fantastic irony of mathematics-- well-define Infinity but lose out on multiplication and powerset and addition #296; Correcting Math
Archimedes Plutonium
mathematics. The human civilization lived through many ironies of
physics such as the idea that Earth is
a roundish ball that travels in space yet the people
on the other side do not "fall off".
But mathematics never faced up to any ironies, until now.
In order to well-define Finite versus Infinite we have to
get help from Physics to tell us what number is large enough to
include all of Physics experimentation and that number is the largest
Planck Unit of 10^500. So beyond that number can be no physics since
we cannot measure beyond it. So we define in a well-defined definition
that Finite is 10^500 and below
(inverse for the microworld).
Thus, since we well-defined Finite it is easy to define
Infinite as beyond Finite with an Incognitum territory between Finite
and Infinite. This Incognitum is a sort of
picking grounds since we all know that 10^500 +1 is
still "feeling like Finite". So we have a territory that
blends its way from Finite into Infinite and the Incognitum is from
10^500 to that of 1000....0000 where this number is 10% of
9999....9999+1.
Now the number 9999....9999 which corresponds to
1111....1111 in binary is the Infinity-Number. There is no
number larger than is number and whenever we say Infinity we can
interchange it with 9999....99999.
This Infinity-Number is also what happens when you
write the Series 1 + 1 + 1 + . . . .+ 1. That Series is
equal to 9999....9999. And the Infinity Number is the
number that Peano Axioms reaches through its Successor Axiom. Only,
Peano and his followers never
realized that such happened.
But what I want to discuss is the fact that once you well-define
Finite, then you well-define Infinity and by doing so, you are forced
to have to admit that multiplication and addition are restricted. That
you cannot, or that it is impossible to have a multiplication
and addition for things like this:
9999....9999 x 7777....77777 or this 9999....999 + 88
In other words, math never faced such a predicament.
Either math well defines Finite versus Infinite and thus
has to restrict the definition of addition and multiplication. Or that
mathematics never well-defines
Finite versus Infinite and thus is left to having no
restrictions on multiplication and addition.
Now most people with commonsense would agree with me, that how can you
have a full licence of multiplication and addition on infinity itself.
The thread that I barged in on was titled Re: Powerset
and it was powerset on Infinite sets. And that is how
Cantor alleges there to be transfinite numbers or all types of layers
of infinity, or hierarchies of infinity.
But if a commonsense person thinks about it, how can one be
"multiplying the endlessness?" Or how can one
be adding to something considered "endless".
So it makes sense that we are forced to well-define Finite. We cannot
escape that chore or job. And the moment we define Finite, is the
moment we pick a
large number, for there is no escaping that picking or
selection of a large number. And thus, once we well define Finite we
instantly have Infinity well-defined as
beyond the Finite.
And here is the irony. Since we have a well-defined
Finite versus Infinite, our definition of the operations of
addition and multiplication cannot extend into the Infinite realm.
Addition and Multiplication are only well-defined and feasible in the
Finite and Incognitum realm,
but addition and multiplication melt away like snow
in the Infinite realm.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Nam Nguyen
>
> But mathematics never faced up to any ironies, until now.
>
> In order to well-define Finite versus Infinite we have to
> get help from Physics to tell us what number is large enough to
> include all of Physics experimentation and that number is the largest
> Planck Unit of 10^500. So beyond that number can be no physics since
> we cannot measure beyond it. So we define in a well-defined definition
> that Finite is 10^500 and below
> (inverse for the microworld).
Heck, 10^500 is an overblown value: you just need 1 - for _one_ universe.
There "can be no physics since we cannot measure beyond it", i.e. beyond
1 universe!
The irony is, AP, you forgot you've always stated that in your signature:
the "whole entire Universe is just _one_ big atom."!
bert
>
> - Show quoted text -
Nam When motion and gravity being the same in our every day thinking
it will be a big step in understanding how the universe was
created.How it evolved. Bert
Jens Stuckelberger
> [The usual ramblings.]
For how many years have you been posting your junk now? 15? 20?
Are you going to get a life any time soon?
David R Tribble
> Now the number 9999....9999 which corresponds to
> 1111....1111 in binary is the Infinity-Number. There is no
> number larger than is number and whenever we say Infinity we can
> interchange it with 9999....99999.
>
> This Infinity-Number is also what happens when you
> write the Series 1 + 1 + 1 + . . . .+ 1. That Series is
> equal to 9999....9999. And the Infinity Number is the
> number that Peano Axioms reaches through its Successor Axiom.
So is the Infinity-Number 999...999 evenly divisible by 9?
(It certainly looks like it is.)
If so, then can we assume that the binary Infinity-Number
111...111 is also evenly divisible by 9 (binary 1001)?
And does that mean that the number of terms summed in
1+1+1+...+1 is also a multiple of 9?
Ostap S. B. M. Bender Jr.
What is '9999....9999'? What is '7777....77777'?
Archimedes Plutonium
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > Now the number 9999....9999 which corresponds to
> > 1111....1111 in binary is the Infinity-Number. There is no
> > number larger than is number and whenever we say Infinity we can
> > interchange it with 9999....99999.
> >
> > This Infinity-Number is also what happens when you
> > write the Series 1 + 1 + 1 + . . . .+ 1. That Series is
> > equal to 9999....9999. And the Infinity Number is the
> > number that Peano Axioms reaches through its Successor Axiom.
>
> So is the Infinity-Number 999...999 evenly divisible by 9?
> (It certainly looks like it is.)
Why would you not think it was?
>
> If so, then can we assume that the binary Infinity-Number
> 111...111 is also evenly divisible by 9 (binary 1001)?
>
In three divisions the cycle starts all over, so yes.
What is with these oversimplistic questions that are going nowhere.
> And does that mean that the number of terms summed in
> 1+1+1+...+1 is also a multiple of 9?
Type of question a robot would ask
Archimedes Plutonium
Archimedes Plutonium wrote:
> David R Tribble wrote:
> > Archimedes Plutonium wrote:
> > > Now the number 9999....9999 which corresponds to
> > > 1111....1111 in binary is the Infinity-Number. There is no
> > > number larger than is number and whenever we say Infinity we can
> > > interchange it with 9999....99999.
> > >
> > > This Infinity-Number is also what happens when you
> > > write the Series 1 + 1 + 1 + . . . .+ 1. That Series is
> > > equal to 9999....9999. And the Infinity Number is the
> > > number that Peano Axioms reaches through its Successor Axiom.
> >
> > So is the Infinity-Number 999...999 evenly divisible by 9?
> > (It certainly looks like it is.)
>
> Why would you not think it was?
>
Okay, it dawned on my whilst doing another post as to why Tribble was
asking these simplistic questions. At least this is my guess as to the
simplistic questions.
In an earlier post I said that the multiplication of 9999...9999 x 3
was
not a larger number than 9999....99999 because the number 3 was
really 0000....00003 and that number is close to zero, so that
the multiplication of 9999....99999 x 3 was going to be
0000....99997 or about zero percent of 9999....9999
So probably, Tribble is thinking that division is all the same as in
the
old math and so how can we have 9999....9999 x 3 = almost zero
whilst 9999....99999 / 3 is 3333....33333 ?
And my answer is this; as in previous post. That mathematics is
confined
to the region of Finite and once it goes beyond the finite territory
it is on
shaky grounds. That multiplication, division, add subtract no longer
are
upheld as a Algebra but is rather lost.
So that mathematics is valid only in the region of Finite for an
Algebra
of its operations.
Just as we can never tell from any mathematical system whether the
number
9999....99997 is definitely a prime or composite. We also lose out on
Algebra
in the Infinite territory.
We can define multiplication on Infinite Integers by saying that
multiplication
is restricted as in the 100 Model to multiplication that does not
exceed 9999...999
The restriction applies to add, subtract and divide also.
And we define the decimal place value of 9999....99999 by the 100
model
so that 9999....9999 x 0000....0003 = 00000....9997 and that
33333....33333 x 30000....0000 = 99999....99999
> >
> > If so, then can we assume that the binary Infinity-Number
> > 111...111 is also evenly divisible by 9 (binary 1001)?
> >
>
> In three divisions the cycle starts all over, so yes.
>
> What is with these oversimplistic questions that are going nowhere.
>
> > And does that mean that the number of terms summed in
> > 1+1+1+...+1 is also a multiple of 9?
>
> Type of question a robot would ask
Okay the answer to the 1+1+1+....+1 is that it is equal to
9999....9999 because it
reaches that number. The number 2 + 2 + 2+ ....+2 is also equal to
9999....9999 because of
the definition of restrictive addition in that no number can exceed
the infinity number.
The trouble that Tribble is having, if my anticipation is correct, is
that Algebra of add,
subtract multiply and divide give out or are crushed once Finite
versus Infinite is well-defined.
So you cannot apply your old musty worn out ideas of add, subtract
multiply divide on Infinite
entities.
Algebra is confined to only the Finite realm of mathematics and
becomes nonsense when
applied to the infinite-realm.
