concept of prime and composite no longer exists beyond the Finite
realm. Example: in physics when you are travelling at the speed of
light, your front face is no longer distinguishable from your back
and that if you are viewed, your front and back are one and the same.
This is called
quantum strangeness. It occurs because Duality comes into play. The
same thing happens
to mathematics in that once you precisely define Finite, beyond
finite, your concepts fall
apart such as prime such as the powerset.
This thread started out with powersets on infinite sets. Well, in a
mathematics that never defined finite versus infinite it is utterly
preposterous to even consider a powerset on
a infinite set of 9999...99999 elements. Preposterous to even use
Aristotliean logic of a
powerset on a infinite set of elements.
When LWalk comes back to define "ellipsis" better. I hope he keeps in
mind to start
with a precision definition of Finite first. Because to me Planck
Units of 10^500 is a
good starting point and that all numbers that are finite should have a
two-dot ellipsis
such as 100..-3 where the -3 is in the 10^500 decimal place. Now for
an infinite
integer the ellipsis is three-dots such as this: 6888...87 where the 7
is the backview
and the 6 is the frontview and where the ellipsis indicates an
infinity of 8s in the middle.
Modern math has been extremely sloppy in never defining with precision
what finite is
and how it relates to infinity and due to that dereliction or
sloppiness, the ellipsis
in math was never defined.
Archimedes Plutonium
mike3 wrote:
> >
>
> Hmm. So beyond some magic threshold, rules like those of arithmetic
> are no longer "trustworthy"?! So I guess you can't trust that
snipped the venom
voice without a mind.
Okay look at some Quantum Mechanics with addition. As we travel closer
to the speed of light we add on more and more distance per time. But
our mass
seems to get heavier and heavier. Infinite mass as we hit the speed of
light.
So addition is no longer valid beyond a precision-definition of
finite.
Lets look at it in the Microworld. We approach absolute zero
temperature in finite steps
of cooling. We seem to get closer and closer to absolute zero but
never as close
as 10^-500 K. As we get colder, it takes infinite energy to go to 0 K.
And along that
coldness trail we run into all sorts of Quantum Strangeness. Here
again, addition
beyond the finite world or finite mathematics, we lose Aristotliean
Logic. We lose the
logic that was mathematics itself and we enter into a logic that
destroys the math
we entered with.
Finite is a dual of Infinity, and the two cannot be handled by
Aristotliean Logic.
I really need to find out how to spell "Aristotliean", and funny how
my detractors
and hatemongers pick on almost every other thing I make a typo but the
typo that
I really want fixed.
Finite is a dual of Infinity, and when doing math or physics, we can
only handle the
Finite with Aristotliean Logic, and we can only make guesses about
what happens
beyond the finite realm. Since physics gives out at the Planck Units,
then mathematics
is no longer assured of its answers beyond 10^500.
Archimedes Plutonium
Archimedes Plutonium wrote:
>
> If all numbers were finite numbers, then the set of all primes is not
> infinite, because
> every Finite number has a successor.
>
I am not happy with that paragraph and should clarify it better. The
idea is that
you cannot have an infinite set if every one of its elements is a
"finite number".
And in fact, it must have an infinity of infinite numbers.
If all the elements of an infinite set were finite numbers then those
numbers cannot
be endless, since every one of those numbers is finite.
So this is what I mean that noone from Pythagoras to VonNeumann ever
provided a
precision definition of Finite versus Infinite. When you define finite
as 10^500 or below, then you can talk about what is a infinite set
versus a finite set.
Archimedes Plutonium
Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
>
> >
> > If all numbers were finite numbers, then the set of all primes is not
> > infinite, because
> > every Finite number has a successor.
> >
>
> I am not happy with that paragraph and should clarify it better. The
> idea is that
> you cannot have an infinite set if every one of its elements is a
> "finite number".
> And in fact, it must have an infinity of infinite numbers.
>
> If all the elements of an infinite set were finite numbers then those
> numbers cannot
> be endless, since every one of those numbers is finite.
>
Maybe I hit the barebone essentials of the duality of finite to
infinite in the above
sentence-- If all the elements of a infinite set were finite numbers,
then that set
cannot be infinite because it cannot be endless since every member is
finite.