This thread that busted in on was titled Powerset and was about the
powerset on infinity. Well,
if a commonsense person ever thought about it well, would consider
that how can one multiply
or increase that which is already infinity? This is akin to the
ancients trying to add angels on
the head of a needle. A game in pretentious preposterousness
prehension.
P.S. The reason I am adamant in defining a Incognitum between the
Finite and Infinite is that
this zone (about 10% of all the numbers from 0 to 999....9999) allows
us to use Algebra without the restrictions. Just as in the 100 Model
we can call 99 the largest number and so
10 and 9 and 11 and 9 are the extent of where multiplication is still
feasible.
So in this new world of mathematics where Finite versus Infinite are
well-defined means we
have to drop alot of Algebra for it simply is inconsistent with the
rest of mathematics.
pnyikos
On Jan 13, 11:28 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> Archimedes Plutonium wrote:
>
> > But mathematics never faced up to any ironies, until now.
It faced up to Zeno's Paradoxes, much more deserving of the term
"ironies", centuries ago.
> > In order to well-define Finite versus Infinite we have to
> > get help from Physics
Or from our own intuition, which tells us "I am, I exist" [a la
Descartes] and "Time is".
> > to tell us what number is large enough to
> > include all of Physics experimentation and that number is the largest
> > Planck Unit of 10^500. So beyond that number can be no physics since
> > we cannot measure beyond it. So we define in a well-defined definition
> > that Finite is 10^500 and below
> > (inverse for the microworld).
I suppose one could make a case for calling this "physics infinity"
but it just defines 10^500 (in the von Neumann convention, which
Archimedes apparently uses) as far as we mathematicians are concerned.
Even that case is weak: if the "many worlds of quantum physics"
philosophical speculation becomes physically plausible, 10^500 is
dwarfed by the number of parallel universes that have branched off
from ours since ours began. I'm referring to the speculation that
says that every time an undetermined event takes place, universes fork
off from each other according to all possible outcomes of the event.
> Heck, 10^500 is an overblown value: you just need 1 - for _one_ universe.
> There "can be no physics since we cannot measure beyond it", i.e. beyond
> 1 universe!
>
> The irony is, AP, you forgot you've always stated that in your signature:
> the "whole entire Universe is just _one_ big atom."!
I think your reply hit just the right note where "AP" is concerned,
Nam. I decided to complement it with something that gets a little
philosophical traction.
> > Archimedes Plutonium
> >www.iw.net/~a_plutonium
> > whole entire Universe is just one big atom
> > where dots of the electron-dot-cloud are galaxies
Peter Nyikos
Professor, Dept. of Mathematics
University of South Carolina
http://www.math.sc.edu/~nyikos/
Archimedes Plutonium
pnyikos wrote:
> This thread originated on sci.math
>
> On Jan 13, 11:28 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > Archimedes Plutonium wrote:
> >
> > > But mathematics never faced up to any ironies, until now.
>
> It faced up to Zeno's Paradoxes, much more deserving of the term
> "ironies", centuries ago.
>
Not really. They should have wrestled with the definition of a well-
defined
Finite versus Infinite starting with the Calculus. The problem with a
"precision Finite"
any earlier than the Calculus would have been far too out of place in
terms of
the advancement or technology of the times. We really could not do
disease well
enough until after the microscope was well in place for bacteria. And
likewise for
defining Finite with precision could only start with the Calculus and
its microworld of
numbers such as infinitesimals. And unfortunately the history of math
ignored defining
Finite and zoomed right into defining Infinity as if Finite was not
needed to have a
precision definition.
In order to define Finite precisely we really had to wait for the
technology to come
of sophistication in physics of Quantum theory.
But I think the concept of infinity was never really tackled until the
advent
of the Calculus, and there it was of a micro-infinity. So Zeno's
tackling was
just a sign of things to come.
Honestly, I think that the precision definition of Finite had to wait
until someone
recognized that Physics is far above mathematics and guides
mathematics.
So all of the Cantor stuff and the Peano axioms were way to early to
even
start on a precision definition of Finite.
> > > In order to well-define Finite versus Infinite we have to
> > > get help from Physics
>
> Or from our own intuition, which tells us "I am, I exist" [a la
> Descartes] and "Time is".
>
Not so. For apparently your well-defining of Finite is the same old as
the current establishment:
Finite : a number is finite if it repeats endlessly to the leftward
string in
zeroes such that 55 is finite because it is 0000....000055.
If that is not your definition of Finite number, feel free to state
otherwise.
But as far as I know, all modern day mathematicians assume that Finite
is what I just listed. And they are all wrong.
So I am asking you Peter to give math a well-defined definition of
Finite, a precision definition. Don't go waltzing around with
philosophy.
> > > to tell us what number is large enough to
> > > include all of Physics experimentation and that number is the largest
> > > Planck Unit of 10^500. So beyond that number can be no physics since
> > > we cannot measure beyond it. So we define in a well-defined definition
> > > that Finite is 10^500 and below
> > > (inverse for the microworld).
>
> I suppose one could make a case for calling this "physics infinity"
> but it just defines 10^500 (in the von Neumann convention, which
> Archimedes apparently uses) as far as we mathematicians are concerned.
>
You say "we mathematicians", but that must be a pitifully poor lot
since you
are not able to recognize when you have a "ill-definition of Finite"
and when
you have a "precision definition of Finite". So why the braggadocio
of
"we mathematicians."
Peter, I want you to well-define Finite first, only then, can you
embark on
defining infinity, for that stands to reason. Since what you use as
"finite" is
poorly-defined, then your infinity can only be another garbled mess.
Stands to reason, Peter, that we cannot define Infinite in a well-
defined definition
until we have a well-defined Finite. Would you not agree?
> Even that case is weak: if the "many worlds of quantum physics"
> philosophical speculation becomes physically plausible, 10^500 is
> dwarfed by the number of parallel universes that have branched off
> from ours since ours began. I'm referring to the speculation that
> says that every time an undetermined event takes place, universes fork
> off from each other according to all possible outcomes of the event.
>
Now you zoomed off into philosophy.
Ask yourself this question since you like philosophy more than
anything.
Since all the sciences such as chemistry, geology, sociology
ultimately reduce
to physics, then mathematics is also a subset of physics. Because math
cannot
exist without life and since Biology reduces to physics, means that
math is a
tiny subset of physics. In other words, math is a byproduct of living
activity, for
without living organisms, the world would have no mathematics. Buy
that ?
I simply want you to well-define Finite. You have been in mathematics
all these
years yet you never bothered to think about how poorly and ill defined
was your
concept of Finite. You, as well as millions of others have simply
assumed that
Finite meant repeating zeroes leftwards such as 2645 is Finite because
it is
0000....0002645. But what is 002645....55 Is that also finite for you?
Or how
about 00000....2645....2645 is that Finite for you also since it
conforms to your
assumed definition yet it looks awfully much like a infinite-integer
not a finite integer?
You see Peter, you have never well-defined Finite, and you assumed it
was
something akin to zeroes leftwards of a number.
And when you talk about the Peano Natural Numbers of the set
{0,1,2,3,4, ....}
you assumed all those numbers were Finite, yet you never precisely
defined
finite. You left it to each reader to assume whatever "finite" meant.
And if any
of your students differed from your "assumption of finite" they would
see it
on their test scores.
> > Heck, 10^500 is an overblown value: you just need 1 - for _one_ universe.
> > There "can be no physics since we cannot measure beyond it", i.e. beyond
> > 1 universe!
> >
> > The irony is, AP, you forgot you've always stated that in your signature:
> > the "whole entire Universe is just _one_ big atom."!
>
> I think your reply hit just the right note where "AP" is concerned,
> Nam. I decided to complement it with something that gets a little
> philosophical traction.
>
> Peter Nyikos
> Professor, Dept. of Mathematics
> University of South Carolina
> http://www.math.sc.edu/~nyikos/
You contributed nothing to this thread other than display your
ignorance that
you never precisely defined what Finite means. And in fact, you
probably never
realized that you cannot define infinity until you have given a
precision definition
of Finite. Your problem is that you are too much of a philosopher and
not enough
of a scientist. But I do appreciate you posting with a full name, and
thank you
for doing so, since so much of the newsgroups are a wasteland of phony
names and posts that are , thus, not worth reading. I strive to reply
to every
post that is from a true name and address.
Archimedes Plutonium
Usenet does not like to have more than 3 newsgroups per post.
And I do not want to post in any philosophy newsgroup because
philosophy
is not science but that of prescience or when we have not enough
science
we rely on just notions or past beliefs to guide us through. So
philosophy is
more like religion than it is like physics or math.
AP
Archimedes Plutonium
Now I am hoping that when Lwalk comes back, that he can lend some
insights into this
Incognitum concept. And thinking of a better name for it as that of
Algebra's-Upper-Bound.
So let me briefly walk through the reasoning of this.
(1) Want to well-define or precisely define Finite versus Infinite.
(2) That implies we have to select a large number by using Physics
and thus define Finite as 10^500. No other way around this.
(3) Now that Finite is precisely defined at 10^500 means that
the Infinite is beyond 10^500 but the number 10^500 + 1 does
not feel infinite, so can we do something about this "feeling"?
(4) Yes, we define a region between Finite and Infinity as the
Incognitum,
since we already know that some large prime numbers exceed 10^500.
(5) Now is this Incognitum a "ad hoc" structure?
(6) No, the Incognitum is demanded for by mathematics itself, because
infinite-numbers cannot be algebraically multiplied, divided, added,
or
subtracted as can the Finite numbers. So Algebra itself breaks down
as it goes past Finite. Where does Algebra break down exactly when
10^500 is Finite?
(7) Algebra breaks down using the decimal numbers and using the 100-
Model,
that Algebra breaks down at 10% of the numbers from 0, 1,2, to
999....999+1.
Which in the 100-Model means that we stop multiplying when the
multipliers
exceed 10. In other words, with the counting-numbers from 0 to 100, we
can no longer multiply when the product is going to exceed 99 where we
establish that 99 is the last and largest number, ditto for
9999....99999.
(8) In the 100-Model we can multiply 11 x 9 =99 and 10x9=90 but we are
forbidden to multiply 12x9 for that exceeds 99. Likewise for Infinite
Integers
we can multiply 1000....00000 x 099999....9999 which is equal to
999....999
but we cannot multiply 9999....99999 x 8888....88888. We can multiply
09999....99999 x 08888....8888 which is like 9 x 8 in the 100-Model.
(9) So we see that Algebra falls apart at about 10% of the full range
of
decimal representation of numbers. What it is in binary, I have no
full
knowledge as yet-- perhaps it is 1000....0000 versus 1111....1111 as
9999...9999.
(10) So this Incognitum is not "ad hoc" but is descriptive of a region
of
numbers wherein Algebra breaks apart. And from the looks of it, it is
the number
10000....0000 where the endpoint of Algebra ceases to have the
operations. That
number is 10% of 99999....9999 +1.
(11) However, I must admit that if we were to consider the Powerset as
an Algebraic
significant concept that the point in which Algebra breaks down is far
sooner than the
number 1000....0000 = 10% of 9999.....9999 +1. For consider the 100-
Model, in that
Algebra breaks down beyond 11 since 12x9 exceeds 99. But if the
Powerset were
a vital concept of Algebra along with multiplication, then the
Powerset would cause a
breakdown of Algebra far earlier than the number 11.
So I think the name Incognitum can be improved to mean the region or
territory of mathematics wherein Algebra no longer exists in any
meaningful capacity. That Algebra
is a smaller department of mathematics than was previously thought.
And this makes
alot of commonsense because we all know that Infinity has a meaning of
"never ending"
and so how can anyone dare to suggest in a multiplication of one
endless string by another endless string where that multiplication is
going to behave as if two finite strings were involved.
Archimedes Plutonium
finite steps wherein
the pattern is discerned. Take for example the multiplication of two
infinite-integers such:
2222....2222 x 4444....444 = ?
22 x 44 = 968
222 x 444 = 98568
2222 x 4444 = 9874568
So that enough of the pattern emerges that we can say the
product is 98.....68. If we want to refine the answer further
we carry out the steps further.
Now in this manner of defining multiplication on infinite-integers we
have to ask
how we define multiplication itself on infinite-integers in light of
the idea that
multiplication is not well-defined on infinite sets in that once
Finite is precisely defined
that multiplication and the other operations are lost in the infinite
region.
Now there should be an easy proof of this in mathematics, of a
statement that says
something like this: Theorem: once Finite is given a precise
definition, then the Algebraic
operations of add, multiply, divide, subtract have a limited range. So
that whenever we
have a precision definition of Finite, we lose out on the Algebraic
operators.
This very much reminds me of Quantum Physics of the Uncertainty
Principle in that
when you precisely have the momentum you no longer have a position,
and vice versa.
In other words, you can only have one precisely and never the both
simultaneously.
So now I defined multiplication on Infinite Integers as above a
stepwise unfolding of the
pattern. But I still have problems overall.
Such as the problem of what is the multiplicative identity? Is it
0000....0001 or is it
10000....0000?
So does 9999....99999 x 0000....00001 equal 99999....99999 or does it
equal
000000....99999. Does 0999....9999 x 1000....0000 = 9999....9999 or
does it
equal 9999....00000?
I keep using the 100-Model where I analogize 99 as 9999....9999 and
where 10 becomes
the analog of 1000....0000. So in the 100-Model multiplication fizzles
out at 11x9 =99
and we are forbidden to multiply 12x9 for no product can surpass 99.
So does Multiplication become impossible to precisely define on an
infinite set with
infinite-numbers? Probably so, if the Uncertainty Principle of Physics
is present in
mathematics. And it probably is.
Now that is an odd twist to those in mathematics who devoted most of
their time on
Algebra. Algebra as the heavyweight of mathematics, but here it looks
as though
Algebra is confined to only Finite sets with finite numbers. And it
used to be where
algebraists frowned on Finite Algebra, scoffed at Finite Group theory
or Finite Field
theory. Who looked at Finite Algebras as abnormal.
And here, where I uncover the idea that once you define Finite with
precision that
you no longer can have a Infinite Algebra on an infinite set or with
infinite-numbers.
Now let me touch base again with binary representation, because
humanity is too
much biased by its 10 fingers and decimal system. The binary is
closest to theoretical
math and not a biased outlook. If we were to make an infinite chart of
all the binary integers
we would have a huge square array and in the lower left hand corner we
would have
00000......000000
And in the upper right hand corner we would have the last and largest
binary infinite
integer as 11111.....11111 which corresponds to the decimal of
9999....99999.
Now the question is in this huge binary array does the number
100000.....00000 rest
at? I consider it the analog of 50000.....0000 which is halfway in
between 0
and 9999....9999 +1
So somewhere in this huge square array of binary infinite integers
rests the number
10000....00000.
On a globe the longitude lines and the equator would meet at
5000....0000 and that number
would designate the equator line and I expect 1000....000 in binary is
the equator-line.
Now what I want to obtain from this analysis is the idea that half of
all the numbers in binary
are from 0 to 10000.....0000 and the other half are from 10000....0000
to 1111....11111.
Just as in the 100-Model that 50 is halfway in between, and leaving
out 100 itself that all
the numbers in the Model 00, 01, 02, .., 50, 51, ..98, 99, and notice
that as we spin them
180 degrees we get back all the numbers where 00 is spun into 00 and
01 is spun into
10 and where 50 is spun into 05 and 51 spun into 15 and 98 spun into
89. Not sure, but
I think in this spinning we get back every number.
What I am doing is searching for a proof that because you precisely
define Finite, that
multiplication can never be precisely defined over an entire infinite
set having infinite-numbers.
It maybe a totally simple and easy proof-- since you have a last and
largest infinite integer--
999....9999 where infinity is "in the middle region of the number"
means that you cannot
ever have a multiplication of two of these large numbers for it
exceeds the 9999....999. It
maybe as simple as that.
When we never defined Finite, we could get away with defining
multiplication as if 9999....9999
were 0.9999....9999 stuck between 0 and 1 and where multiplication of
any two numbers
between 0 and 1 was confined to a number between 0 and 1. Call this
the (0,1)-Model
in contrast to the 100-Model of multiplication.
So where I defined multiplication as a stepwise multiplying of finite
numbers to discern a
pattern. Here I am taking the operation of Multiplication itself to
discern a pattern
in the 100-Model or the (0,1)-Model. So is Multiplication as an
operation prone to
stepwise Models?
The only thing I am reasonably sure of, is that Physics no longer
exists beyond
the Planck Unit of 10^500 because no experiments can reach 10^500,
just as no
experiments of cold temperature can reach 10^-500 Kelvin. So if there
is no physics
beyond 10^500, that there is no meaningful mathematics beyond those
numbers either.
And we begin to see how mathematics has its own Uncertainty
Principles, that you
precisely define Finite and yet you lose all of Algebra on infinity.
Which would be
sensible to any commonsense person, when asked what is infinity
multiplied by
89? Does it even make sense to multiply an infinite set containing
infinite numbers
or does it even make sense to have a powerset on an infinite set?
Seems like none
of these questions have ever been asked before, or at least, never
taken seriously.
So much of modern mathematics was just taken as assumption. Rarely was
there
ever "precision" demanded on mathematics. And, after all, that is the
primary duty,
the job, the work of mathematics-- precision, precision, precision.
Archimedes Plutonium
I want to prove that the only way to well-define or precisely define
Finite is to
pick a large number and say that is the end of Finite. But once that
is accomplished
the Infinity no longer has a mathematical Algebra because
multiplication over the
infinite set containing infinite-numbers is impossible to have a
precise multiplication
or addition operators.