In other words, from Pythagoras to VonNeumann, we have ignored that
internal
contradiction. When trying to prove Fermat's Last theorem or Riemann
Hypothesis
or Goldbach Conjecture we have pretended that the Natural Numbers are
all finite
and yet that set is an infinite set. So we started with a
contradiction.
Once finite is defined as 10^500, do we release ourselves of a failed
mathematics
and begin to do a true precision mathematics.
Archimedes Plutonium
the one hand
says that the Natural Numbers are all of them "finite numbers" and
that such a set of
Natural Numbers forms an infinite set. I worry that such a person
really is not a
mathematician because they cannot see the error of their ways. That
how can you
have a infinite set of items when every one of those items is a finite-
number.
I wonder why that does not bother the people who consider themselves
as being a
mathematician. Why their inconsistency and their self-contradictory
statements
does not bother them.
A set like this {2,3,4} is composed of three finite-numbers and noone
is fooled into
thinking the set is infinite. But a set like this {1,2,3,... where all
members are finite
numbers} seems to fool all the mathematicians from Pythagoras to
VonNeumann,
but not fooling Archimedes Plutonium.
If you claim that a set is infinite and you simultaneously claim that
each member is
a finite-number. It is inescapable that you have a self-contradiction
and a inconsistency.
I guess the only reason it escaped the minds of all the mathematicians
from Pythagoras
to Plutonium, is that they all thought that the word "endless" for the
Natural Numbers
1,2,3,... covered up or made up for the fact that all the Natural
Numbers were deemed
to be finite-numbers.
So what I am doing in this thread is pointing out the flaw of modern
mathematics of its
inconsistency. It is inconsistent because noone dared to define Finite
with precision.
That is understandable because noone prior, thought that Physics was
more important than
mathematics. That only Physics can well define Finite. And that only
physics can show
mathematicians that the logic of mathematics is both of Aristotliean
for finite and becomes
a nonlinear-duality logic for infinity. In other words, mathematics is
only valid for the realm of
Finite of about 10^500 or less.
Most of modern day mathematics is trash and has become more of
philosophy or religion
and has abandoned maths primary job-- precision.
The Calculus is good, but the endless prittle prattle of infinity and
infinite sets is all miserable
trashable nonsense. Many of the modern day so called proofs or
theorems of mathematics
are trashable, built on quicksand.
Truly amazing that you can have modern day mathematicians entering
something like Fermat's Last Theorem or the Goldbach Conjecture
thinking that the definition of finite is
"ending in zeroes to the leftward string" and thinking that the
Natural Numbers are all
individually finite-numbers and yet the collection of all these finite-
numbers forms an
infinite set. No wonder FLT or Goldbach were never proven, because the
whole enterprize
was chock full of inconsistency and contradiction.
And to this very day, you walk down the halls of a University and ask
a math professor
how come he/she believes the Natural Numbers can all be finite-numbers
and yet simultaneously be an infinite set. And ask that math professor
why he/she could never
give a precision definition of finite.
While physics is in the modern century, the new century of 21st
century, mathematics
seems to be at least a century behind and far into depths of
ignorance.
Robert
wrote:
> Robert wrote:
> > what is 10^500 + 1?
>
> > there be dragons.
>
> Pray tell, is it prime? Is this number
>
> 10000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000001
>
> A prime number?
No idea. Is that the test for whether numbers have meaning? Whether
you can tell if they're prime. No.
> And with a definition, a precise definition of Finite as the Planck
> Unit
> of 10^500
>
> Then 10^500 +1 is no longer a finite number but is in the Incognitum
> where numbers have "no meaning" because there is no longer a reliable
> physics going on to test and experiment.
You can reliably tell that 10^500 + 1 is an odd number. Totally
trustworthy, with total cognition. You can do mathematics with numbers
bigger than 10^500. Not all mathematics it's true, but you shouldn't
have a tantrum about that.
Archimedes Plutonium
Robert wrote:
> On 6 Jan, 19:33, Archimedes Plutonium <plutonium.archime...@gmail.com>
> wrote:
> > Robert wrote:
> > > what is 10^500 + 1?
> >
> > > there be dragons.
> >
> > Pray tell, is it prime? Is this number
> >
> > 10000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000001
> >
> > A prime number?
>
> No idea. Is that the test for whether numbers have meaning? Whether
> you can tell if they're prime. No.
>
I just used the concept of prime, since I started this book with the
correction
of the Infinitude of Primes proof.
Let me give an example of the "test" that uses just simply addition. A
test that
beyond Finite, mathematics breaks down and is undecidable.
We define Finite as to the largest Planck Unit which is about 10^500
and a little
ways beyond we call Incognitum but is meaningless for Physics and thus
meaningless for mathematics. Beyond the Incognitum is Infinity
So we can construct meaningful mathematics all within the boundaries
of 10^500
and we can poke around in the Incognitum and be kind of assured of
some
carryover meaning in that realm and find no "undecidables"
But now look at Infinite-Integers such as
9999....99999 and 9999...99998 and 9999....9997
If I tested those three Infinite numbers for primeness I can be
assured that the first is
divisible by 9 or 3 and the second by 2, but the third number is
forever undecidable
as to being prime or composite.
Now let me do some adding. If I add 1 to 9999....9997 I get
9999...9998 and add 1 to
that gives me 9999...99999. But if I add 1 to 9999....99999 our
Aristotliean Logic
that we use for all of mathematics no longer holds up. We are already
at infinity with
an infinite decimal place value. So adding 1 to 9999....9999 leaves us
with an
undecidable addition.
What I have done to 999...9999 +1 is call it an imaginary number as
the South Pole
number where 9999...9999 is one unit shy of the South Pole.
So just as it is impossible for the mathematics that exists at present
with its
Aristotliean Logic to decide whether 999...9997 is prime or composite
and it is
impossible to decide what 999...999 + 1 is. Physics does have an
answer to these
questions. Physics has an axiom of duality. That Physics needs
infinity, only as
a dual to finite. But whenever Physics has a infinity in its
calculations, it renormalizes
them away. Physics gets rid of the infinities for they are nothing but
a nuisance even
though that exist as "potentials" not as "realities". And since
mathematics is a
subset of physics, that mathematics can only recognize infinity
exists, but leave it
or drop it at that. We know the primes are an infinite set, but it is
pointless to
dive into it any further than 10^500.
> > And with a definition, a precise definition of Finite as the Planck
> > Unit
> > of 10^500
> >
> > Then 10^500 +1 is no longer a finite number but is in the Incognitum
> > where numbers have "no meaning" because there is no longer a reliable
> > physics going on to test and experiment.
>
> You can reliably tell that 10^500 + 1 is an odd number. Totally
> trustworthy, with total cognition. You can do mathematics with numbers
> bigger than 10^500. Not all mathematics it's true, but you shouldn't
> have a tantrum about that.
>
I can reliably tell any number from 1 to 9999....999 as to whether it
is even or odd.
But I cannot tell whether 9999...9999 +1 or 9999...9999+2 is even or
odd since I
have already exhausted the infinite place value.
I have never done a Renormalization in Quantum Electrodynamics. I
wonder if such a
task sheds light on something like 9999...999+1
The Complementarity Principle of Quantum Mechanics tells us that the
knowledge
of a physics experiment is either interpreted solely of its particle
nature with no
reference to its wave nature or vice versa. We can thus transpose that
principle onto
mathematics that a problem in mathematics is all interpreted via
Aristotliean Logic
with no reference to Nonlinear Logic. So the message of this from
Physics tells us
that mathematics at best, can only exist as a science of precision
with a definition
of Finite as 10^500 and where we obtain "precision" only within those
confines
of 10^500. If we try to have "precision" hold from 0,1,2,3, ....
9999...9999 mathematics
breaks apart into contradiction and undecidability.
Archimedes Plutonium
Archimedes Plutonium
to
sci.math in February with a well-defined ellipsis. I am thinking that
I already
gave a well-defined ellipsis by saying that we define Finite as
10^500.
And little beyond 10^500 is Incognitum and further is the Infinity but
where
mathematics is stuck only in the range of Finite. Thus whenever we use
ellipsis as a shorthand with two-dots such as 0.999..-9 the two-dots
indicates that the -9 is in the 10^-500 place value.
For an ellipsis of three-dots such as 299...995 indicates an Infinite-
Integer
with frontview of 2 and backview of 5 and where the three-dots means
an infinitude of 9s in the middle.
And since only "precision mathematics" can exist within the confines
of
10^500 there is no need for further defining of the three-dot
ellipsis.
There is one and only one type of infinity and so the ellipsis has
meaning only
for the two-dots ellipsis.