Proof: Infinite set with infinite-numbers always has a largest
infinite number, since
every number and every set has both a FrontView with a BackView.
Infinity in
mathematics is always able to be formed into the middle of the set or
the middle
of the number. Just as a Finite set or finite-number has a FrontView,
a BackView
and a Middle region. So in infinity, we just conveniently tuck the
infinity into the middle
and in this manner is able to perform alot more work on sets and
numbers and ideas.
Since an infinite set or infinite number has both a FrontView and
BackView with infinity
in the middle ground, depending on which end one takes we can call one
end the start
and the other end the finish. Now the proof is that with Finite
numbers or Finite Sets,
we can always add or multiply and achieve a new and larger number. But
with an Infinite
Set or infinite-number we come to the impossible where we have the
largest infinite number
and unable to multiply two such large numbers or the largest by itself
multiplied cannot achieve a newer larger infinite number. With an
infinite-set, we cannot do a powerset
because the cardinality of the powerset is the same as the orginal
infinite set.
A bit uneasy about the above, but it makes sense.
Archimedes Plutonium
pnyikos wrote:
> This thread originated on sci.math
>
> On Jan 13, 11:28 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > Archimedes Plutonium wrote:
> >
> > > But mathematics never faced up to any ironies, until now.
>
> It faced up to Zeno's Paradoxes, much more deserving of the term
> "ironies", centuries ago.
>
> > > In order to well-define Finite versus Infinite we have to
> > > get help from Physics
>
> Or from our own intuition, which tells us "I am, I exist" [a la
> Descartes] and "Time is".
>
So, Peter and Nam, instead of mocking me. Why not examine your own
definition of Finite versus Infinite. Just as Socrates said a long
time
ago: "The unexamined life is not worth living."
Here I am in a gigantic thread of over 303 posts, examining Finite
versus
Infinite, and then there is Nam and Peter, mocking the examination.
So state your definition of Finite, Peter Nyikos and state your
definition of
Finite Nam Nguyen. Is it any different from what most mathematician
use--
a number is Finite if it ends in a endless string of zeroes leftward
such as
89 is finite because it is 0000....00089.
So do we have Nam Nguyen and Peter Nyikos, both of which spend much of
their
life in mathematics, and having studied mathematics. And when it comes
time
for both to examine their understanding of the differences between
finite and infinite,
to give a precision definition of "finite" that they seem to only
counter with mockery
of one who actually does examine "finite versus infinite."
> > > to tell us what number is large enough to
> > > include all of Physics experimentation and that number is the largest
> > > Planck Unit of 10^500. So beyond that number can be no physics since
> > > we cannot measure beyond it. So we define in a well-defined definition
> > > that Finite is 10^500 and below
> > > (inverse for the microworld).
>
> I suppose one could make a case for calling this "physics infinity"
> but it just defines 10^500 (in the von Neumann convention, which
> Archimedes apparently uses) as far as we mathematicians are concerned.
>
Peter says: "as far as we mathematicians are concerned"
I do not think other mathematicians like it when Peter makes such
comments.
Again, Peter, you entered a very long thread that dives into the
precision definition
of Finite versus Infinite. And for anyone with a milligram of logic
would first off,
offer their definition of what they thought and used as "finite".
You see, Peter and Nam, I have given a precision definition of finite
as 10^500.
But the both of you are still hiding your definition. The both of you
have only
mocked my thoughts.
Peter and Nam, define what you mean by this ellipsis of the Peano
Axiom set
of Natural Numbers:
{0, 1, 2, 3, 4, 5, 6, . . . . }
Peter and Nam, define why 6 is a member of that set but what about
6666....66666
Peter and Nam, do you buy that every number in that set is a "finite
number"? And if
so, why is it that Peter wants the definition of "finite" to have a
flavor of intuition, citing
Peter's above.
Peter and Nam, how is it that Successor Axiom of Peano Axioms does or
does not
produce the number 9999....9999 in the above set?
Does this Series 1+1+1+1+ . . . .+1 produce the number 9999....9999,
and if not, why
not?
The this series 9999....999 + -1, + -1, + -1, +-1, + .... + -1
produces 0000....0000, and
if not, why not?
So, Nam and Peter, why is it that you can accept this set as the
Natural Numbers from
the Successor Axiom
{0, 1, 2, 3, ....}
but why you cannot accept this set from the Successor Axiom:
{0 , 1, 2, . . . . , 9999....998, 9999....999}
Is it that neither Nam nor Peter can rouse themselves of examination,
and rather
live a life of mocking those that do open up to examination?
Further examination: Why not have both FrontView with BackView in all
numbers where
infinity can settle in the Middle if it is an infinite-number? Why not
have both FrontView
and BackView since the old-math has frontview on Reals with no
backview and the
Hensel P-adics and Peano Natural Numbers have backview but no
frontview, so obviously
it is logical that numbers should have both.
So why not examine this stuff with seriousness, Peter and Nam, instead
of coming
in and mocking. Why not list your definition of Finite and Infinite at
the doorway
and then go from there.
> Even that case is weak: if the "many worlds of quantum physics"
> philosophical speculation becomes physically plausible, 10^500 is
> dwarfed by the number of parallel universes that have branched off
> from ours since ours began. I'm referring to the speculation that
> says that every time an undetermined event takes place, universes fork
> off from each other according to all possible outcomes of the event.
>
> > Heck, 10^500 is an overblown value: you just need 1 - for _one_ universe.
> > There "can be no physics since we cannot measure beyond it", i.e. beyond
> > 1 universe!
> >
> > The irony is, AP, you forgot you've always stated that in your signature:
> > the "whole entire Universe is just _one_ big atom."!
>
> I think your reply hit just the right note where "AP" is concerned,
> Nam. I decided to complement it with something that gets a little
> philosophical traction.
>
> > > Archimedes Plutonium
> > >www.iw.net/~a_plutonium
> > > whole entire Universe is just one big atom
> > > where dots of the electron-dot-cloud are galaxies
>
> Peter Nyikos
> Professor, Dept. of Mathematics
> University of South Carolina
> http://www.math.sc.edu/~nyikos/
So, Peter and Nam, why is it that you enter mathematics and make it a
career, and yet
when the topic of "finite versus infinite" is raised, why is it that
you cannot disclose
or offer up your definition of "finite" that you use each and every
day you engage in
mathematics? Is it because mocking those that do examine the
definition, is easier for
Nam and Peter than to make the effort to examine Finite?
Or is it as Peter says "we mathematicians" have been so much spoonfed
what to believe
in mathematics, that "we mathematicians" are more like parrots of math
than are Socrates
wanting to open up for examination?
Marshall
<plutonium.archime...@gmail.com> wrote:
>
> I want to prove that the only way to well-define or
> precisely define Finite is to pick a large number and
> say that is the end of Finite.
Anyone can define anything to be anything. The idea that
there is only one right definition of something is a failure
to understand what definitions are.
Marshall
Nam Nguyen
Totally agreed with you on this. (Not that AP's "precise" definition
of "Finite" would make a lot of mathematical sense anyway).
So are you with me that the currently widely accepted definition of
the "natural numbers" is *not* the only right definition?
For instance, the following 2 definitions would be equally the right
ones (as well as the current one):
Let F be the formula "There are infinite counter examples of GC"
Def 1: The natural numbers = the current definition + that F is true.
Def 2: The natural numbers = the current definition + that F is false.
Right?
Jesse F. Hughes
> Marshall wrote:
>> On Jan 15, 11:43 pm, Archimedes Plutonium
>> <plutonium.archime...@gmail.com> wrote:
>>> I want to prove that the only way to well-define or
>>> precisely define Finite is to pick a large number and
>>> say that is the end of Finite.
>>
>> Anyone can define anything to be anything. The idea that
>> there is only one right definition of something is a failure
>> to understand what definitions are.
>
> Totally agreed with you on this. (Not that AP's "precise" definition
> of "Finite" would make a lot of mathematical sense anyway).
>
> So are you with me that the currently widely accepted definition of
> the "natural numbers" is *not* the only right definition?
>
> For instance, the following 2 definitions would be equally the right
> ones (as well as the current one):
>
> Let F be the formula "There are infinite counter examples of GC"
Can you specify that formula in the language of PA? For simplicity's
sake, let's assume that P(x) is a first order formula in PA such that
P(x) <=> x is a counterexample to Goldbach's conjecture.
So your formula F is essentially
(Ex)( x is infinite & P(x) ).
How do you plan on expressing "x is infinite" in the language of PA?
> Def 1: The natural numbers = the current definition + that F is true.
> Def 2: The natural numbers = the current definition + that F is false.
>
> Right?
--
Jesse F. Hughes
"[From now on] no simple dismissal of claims of counterexamples or
counter proofs. I may not consider such claims, but I won't just
dismiss them [...]" -- James S. Harris turns a new leaf
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> Marshall wrote:
>>> On Jan 15, 11:43 pm, Archimedes Plutonium
>>> <plutonium.archime...@gmail.com> wrote:
>>>> I want to prove that the only way to well-define or
>>>> precisely define Finite is to pick a large number and
>>>> say that is the end of Finite.