But I wonder if LWalk can take on this challenge instead:
Here I have two matrices of 99..-9 x 88..-8. And can LWalk
use a computer to write out the precise product?
0.99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
MULTIPLY
0.88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
88888888888888888888888888888888888888888888888888
EQUALS
0.888..-?
In other words, Lwalk fills in what the 10^500 matrix is for the
product above. Plus, tells us how much computer time it took
to fill in that matrix?
Some future questions: how far out on "pi and e" are we in place-
value?
Are we say 10^-500? And if so, what is the 10^-500 matrix of pi x e
and how long of a computer time did it take to deliver the answer?
A
(1 - 10^(-500)) * (8 * \sum_{n=1}^{500} 10^(-n))
= 8 * (\sum_{n=1}^{500} 10^(-n) - 10^(-n-500))
< 8 * (\sum_{n=1}^{500} 10^(-n)) = 0.888...8 (500 digit 8s)
and
8 * (\sum_{n=1}^{500} 10^(-n) - 10^(-n-500))
> (8 * \sum_{n=1}^{500} 10^(-n)) - 10^(-500) = 0.888...87 (499 digit
8s)
Looks like your answer, to 500 decimal places, is 0.888...87 (with 499
digit 8s).
This uses ordinary algebra methods which are quite available to high
school students. A computer is unnecessary. It is interesting that you
ask others to use their computers to solve a mathematical problem for
you which is quite elementary and which every student of college
mathematics would be expected to be able to solve by hand.
Marshall
<plutonium.archime...@gmail.com> wrote:
>
> Now let me do some adding. If I add 1 to 9999....9997 I get
> 9999...9998 and add 1 to that gives me 9999...99999. But if
> I add 1 to 9999....99999 our Aristotliean Logic
> that we use for all of mathematics no longer holds up. We are already
> at infinity with an infinite decimal place value. So adding 1 to
> 9999....9999 leaves us with an undecidable addition.
So there's this number system that you made up that
has this difficulty. Sure. But that doesn't mean that
the natural numbers have this problem. And in fact
they certainly don't have this problem.
There are no natural numbers that you cannot add 1 to.
This shows that the difficulty you are talking about is
specific to your personal system. So it's fine for you
to say "my personal math can't handle numbers above
10^500" but it's not accurate to say that there is any
problem with natural numbers above that range.
Marshall
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
--- quoting http://news.bbc.co.uk/2/hi/technology/8442255.stm ---
Pi calculated to 'record number' of digits
By Jason Palmer
Science and technology reporter, BBC News
Pi appears in a wide range of formulae and natural phenomena
A computer scientist claims to have computed the mathematical constant
pi to nearly 2.7 trillion digits, some 123 billion more than the
previous record.
Fabrice Bellard used a desktop computer to perform the calculation,
taking a total of 131 days to complete and check the result.
This version of pi takes over a terabyte of hard disk space to store.
Previous records were established using supercomputers, but Mr
Bellard claims his method is 20 times more efficient.
The prior record of about 2.6 trillion digits, set in August 2009 by
Daisuke Takahashi at the University of Tsukuba in Japan, took just 29
hours.
--- end quoting http://news.bbc.co.uk/2/hi/technology/8442255.stm ---
Alright, it is looking real good here for that pi and e are easily
known to 500 digits
and that it would be a breeze for any computer to be rigged to reveal
what
pi x e is to 10^-500 place value.
I would guess the above 0.99..-9 x 0.88..-8 would take a few minutes
and that
pi x e would take slightly longer than a few minutes for the computer
to spit out
the product answer in full.
Now that also sheds light onto a bigger and major question. Once we
confine all
of Mathematics to the Planck Unit of 10^500 and its inverse, then the
proof of
conjectures such as Riemann Hypothesis take less than a day for a
supercomputer
to reveal that RH is true for 10^500 and its inverse and thus true for
mathematics.
In other words we forget the folly of statements that such and such is
true to
infinity when mathematics no longer is mathematics beyond the Planck
Units.
So I will have achieved the goal that given a well defined statement
in mathematics,
that a proof or disproof of that statement is to be obtained in a
day's worth of supercomputer
effort. So that the science of precision-- mathematics, as soon as you
deliver a
well-defined statement, it can be proven true or false before the sun
goes down on
that day. This is how it should be. And the only reason it was never
like this in
mathematics before, is because noone ever had a proper historical
perspective of
mathematics as a science and how it relates to physics.