>>> Anyone can define anything to be anything. The idea that
>>> there is only one right definition of something is a failure
>>> to understand what definitions are.
>> Totally agreed with you on this. (Not that AP's "precise" definition
>> of "Finite" would make a lot of mathematical sense anyway).
>>
>> So are you with me that the currently widely accepted definition of
>> the "natural numbers" is *not* the only right definition?
>>
>> For instance, the following 2 definitions would be equally the right
>> ones (as well as the current one):
>>
>> Let F be the formula "There are infinite counter examples of GC"
>
> Can you specify that formula in the language of PA?
Sure. Assuming we have a P(x), the statement F = "There are infinite examples
of P" can be formally written [or "translated"] in L(PA) as:
F = Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))]
Naturally x is a counter example of GC iff ~GC(x), and GC(x) iff x satisfies
GC.
Nam Nguyen
Of course by "There are infinite counter examples of GC" I meant the set
of such counter example would be infinite, not each number is an infinite
number. (A natural number is actually neither finite nor infinite!)
Nam Nguyen
I can be said that the key shortcoming of Godel's work is that it doesn't
recognize the truth of the meta statement:
(I) For any concept as strong as that of the natural numbers, there are
concepts such as F = "There are infinite [number of] counter examples
of GC" that would be independent from [the original concept].
It's truly an irony that this statement rhythms (i.e. sounds like) a statement
in his Incompleteness work, when he tried to point out the weakness of relying
on one "giant" formal system for proving _all useful_ mathematical/arithmetical
properties and relations.
In other words, in pointing out the Incompleteness of mathematical provability of
any one "giant" formal system, Godel ignored the Incompleteness of knowledge
of any "giant" definition of "The Natural Numbers".
Jesse F. Hughes
> Of course by "There are infinite counter examples of GC" I meant the set
> of such counter example would be infinite, not each number is an infinite
> number. (A natural number is actually neither finite nor infinite!)
Ah. Perhaps you should write, "There are *infinitely many*
counterexamples of GC," to clear up the confusion.
--
"And yes, I will be darkening the doors of some of you, sooner than you
think, even if it is going to be a couple of years, and when you look
in my eyes on that last day of work at your school, then maybe you'll
understand mathematics." -- James S. Harris on Judgment Day
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> Of course by "There are infinite counter examples of GC" I meant the set
>> of such counter example would be infinite, not each number is an infinite
>> number. (A natural number is actually neither finite nor infinite!)
>
> Ah. Perhaps you should write, "There are *infinitely many*
> counterexamples of GC," to clear up the confusion.
>
Agree.
A
This is an extremely nonstandard (and poor) definition of finite. Can
you cite anywhere where this definition is used in print? The only
place I have seen it before is in your own posts.
I have posted this before, but I might as well post it again: a set S
is defined to be "finite" if there exists no bijection between S and a
proper subset of S. The natural numbers are precisely the isomorphism
(bijection) classes of finite sets; the disjoint union of finite sets
induces the addition operation on the natural numbers, and the
Cartesian product of finite sets induces the multiplication operation
on the natural numbers.
If you want to invent some strange definition of the word "number"
that requires distinguishing "finite numbers" from "infinite numbers"
then the burden is on you to make that precise. Certainly there are
finite cardinal numbers and infinite cardinal numbers; but this does
not seem to be what you mean.
The burden is on you to explain what 6666....66666 is supposed to
mean!
6 = 6 * \sum_{n=0}^0 10^n
66 = 6 * \sum_{n=0}^1 10^n
666 = 6 * \sum_{n=0}^2 10^n
6666 = 6 * \sum_{n=0}^3 10^n
Is 6666....66666 supposed to be the limit of the sequence
6 * \sum_{n=0}^m 10^n
as m goes to infinity?
> Peter and Nam, do you buy that every number in that set is a "finite
> number"? And if
> so, why is it that Peter wants the definition of "finite" to have a
> flavor of intuition, citing
> Peter's above.
>
> Peter and Nam, how is it that Successor Axiom of Peano Axioms does or
> does not
> produce the number 9999....9999 in the above set?
>
> Does this Series 1+1+1+1+ . . . .+1 produce the number 9999....9999,
> and if not, why
> not?
>
> The this series 9999....999 + -1, + -1, + -1, +-1, + .... + -1
> produces 0000....0000, and
> if not, why not?
All of these questions are answered in the real analysis courses taken
by undergraduate mathematics majors in college. I suggest you take the
time to read a textbook on this subject, instead of asking for someone
in sci.math to teach it to you, and then arguing and insulting anybody
who does you the favor of trying to answer your questions.
Archimedes Plutonium
A wrote:
> This is an extremely nonstandard (and poor) definition of finite. Can
> you cite anywhere where this definition is used in print? The only
> place I have seen it before is in your own posts.
>
And did you see in my own posts where I frequently mentioned the idea
that everyone in mathematics, assumes they know what finite is versus
infinite? So when everyone is assuming what finite is, then no-one is
going
to be publishing what a finite-number is versus a infinite-number.
Does
that make sense to you?
I once made a survey of a department of mathematics at two expensive
and prestiges Universities in the 1990s. And the professors were
glad to oblige in the survey because the question sounded so innocent.
Asking as many professors what their definition of "finite number"
was. The background of this was that I invented the Infinite Integers
circa 1991 and I wanted to know what the definition of Finite Integer
was.
All of them (about 10 professors) came up with the same answer:
A finite number such as the Counting Numbers are finite because
they repeat in zeroes to the leftward string, so that 96 is finite-
number
because it is ....000096
And quite a number of those professors also bolstered their definition
of
finite-number by using the Reals as possessing a finite leftward
string
and that "finite" was given because the leftward string repeats in
zeroes
so that the "8" in 8.9999.... is finite because it is .....000008.
Some also went so far as to talk about finite for the rightward string
on
Reals in the case of 9.999000.... versus 9.9999....
So modern day mathematics throughout the world, never sat down, never
gave a precise well-defined definition of Finite Number and instead
assumed that Finite-number meant repeating zeroes of the leftward
string, and because they never
did that, well-define Finite-number, none of them could see that their
Peano axioms are inconsistent and flawed.
> I have posted this before, but I might as well post it again: a set S
> is defined to be "finite" if there exists no bijection between S and a
> proper subset of S. The natural numbers are precisely the isomorphism
> (bijection) classes of finite sets; the disjoint union of finite sets
> induces the addition operation on the natural numbers, and the
> Cartesian product of finite sets induces the multiplication operation
> on the natural numbers.
>
Yeh,yeh, I saw it before some year or two ago, and another poster
interceded
to tell you how misguided you are. How misguided you are in jumping
from
finite-number to finite-set. Because this discussion is not about
sets, it is about Finite-numbers versus Infinite-Numbers. And even
so,
the cardinality of a set fetches either a finite-number or a infinite-
number.
So tell me, since you cannot even stay focused from one paragraph to
the
next, how in the world did you ever study any math. Back into my
killfile, for
your posts are a waste of time.
A
I find this extremely hard to believe. Can you provide some evidence
that this actually happened?
> So modern day mathematics throughout the world, never sat down, never
> gave a precise well-defined definition of Finite Number and instead
> assumed that Finite-number meant repeating zeroes of the leftward
> string, and because they never
> did that, well-define Finite-number, none of them could see that their
> Peano axioms are inconsistent and flawed.
>
> > I have posted this before, but I might as well post it again: a set S
> > is defined to be "finite" if there exists no bijection between S and a
> > proper subset of S. The natural numbers are precisely the isomorphism
> > (bijection) classes of finite sets; the disjoint union of finite sets
> > induces the addition operation on the natural numbers, and the
> > Cartesian product of finite sets induces the multiplication operation
> > on the natural numbers.
>
> Yeh,yeh, I saw it before some year or two ago, and another poster
> interceded
> to tell you how misguided you are.
I have no recollection of anyone interceding except yourself. Please
cite the thread in which this happened.
> How misguided you are in jumping
> from
> finite-number to finite-set. Because this discussion is not about
> sets, it is about Finite-numbers versus Infinite-Numbers. And even
> so,
> the cardinality of a set fetches either a finite-number or a infinite-
> number.
> So tell me, since you cannot even stay focused from one paragraph to
> the
> next, how in the world did you ever study any math. Back into my
> killfile, for
> your posts are a waste of time.
You already wrote some time ago that you would no longer respond to my
posts; yet you still do. Have fun continuing to abuse anyone who tries
to help you out.
>
> Archimedes Plutoniumwww.iw.net/~a_plutonium
David Bernier
Seconded. Without standard quoting conventions or the equivalent,
we don't know if Archimedes Plutonium is paraphrasing, re-wording,
re-phrasing or interpreting & "re-wording". It's pretty hard
to figure out what the original source material actually was ...