Math has these characteristics:
(1) subset of physics
(2) science of precision
(3) because it is a subset of physics, its domain or range of validity
is only as
far as what numbers are meaningful in physics and since the Planck
Units
end at 10^500 that mathematics ends at that number also.
Now let me put in a word or two about geometry, since mathematics is
borne
of physics since atoms are numerous giving rise to arithmetic and
atoms have
shape and size giving rise to geometry. So that numbers have no
meaning beyond
10^500 and where mathematics stops, so let us apply that to geometry.
Can we picture a 10^500 sided polygon or inscribed inside a circle?
How about a
square whose side is 10^-500 meters? How about a fractal of 10^500
segments?
It is estimated that there are 10^78 atoms in all the Universe and so
what would
happen if we filled the Universe with 10^500 atoms?
Archimedes Plutonium
science posts sent off.
And yesterday was the first day in my life that I experienced
frostbite. It happened real
quickly with a blizzard wind. It happened on the cheeks of my face,
around the nose
and jaw. It feels like a sunburn with reddened skin. Looks like I need
to buy a face-mask
for just those occasions.
But anyway, the definition of ellipsis that I am veering towards is
that of two-dots signifies
a finite string of digits where finite is defined as 10^500 or below.
And for the three-dot
ellipsis signifies infinite string of digits and where infinite is
defined as far beyond 10^500.
Now using the ellipsis let us discuss the definition of Finite that
was used throughout the
history of mathematics. That definition was never given or stated in
Peano axioms and a
pity shame it was not stated because it made mathematics inconsistent.
That assumed definition of finite was that a number was finite if it
ends to infinity in a string of zeroes
leftwards. An example instantly illustrates this definition. The
number 765 is finite because
it is 0000...000765. So here we use the ellipsis to indicate an
infinity of zeroes and all the digits to the left of "7" are zeroes.
And the three-dot ellipsis can be used several times in a number for
example, 12...233...344
indicates an infinity of 2s and an infinity of 3s for that infinite
number.
And the definition of ellipsis includes the idea that we have an
ellipsis for a shorthand. We
obviously cannot write out a infinite number of all its digits so we
use a shorthand of
three-dots. And for finite numbers that are long we also use this
shorthand only we use
two-dots.
Now no mathematician ever gave a well-defined concept of Finite until
Physics defines
it as the largest number useable by Physics-- the Planck Units.
Now let me examine why the Peano Axioms of the Natural Numbers are
flawed and inconsistent and contradictory.
Peano never defined Finite but assumed it meant that the number ended
in a leftward string
of all zeroes, as noted above for 765 example.
Peano and later mathematicians all assumed the Natural Numbers as a
set was an infinite set
and they assumed it was infinite because you can endlessly add 1. So
the endless process planted into the minds of mathematicians that the
Natural Numbers formed an infinite set.
But, also, Peano assumed and all later mathematicians train-locked-
assumed with him that
every Natural Number of the Peano Axioms was a finite number. Even
though Peano never
defined Finite.
So, here is the contradiction, the inconsistency of the Peano Axioms.
If every Natural Number
was a Finite number then the set of all these Finite numbers cannot be
an infinite set. So Peano and all the followers focused only on the
idea that since the adding of 1 is endless that
the total set of all the Natural Numbers would be endless and thus
infinite. However, keep in mind, there was the other assumption that
each and every Natural Number had to be a Finite
Number.
So to Peano and his followers this set {0, 1, 2, 3, 4, 5, ...} was an
infinite set and where every
number in that set had to be a finite number. Can you see the problem,
can you see the contradiction and can you see the inconsistency?
Probably not. The contradiction is simply the fact that if every
element or member of a set is a Finite member, then the overall set
can
never be infinite. In order for a set to be infinite, it must contain
at least one member or element that is itself infinite.
Now most people would still not grasp that idea. So I will explain it
in another way.
We can add piecewise two elements of a set of numbers to form a more
compact set.
Using the Natural Numbers {0, 1, 2, 3, 4, 5, ...} I form the newer
compact set by adding
together adjacent numbers {0+1=1, 2+3=5, 4+5=9, ...} So is this newer
compact set still
infinite? According to Peano and his followers it is still infinite.