David
Archimedes Plutonium
David Bernier wrote:
> A wrote:
(snipped)
> >
> > I find this extremely hard to believe. Can you provide some evidence
> > that this actually happened?
>
> Seconded. Without standard quoting conventions or the equivalent,
> we don't know if Archimedes Plutonium is paraphrasing, re-wording,
> re-phrasing or interpreting & "re-wording". It's pretty hard
> to figure out what the original source material actually was ...
>
> David
A private survey is a private survey. But it is easy to check and
verify in
a roundabout method. We look at the entire math literature for anyone
who has tried to define finite-number with any precision. Even "A"
says
the literature is empty, dearth, zip, zero of a definition of "finite-
number."
Now you ask anyone in mathematics or you consult the math literature
for what kind of numbers that the Peano Natural Numbers are, and the
literature
is overflowing with the announcements that the Natural Numbers are
each and every
one of them a "finite-number".
I find that horribly odd and unsettling that mathematics, the science
of precision
is replete of the fact that the Peano Natural Numbers are all "finite
numbers" yet
noone, ever bothered to define "finite-number". That is worse than
going to buy
a brand new pickup truck and find out that the company forgot to
install a engine.
Look up MathWorld for definition of "finite number". Look up
Wikipedia for definition of "finite number". Type in "finite-number"
in Google
to see if anyone defines it with precision.
What you end up with is zero, zip. Because everyone who has been
trained
in mathematics puts the definition of "finite-number" as one of those
assumed items, a fuzzy assumed piece of hardware in mathematics, so
long as noone asks poignant questions about your assumed "finite
number definition"
everyone goes their merry inconsistent ways.
In fact, the only place in all of math literature that defines "finite-
number" is not even
in math but the Webster's New World 4th ed, 2002, dictionary:
Finite: (Math) (a) capable of being reached, completed, or surpassed
by counting (said
of numbers or sets) (b) neither infinite nor infinetesimal (said of a
magnitude)
So that definition of Finite by Webster's falls apart because
999....9999 is reached by
1 + 1 + ....+ 1, and completed. But it is not surpassed by counting,
but that 9999....998
is surpassed by counting so it would be finite according to Websters.
So that definition fails, but few mathematicians expect a dictionary
to be at the cutting
edge of math definitions.
Now there is another aspect to the previous post where I lambasted "A"
about his
set definition of finite-set. I forgotten who it was who defined
"infinite set" as a set that
can be placed in a 1-1 correspondence with a proper subset, (I think
it was Riemann but
my memory is not as good as when I was younger, and it seems as though
Riemann was
far brighter than to be making such a goofy definition of infinite
sets, so I am hoping it was
not Riemann) and that a finite-set would
thus be unable to perform that feat. And as I wrote about a year or
two ago, the flaw of
that definition of infinite-set and finite-set. The question becomes
which is more important
of a concept, a finite-number or finite-set? I believe numbers come
first or are more
primal than is set theory. Regardless, here is a counterexample that
tells us that finite-set
and infinite-set based on the notion of a 1-1 correspondence with a
proper subset is fakery.
The infinite set of singlet numbers {0, 1, 2, 3, . . . . 9999....998,
9999....999}
Now the reason Cantor and others got away with the fake definition of
infinite-set as a
1-1 correspondence with a proper subset, is because they never defined
finite versus
infinite for numbers and always just looked at the Peano Natural
Numbers as this:
{ 0 , 1, 2, 3, . . . .} and those four dot ellipsis just swept the
fake definitions away.
So try putting the even Numbers into a 1-1 correspondence with all the
numbers in the
set
{0, 1, 2, 3, . . . . 9999....9999}
You see, your 1-1 correspondence with a proper subset fizzles away
into the trash dump.
The reason the junky definition of infinite-set worked for Cantor et
al and for "A", is because
they kept looking at
{0 , 1, 2, 3, . . . . } at that four-dot ellipsis to rescue them with
a 1-1 correspondence. But when
you realize that the Successor Axiom goes to 999....9999 then your
convoluted infinite-set
definition is destroyed.
So, David Bernier, you have been in math for a good long time, and
what is your understanding
of defining a Finite-Number versus an Infinite-Number. What is your
definition? What is your
definition when you first learned the Peano Axioms of the Natural
Numbers and what is your
definition of finite-number since it is key to the Peano Natural
Numbers.
If I did a survey on sci.math as to what posters defined as a finite-
number versus infinite-number, apparently in all my threads, only "A"
has even offered a definition-- but unfortunately
off on a tangent of finite-set when finite-number was required.
I think it is obvious why noone in sci.math has given a definition of
finite-number except AP with 10^500 and the reason for this lack by
others is that they are so ashamed of the fact that
they assumed what is meant by "finite-number" and too embarrassed to
admit that they assumed it all these years and lacking of wit to give
a precise definition. Or like "A" veering off
on a tangent with finite-sets rather than focused on finite-number.
So, David, what is your definition, a precise definition of finite-
number, for surely you must have had some notion in all these years of
doing mathematics, or have you left the concept
of finite-number just totally assumed in the aether?
Archimedes Plutonium
Marshall
> A wrote:
> >>
> >> All of them (about 10 professors) came up with the same answer:
>
> >> A finite number such as the Counting Numbers are finite because
> >> they repeat in zeroes to the leftward string, so that 96 is finite-
> >> number because it is ....000096
>
> >> And quite a number of those professors also bolstered their definition
> >> of finite-number by using the Reals as possessing a finite leftward
> >> string and that "finite" was given because the leftward string repeats in
> >> zeroes so that the "8" in 8.9999.... is finite because it is .....000008.
> >> Some also went so far as to talk about finite for the rightward string
> >> on Reals in the case of 9.999000.... versus 9.9999....
>
> > I find this extremely hard to believe. Can you provide some evidence
> > that this actually happened?
>
> Seconded.
I dunno; I could kinda see it. Can't you just imagine some
professor talking to AP for a bit an realizing just the sort
of fellow he was dealing with, and dialing his verbiage
way, way down, to the point where he's saying something
about digit strings because he figured that was about all
that was likely to get through? Or maybe he started with
something better, but AP kept steering the conversation
in the direction of digit strings. Or maybe the digit string
part of the conversation was the only part that AP was
able to remember; you must have noticed how about every
ten posts or so someone responds to his challenge to give
a good definition of "finite" and he doesn't seem to notice.
Marshall
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
> set definition of finite-set. I forgotten who it was who defined
> "infinite set" as a set that
> can be placed in a 1-1 correspondence with a proper subset, (I think
> it was Riemann but
> my memory is not as good as when I was younger, and it seems as though
> Riemann was
> far brighter than to be making such a goofy definition of infinite
> sets, so I am hoping it was
> not Riemann) and that a finite-set would
> thus be unable to perform that feat. And as I wrote about a year or
> two ago, the flaw of
> that definition of infinite-set and finite-set. The question becomes
Wrong, it was Dedekind, as Wikipedia writes:
In mathematics, a set A is Dedekind-infinite if some proper subset B
of A is equinumerous to A. Explicitly, this means that there is a
bijective function from A onto some proper subset B of A. A set is
Dedekind-finite if it is not Dedekind-infinite.
--- end quoting Wikipedia ---
Also, looking at Wikipedia defining finite-number is the pitiful
definition of "not infinite."
This is probably the worst cesspool in mathematics where definitions
are given which
are void of meaning--
Transcendental-number -- not algebraic
Finite-number -- not infinite
When mathematics does something like this, you wonder if everyone went
on vacation
and left the shop to be attended by goats and dogs.
Definitions like this in mathematics, is like defining a human male as
not a female. Or,
a human female is not a male. A proper definition is something
intrinsic to the object
such as Human male has xy chromosome and female has xx.
Surprizing that mathematics has reached and soared to the heights of
Game theory or Calculus or NonEuclidean geometry, yet, simultaneously
burdened by definitions that morons
would come up with.
Archimedes Plutonium
"shut-up" once
asked to define precisely what is a "finite number versus an infinite-
number".
Occasionally, some misguided person is going to belch out "finite-set"
not knowing
that we are talking about numbers, not sets. And even so, if we saddle
numbers
with a set-theory definition derived from finite-set and infinite-set
we have
counterexamples to show them that set-theory is flawed.
So I asked : Nam Nguyen for his most precise definition of "finite
number" and he seems to have run away into hiding. I have asked Peter
Nyikos for his best precise definition
of "finite-number" and he seems to have run away into hiding. I have
asked David Bernier for his best precise definition of "finite number"
and no reply.
So, I can take this survey right to sci.math itself, and without
having to ask anyone who
posted to sci.math of its entire history of posts, I can conduct a
survey to see how many
people understand and accept the definition of "finite-number versus
infinite-number"
as being the definition: a finite-number repeats in an endless string
of zeroes leftward
so that 9999 is finite because it is .....00009999 whilst 9999....9999
is infinite. The immediate
contradiction of such a definition is that what about a number such as
00009999....999
which is finite according to that definition.