So I continue to compactify
the prior set {6, 14, ...} and according to Peano and followers that
set is also infinite. But
I compactify infinitely many times and what do I end up with? Well,
since each and every
Peano Natural Number is a finite number, what I end up with is a set
that has one number
and that number is a finite number.
So Peano Axioms destroys the set theory arithmetic of the union of
elements.
So how can one correct the Peano axioms so they are consistent? It is
easy. It is to
define Finite with a well-defined definition such as saying all
numbers less than 10^500
are finite and beyond is incognitum and far beyond is infinite. There
you have a consistent
Peano axioms. And the 10^500 comes out of Physics of the largest
number in physics
that can still carry on experimentation. And I have always thought of
mathematics as
a paper and pencil experiment. None of us can count to 10^500 and
still be alive.
And the most funny part of this story, is how long it took for anyone
to notice the huge
flaw of the Peano Axioms. Those axioms are about 150 years old, and
one would think
that no inconsistent and contradictory set of axioms could pass
undetected for 150 years.
One would think that as fast as I could get frostbitten in 1/2 hour
time that the Peano
axioms as flawed would have been realized by some astute student
learning them for the
first time. But I guess noone put these two together (i) every Peano
Natural was finite
(ii) finite was not well-defined.
Archimedes Plutonium
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.
Archimedes Plutonium
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?
mike3
<plutonium.archime...@gmail.com> wrote:
<snip>
Two points:
1. Mathematics must not deal with anything abstract. Only non-abstract
things
in physics can be dealt with.
2. Physics -- incl. QM -- is described with math, which is in turn
based on logic.
So much for "destroys the math".
Archimedes Plutonium
as to give
a basis for the discussion. And the Peano Axioms are so riddled with
contradiction
and inconsistency that I need to replace them with a new system.
The major flaws of the Peano Axioms are the assumption of finite and
infinite
with never a well-defined finite, nor infinite. This causes the Peano
Axioms to
be inconsistent. There are minor flaws in the Peano Axioms such as
never a
"metric ruler" established for the creation of the Successor Function
as endless
adding of 1. So we need to create as given axiom, not just the
existence of
0 but the existence of a metric ruler of 0 and 1 so that we use the
metric distance
of "1 unit" for the Successor Function. Otherwise, the Peano Axioms
could just as well
be the set {0, 1/2, 1, 1.5, ...}. This is a minor error. Another error
is the need for inclusion
of a Mathematical Induction. Math Induction comes out of the Successor
Axiom and no
need to have a redundant axiom.
So in the next few days I shall offer the AP-Natural-Numbers-Axioms.
--- quoting Wikipedia on the Peano Axioms ---
The first four axioms describe the equality relation.
1. For every natural number x, x = x. That is, equality is
reflexive.
2. For all natural numbers x and y, if x = y, then y = x. That is,
equality is symmetric.
3. For all natural numbers x, y and z, if x = y and y = z, then x =
z. That is, equality is transitive.
4. For all a and b, if a is a natural number and a = b, then b is
also a natural number. That is, the natural numbers are closed under
equality.
The remaining axioms define the properties of the natural numbers. The
constant 0 is assumed to be a natural number, and the naturals are
assumed to be closed under a "successor" function S.
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
Peano's original formulation of the axioms used 1 instead of 0 as the
"first" natural number. This choice is arbitrary, as axiom 5 does not
endow the constant 0 with any additional properties. However, because
0 is the additive identity in arithmetic, most modern formulations of
the Peano axioms start from 0. Axioms 5 and 6 define a unary
representation of the natural numbers: the number 1 is S(0), 2 is S(S
(0)) (= S(1)), and, in general, any natural number n is Sn(0). The
next two axioms define the properties of this representation.
7. For every natural number n, S(n) ≠ 0. That is, there is no
natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n.
That is, S is an injection.
These two axioms together imply that the set of natural numbers is
infinite, because it contains at least the infinite subset { 0, S(0), S
(S(0)), … }, each element of which differs from the rest. The final
axiom, sometimes called the axiom of induction, is a method of
reasoning about all natural numbers.
9. If K is a set such that:
▪ 0 is in K, and
▪ for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
The induction axiom is sometimes stated in the following form:
If φ is a unary predicate such that:
▪ φ(0) is true, and
▪ for every natural number n, if φ(n) is true, then φ(S(n)) is true,
then φ(n) is true for every natural number n.
--- end quoting Wikipedia on what the Peano Axioms are ---
And funny how the concept of finite and infinite are never mentioned
in
any of the axioms of Peano.
Archimedes Plutonium
>
> 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
Axioms and to give
mathematics, for the first time, a well-defined Finite, Incognitum and
Infinite.
This should be a book in itself, the Correcting the Peano Axioms.
The Peano Axioms yields this set of numbers:
{0, 1, 2, 3, 4, 5, ... }
That set never defined finite nor infinite, yet Peano and followers
still use the ellipsis
of infinity, yet they never sat down to define what finite nor
infinite meant. They left
it to every student to define finite and infinite in their own vagary
ways and means.
You ask any college professor of Cambridge, Harvard, Stanford,
Princeton to define
finite in mathematics and the chances are that they will say " a
finite number is a
number that repeats in zeroes leftward to infinity so that a number
743 is finite because
it is 000...00743
That definition of finite is the best the mathematics community ever
did for defining
Finite versus Infinite.
So anyone can now see quite plainly, quite clearly why the entire
Peano Axioms program
is filled with flaws, holes, is contradictory and inconsistent.
Here is the AP Natural Numbers, and this is the set of Natural Numbers
that replaces
the inconsistent Peano Natural Numbers:
{0, 000...001 = 1 , 000...002 = 2, 000...003 = 3, . . . , 999...998,
999...999 }
Peano Axioms are inconsistent because they fail to define Finite, and
they insist
that all their numbers are "finite numbers" plus, the cream-on-top,
they insist that
their set is an infinite set of finite numbers.
So Peano and followers have several contradictions, namely these two:
First Contradiction: Since Peano and followers used the above assumed
definition of
Finite as a infinite string of zeroes leftward of a number such as 55
is finite because
it is 0000...00055. Then the contradiction or inconsistency arises to
to the fact that
5000...0000 is not finite, nor is 005000...000 finite, nor is
0000...5...0000 finite. So
we see that Peano and Followers definition of Finite that they used to
construct their
axioms and hence their definition of infinite was stuck in the mud
from the very beginning.
For their definition of Finite and Infinite are inconsistent and
contradictory.
Second Contradiction: Peano Natural Numbers are all deemed as finite-
numbers,
so that being all finite numbers, the set itself cannot go to infinity
because the
addition of nothing but finite numbers yields a finite number result.
To be an
infinite set, when we add together all the numbers of such a set, the
sum is an
infinite number, not a finite number.
The upshot is that only when we **select** a large number and call it
Finite, can we
thus well-define Finite and hence well-define Infinite.
Now I need some help from Lwalk as to defining the Incognitum. This is
a new term
for mathematics in that I need a bridge between Infinite and Finite.
Because, clearly
we can sense that although I defined Finite as 10^500 because there is
no more physics
Experiments beyond the Planck Unit of 10^500, just as we are never
going to get to
10^-500 Kelvin temperature. Although we sense that the number 10^500 +
1 is still a
Finite number and is not an Infinite number. So I have devised the
Incognitum to bridge
ourselves a little past the Finite but obviously recognizable as not
an infinite-number.
For the number 10^500 +2 is different from the number 5555...5555
which is unfailingly
an infinite number.
So I want to Well-Define the Incognitum. I am soliciting the help of
Lwalk since he is very
good at this chore. I have a gameplan, Lwalk. I think I can well-
define the Incognitum
by the idea of a theorem: when adding any two finite numbers the
answer is another finite
number. So, Lwalk, what I am thinking is that since I well-defined
Finite as 10^500 and below,
can I use the idea of that theorem to well-define the Incognitum as
steps of 10^500 beyond
10^500 itself. So that the number 10^500 + 10^100 is a number that is
not an infinite-number
but an Incognitum number.
Lwalk, I am trying to well-define Incognitum, so that I can well-
define Infinity?
Any help?
I feel that although I could go the easy route-- dismiss the
Incognitum altogether and simply say that anything beyond 10^500 is an
infinite number. That is the easy route. But it does
not assuage my feelings or notions since 10^500 +1 still looks to be
finite.
Archimedes Plutonium
numbers where the
definition of Finite is 10^500 and below (ditto for inverse).