So to conduct a survey of all of sci.math as to how many people have
accepted and endorse
that definition of finite-number is very easy to conduct, because the
Reals of mathematics
is defined as a finite-string leftwards of the decimal point with a
infinite string rightwards.
So in other words, everyone who accepted the definition of a Real
Number has tacitly assumed that a finite-number is a number that ends
in a repeating string of zeroes leftwards.
So even if Nam Nguyen and Peter Nyikos and David Bernier run away and
hide from my question of them defining precisely finite-number versus
infinite-number, because they
accept and use the Reals is admission that they accept the definition
of finite-number
as being a number that ends in an infinite string of zeroes leftwards.
Now if I go through sci.math, I will actually see some posts in which
people have said
Reals are finite strings leftward of the decimal point. This is
tantamount to saying
Reals are finite-numbers left of the decimal point because they repeat
in zeroes to infinity.
This is the reason why ten professors of prestigous colleges on the
East Coast said that
a finite-number is ending in zeroes leftwards, because they were
simply repeating what
they believed was the accepted definition of a Real Number.
Now we can dive into the definition of Series and to make clear what
we mean by a
"string of digits". In that the Series 1 + 1 +1 +1 + . . . . + 1 is
seen as the Successor
Axiom of Peano axioms and it ends up being an infinite number since it
is unbounded.
Now the old-math could never represent that number other than a
sideways 8 as indicating
infinity, but with the FrontView and BackView of numbers I can easily
say that in decimal
number representation that series of adding 1s endlessly equals the
number 9999....9999,
or in binary is equal to 1111....11111. So how we define "string of
digits" in mathematics?
Well we use the Series definition that a String of Digits is either
the leftward portion of
a Real-number from the decimal or the rightward portion of a Real
Number from the decimal
point. So that we can say a Finite-Number is a string that ends in
zeroes to infinity.
So how does Series define a finite-number such as "6"? Well it be like
this:
1 + 1 + 1 + 1 + 1 + 1 + 0 + 0 + 0 + ....+ 0
So, enough of a preliminary, now we can conduct a full survey of
Sci.math going back
further than the year 1993. My history of sci.math goes back only to
August of 1993, but
sci.math goes back further. And we can inspect any post by anyone who
talks about
finite versus infinite and who talks about Real-numbers.
And in fact we can include everyone who wrote a mathematics textbook
to figure out
if the author had it in mind that the meaning or definition of "finite-
number" was a
definition revolving around the idea of string of zeroes leftwards to
infinity.
So, Nam, and Peter, and David, you probably do not have to waste your
time with what
you believe is a finite-number, because you probably already displayed
your understanding
of what that definition is for you by the simple full endorsement of
what a Real-Number is.
You see, I vary from every one else as to what is a Finite-number and
thus what is a
valid-Real-Number. The Real Numbers ends at 10^500 since that is the
end of finite-number
and thus algebra on Reals ends at 10^500.
So a survey of everyone who posted to sci.math and to everyone who
wrote a book on
mathematics and who endorsed the Real-Numbers, everyone of them
accepted, whether
they realized it or not realized it, they accepted the definition of
finite-number as being a
number in which the leftwards string of digits eventually ends in
nothing but zeroes.
All of those people assumed what finite-number was, and that is the
reason mathematics
starting with the Peano Axioms are inconsistent and that mathematics
as a whole is
in a dreadful state of collapse. When mathematics fails to precisely
define its concepts,
it fails to live up to its primary job-- precision, precision,
precision. The reason Goldbach
Conjecture or Fermat's Last Theorem or Riemann Hypothesis or Perfect
Numbers Conjecture
could not and will never be proven is because we never defined finite-
number with precision.
We assumed what finite-number meant.
Archimedes Plutonium
Sorry, typing too fast, for that should have read 0000....9999. The
idea is that when finite-number is defined in this manner
we leave it open as to ambiguity and confusion and so it cannot be a
precise
definition of "finite-number". And this is what I mean that
the only precision definition is to come down hard and select a number
and
call it the boundary mark of Finite and beyond is infinite (and to
sandwich a
Incognitum between Finite and Infinite).
David Bernier
I'd say small finite numbers are ideas. Just like colors are qualities of
objects around us, abstracted from everything else about those
objects, small finite numbers are qualities of small collections of
objects. It doesn't seem reasonable to me that I would have
had in my life more than 10^(10^100) ideas. So it seems there
is also the idea of a generic finite number, i.e. the idea
corresponding to "some finite number". With small finite
numbers, there's the idea of adding them and obtaining
their sum. For two generic finite numbers, there's also
the idea that they can be added to obtain their sum.
In the course of growing-up, at some point there dawns the idea
that any two generic finite numbers can be added to arrive at
their sum, which is also a finite number. From there many are
likely to conclude that there is no largest finite number,
and that there is no end in principle to counting-up from
one by adding one each time.
Some philosophers such as Plato believed in a world of ideas
that exists independently of humans. This is possible, but
I can't imagine how one could argue very convincingly for
it, or prove it. On the other hand, people mention small
finite numbers every day, so it seems very reasonable that
small finite numbers exist as ideas for them. One can ask if
two different mathematicians have equivalent conceptions
of finite numbers. This is generally believed by mathematicians
but it seems that it would take forever to prove that this is so.
I think many mathematicians are more interested in solving
problems than in philosophy. Thus one comes upon
very large numbers mentioned in proofs, for example
the famous first and second Skewes' number(s).
I doubt that one can give a definition of "finite number"
that will be satisfactory to most or all philosophers
with an interest in mathematics.
David Bernier
David R Tribble
>> Now the number 9999....9999 which corresponds to
>> 1111....1111 in binary is the Infinity-Number. There is no
>> number larger than is number and whenever we say Infinity we can
>> interchange it with 9999....99999.
>
David R Tribble wrote:
>> So is the Infinity-Number 999...999 evenly divisible by 9?
>> (It certainly looks like it is.)
>
Archimedes Plutonium wrote:
>> Why would you not think it was?
>
Because you also stated that 111...111(binary) is equal
to 999...999(decimal), so then it should also be divisible
by 9 (or 1001 binary). But it does not look like that can
be true.
Archimedes Plutonium wrote:
> Okay, it dawned on my whilst doing another post as to why Tribble was
> asking these simplistic questions. At least this is my guess as to the
> simplistic questions.
> [...]
> So probably, Tribble is thinking that division is all the same as in
> the old math and so how can we have 9999....9999 x 3 = almost zero
> whilst 9999....99999 / 3 is 3333....33333 ?
Well, no. I was wondering how 999...999(10) and 111...111(2)
could both be evenly divisible by 9.
> Just as we can never tell from any mathematical system whether the
> number 9999....99997 is definitely a prime or composite. We also lose out on
> Algebra in the Infinite territory.
Well, either it is divisible by some other number or it's not.
If you can't determine this in your AP-number system, then
that's a pretty major gap you have. No primes, or at least no
reliable way to tell if a number is prime.
David R Tribble wrote:
>> And does that mean that the number of terms summed in
>> 1+1+1+...+1 is also a multiple of 9?
>
Archimedes Plutonium wrote:
> Okay the answer to the 1+1+1+....+1 is that it is equal to
> 9999....9999 because it
> reaches that number. The number 2 + 2 + 2+ ....+2 is also equal to
> 9999....9999 because of
> the definition of restrictive addition in that no number can exceed
> the infinity number.
Well, now that's curious, because now you're saying that
repetitive adding of even numbers somehow results in an
odd number. After how many terms does 2+2+...+2 become
odd? If n x 2 = 999...999, what is n?
> The trouble that Tribble is having, if my anticipation is correct, is
> that Algebra of add, [...]
No, the trouble I'm having, as always, is making any logical
sense out of your mathematical rules. Simply saying that the
your new system doesn't follow the same rules (ever) does not
sound like a well thought-out, self-consistent set of rules.
Until you give it a backbone of consistency, it serves mainly
as only a source of entertainment.
> Algebra is confined to only the Finite realm of mathematics and
> becomes nonsense when applied to the infinite-realm.
Obviously, you need to read more books on abstract math.
If you had, you wouldn't say such things.
> This thread that busted in on was titled Powerset and was about the
> powerset on infinity. Well,
> if a commonsense person ever thought about it well, would consider
> that how can one multiply
> or increase that which is already infinity? This is akin to the
> ancients trying to add angels on
> the head of a needle. A game in pretentious preposterousness
> prehension.
You really should read more higher-level math books.
The concepts they present are really not that hard to grasp.
Don Stockbauer
1. Potential
2. Actualized
David R Tribble
> So, I can take this survey right to sci.math itself, and without
> having to ask anyone who
> posted to sci.math of its entire history of posts, I can conduct a
> survey to see how many
> people understand and accept the definition of "finite-number versus
> infinite-number"
> as being the definition: a finite-number repeats in an endless string
> of zeroes leftward
> so that 9999 is finite because it is .....00009999 whilst 9999....9999
> is infinite. The immediate
> contradiction of such a definition is that what about a number such as
> 00009999....999
> which is finite according to that definition.