Now can I well-define the Incognitum? This is a little ways beyond the
Finite realm
and although we would never see this in Physics because a number such
as 10^-500
Kelvin is never obtainable, nor is 10^-501 Kelvin. So in physics the
Incognitum would
be unneccessary. But in mathematics we trespass beyond the number
10^500 frequently.
We have prime numbers, alleged prime numbers that are far beyond
10^500. So in
mathematics, a concept of Incognitum-numbers maybe highly suitable. So
can we
well-define the Incognitum?
If so, then we need the three-dot ellipsis to indicate a number in the
Incognitum.
Finally, then we reserve the four-dot ellipsis for Infinite-numbers.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
> Peano Axioms are inconsistent because they fail to define Finite, and
> they insist
> that all their numbers are "finite numbers" plus, the cream-on-top,
> they insist that
> their set is an infinite set of finite numbers.
I keep striving for clarity. The above paragraph offers that clarity,
especially the
last sentence. I want it so clear that a High School student can
understand the problem
and how to fix it.
Here are the Peano & followers numbers:
{0, 1, 2, 3, 4, 5, . . . }
Here are the AP numbers where 10^500 is the last Finite number.
{0, 1, 2, 3, . . 10^500, 10^500 + 1, . . . , 999...999 }
Now think of each number of Peano as a set itself of a Mounds candy
bar.
So that 1 is 1 Mounds, and 2 is two Mounds and 3 is three Mounds.
So that Peano and followers insist that every one of their numbers is
a finite number
and thus a finite number of candy bars of Mounds. But they also insist
that their
set of all Peano Numbers is an infinite set. So, do you see the
Contradiction? Do
you see the Inconsistency? That each number is a finite number of
candy bars
yet the sum total set is an infinite number of Mounds candy bars.
The Peano axioms and Peano numbers cannot deliver an infinite set when
all of its
members are finite-numbers.
Now look at the AP numbers. That set does have infinite-numbers. That
set was
generated by the very same Successor Axiom that Peano used. But the AP
set
has no inhibitions about infinite numbers being manufactured from the
Successor
Axiom. The AP numbers recognizes that the Successor Axiom will zoom
past any
conceit that the numbers have to be all finite. This is a conceit and
deception of the
Peano followers.
So in the AP set of numbers we quickly get to a point where the
numbers are infinite
and thus the set is an infinite set of Mounds candy bars.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
> Now that also sheds light onto a bigger and major question. Once we
> confine all
> of Mathematics to the Planck Unit of 10^500 and its inverse, then the
> proof of
> conjectures such as Riemann Hypothesis take less than a day for a
> supercomputer
> to reveal that RH is true for 10^500 and its inverse and thus true for
> mathematics.
Well RH is far more complex than simple multiplication of pi x e to
the 10^-500 decimal
place value. RH, as it turns out has been verified by computer to only
about 10^13 or
thereabouts. But I am not sure of what is to be deemed as 10^500 in
the Riemann Hypothesis? Do we verify to the number 10^500 or to the
first 10^500 nontrivial zeroes?
So I am a bit cloudy on what is the 10^500 in the Riemann Hypothesis.
--- quoting from: ---
http://www.zetagrid.net/zeta/rh.html
What have we achieved so far?
The result of the computation which verified the first 100 billion
zeros of the Riemann zeta function confirms the calculations made
previously by "other scientists" and extends these to the first 100
billion nontrivial zeros. All these zeros of the form + it have real
part = 1/2 and are simple. Thus, the Riemann Hypothesis is true at
least for all |t| < 29,538,618,432.236.
More details about this computation will soon appear in a paper.
--- quoting the reference "other scientists" in the above quote ---
RESULTS CONNECTED WITH THE FIRST 100 BILLION ZEROS OF THE RIEMANN ZETA
FUNCTION (DRAFT)
SEBASTIAN WEDENIWSKI
ABSTRACT. This report presents the results of recent and ongoing
internet computing inside the intranet of IBM Corporation which
required 1.3 . 1018 floating-point operations: the first 100 billion
nontrivial zeros of the Riemann zeta function (s) are simple and lie
on the line Re(s) = . Thus, the Riemann Hypothesis is true at least
for all |Im(s)| < 29, 538, 618, 432.236.
Date: August 1, 2002
--- end quoting from "other scientists" ---
So maybe we have many years or centuries to wait before we can verify
the truth of RH to
that of 10^500.