I'm curious about that survey you did in 1991. Did any of the
professors mention a book or two where this definition of
"infinite left strings of zero digits" can be found?
I've read a small stack of math books, but I've never seen a
definition like that in any of them.
Or perhaps there is a name associated with that definition,
like there are names attached to almost every other theorem
in math?
And for the record, it's pretty easy to provide an understandable
definition of "finite number" without having to mention sets.
David R Tribble
> I'd say small finite numbers are ideas.
Huh. I stubbed by toe on a seven the other day.
(With apologies to John Derbyshire.)
David R Tribble
>> This thread that busted in on was titled Powerset and was about the
>> powerset on infinity. Well,
>> if a commonsense person ever thought about it well, would consider
>> that how can one multiply
>> or increase that which is already infinity? This is akin to the
>> ancients trying to add angels on
>> the head of a needle. A game in pretentious preposterousness
>> prehension.
>
David R Tribble wrote:
>> You really should read more higher-level math books.
>> The concepts they present are really not that hard to grasp.
>
Don Stockbauer wrote:
> 1. Potential [infinity]
> 2. Actualized [infinity]
When you apply your test to determine *which* infinity it is.
Of course, when your test fails, you have another case to
consider:
3. Indeterminate infinity.
By the way, what is your test for (1) and (2)? I forget which
one you use.
(r.m.k.w.i.m.)
Archimedes Plutonium
David Bernier wrote:
(snipped)
> [...]
>
> I'd say small finite numbers are ideas. Just like colors are qualities of
> objects around us, abstracted from everything else about those
> objects, small finite numbers are qualities of small collections of
> objects. It doesn't seem reasonable to me that I would have
> had in my life more than 10^(10^100) ideas. So it seems there
> is also the idea of a generic finite number, i.e. the idea
> corresponding to "some finite number". With small finite
> numbers, there's the idea of adding them and obtaining
> their sum. For two generic finite numbers, there's also
> the idea that they can be added to obtain their sum.
>
Well, ending and endless are two more ideas and abstractions.
But physics is rather leaning towards there being "no endless"
and that everything seems to come to some sort of end. Physics
is leaning, also, to no absolute time or space, with is akin to
"no endless".
Another aspect of "finite" is that it seems to be associated with
zero or nothingness. Whereas the association of infinity is
endlessness
and everything. But physics enters this also, in that in physics we
can have a large number for the speed of light and in which that speed
is still a "finite speed" yet which the Universe prohibits any speed
to
exceed the speed of light. So here we have a tangible example of where
we can have a Large number to represent all of physics, which is still
finite,
and in which we have no need for endless speeds or infinite speeds.
So by this example of Physics, we can see that the association of
infinity
with endlessness is rather a misguided association. And that infinity
is rather
much like absolute time and space of Newtonian Mechanics, where it was
thought to "have to exist" but it turns out it does not exist and no
need for
absolute time and space.
In this sense, Physics is telling us that infinity was just a idea
that had
no bases for reality and that infinity, like absolute space and
absolute time
is not "endlessness" but rather , infinity is just superlarge. So the
definition
of finite is a large number marks the boundary of finite and beyond
that
is infinity of superlarge.
> In the course of growing-up, at some point there dawns the idea
> that any two generic finite numbers can be added to arrive at
> their sum, which is also a finite number. From there many are
> likely to conclude that there is no largest finite number,
> and that there is no end in principle to counting-up from
> one by adding one each time.
>
Many would falsely conclude that. Let me substitute the word
speed, and speed of light in your above.
" In the course of growing-up, at some point there dawns the idea
" that any two generic finite speeds can be added to arrive at
" their sum, which is also a finite speed. From there many are
" likely to conclude that there is no largest finite speed,
" and that there is no end in principle to counting-up from
" one by adding one each time and thus speeds larger than
" the speed of light, in fact infinite speeds."
> Some philosophers such as Plato believed in a world of ideas
> that exists independently of humans. This is possible, but
Plato was a great thinker but most of his ideas are replaced
or found to be wrong by modern day physics. Modern day
physics has disproven most of the philosophers of the past
of their main ideas. That is why every major college has a department
of physics but only a miniscule few have a philosophy department.
Philosophy is not science, David, so I do not know why you dwell
on philosophy, when I simply wanted your definition of finite-number.
> I can't imagine how one could argue very convincingly for
> it, or prove it. On the other hand, people mention small
> finite numbers every day, so it seems very reasonable that
> small finite numbers exist as ideas for them. One can ask if
> two different mathematicians have equivalent conceptions
> of finite numbers. This is generally believed by mathematicians
> but it seems that it would take forever to prove that this is so.
>
> I think many mathematicians are more interested in solving
> problems than in philosophy. Thus one comes upon
> very large numbers mentioned in proofs, for example
> the famous first and second Skewes' number(s).
>
> I doubt that one can give a definition of "finite number"
> that will be satisfactory to most or all philosophers
> with an interest in mathematics.
>
> David Bernier
Well, you are the philosopher here, and I am the mathematician, since
it is I who wants precision, and it is you who are ducking the
question.
You studied the Peano Axioms and the Peano Natural Numbers and
accepted
them David. You studied the Real-Numbers also, David.
The Peano Natural Numbers are all alleged to be finite-numbers, and
you bought
that David. The Real Numbers are all defined as being finite portion
leftwards
of the decimal point, or a Peano Natural Number leftwards of the
decimal point.
And you bought that also David. So you bought the concept of finite-
number.
And in all of mathematics, there are only two ways of defining finite-
number:
(i) a finite-number is a repeating block of zeroes leftwards to
infinity, so that
94 is finite because it is ....00094
or the other definition
(ii) Pick a large number like the Planck Unit 10^500 can call
everything above that
as infinity and everything equal or below that as finite.
Those are the only two definitions of finite-number in existence, and
David apparently
bought the first for he has bought into the Peano Natural Numbers and
the Reals.
Unless I am mistaken David, you bought (i), but seem unable to admit
it.
I simply asked you to give your definition of finite-number, and not
expecting a lecture
on philosophy that only distantly relates to this thread.
Archimedes Plutonium
David R Tribble wrote:
>
> Well, no. I was wondering how 999...999(10) and 111...111(2)
> could both be evenly divisible by 9.
>
>
1 11
1001 | 1111 111111.....
1001
1101
1001
1001
Archimedes Plutonium
in physics is
a superlarge speed and that no other speeds in physics can exceed it.
So we need
no infinity in Physics. And if infinity did exist in physics, then we
can expect that
speeds could be infinite. But here is a case in physics where a large
speed is
the final speed and no superluminal speed is permissible.
Another physical attribute to look at is the absolute time and
absolute space of
Newtonian Mechanics. Here again we can see infinity in such a
characteristic,
but that Newtonian Mechanics is wrong and Quantum Mechanics does not
allow absolute space and time.
So Physics is veering away from infinity. This tells me that infinity
is not the
concept of "endlessness" but must be something else. I think that
"something
else" is just a superlarge number.
So that the speed of light is a superlarge speed but not an endless
speed.
Likewise space and time can be superlarge numbers but not endless
numbers.
So, in this manner, we can define Finite as that of a large number,
and Infinity
as simply superlarge number. Now some may gripe that the speed of
light is of
a lowly magnitude of 3 x 10^8 m/s and I am defining 10^500 as finite
of the Planck
Unit of Coulomb Interactions in element 109. And I would reconcile
that by taking
the speed of light over all the atoms in the astronomy universe over
the distance
of all the galaxies as an interaction and would thus reach 10^500 in
that manner.
So, what I am trying to say is that we should start to think about
whether the concept of
infinity is tantamount to a concept of endlessness? Just like the idea
of Absolute-Space
and Absolute-time was a concept most humans bought for much of human
history
and where Physics finally showed to be false, likewise, the idea that
infinity is linked
to "endlessness" is probably also false. And that the Universe has no
endlessness, and
that the Universe has just large numbers and superlarge numbers. So
that Infinity just
means a superlarge number.
I like one analogy that I wrote about recently, call it the Virus
analogy. Suppose the Universe
had an infinity of a particular virus. What that means is that the
Universe would be consumed
in viruses and that the Universe would have no place for stars,
galaxies or planets or other life, but only room for an infinity of
viruses. That is if the concept of infinity meant endlessness. This
analogy is great in that it focuses the mind onto why infinity =
endlessness is a gaggle of
nonsense. Because if there were stars and planets and Earth and people
with an infinity of
viruses, well, it would not be infinity as endlessness because the
viruses end with Earth or with the Sun or galaxies. To be endless,
means it is everything.
So I think the concept that is tantamount to infinity comes from
Physics in the speed of light
in that you have a superlarge speed whereas all other speeds are so
much smaller.
And this solves the problem of different infinite-numbers such as
999....9999 versus
5555.....55555, both are infinite and both are superlarge, but if we
looked at them for
"endlessness", well they are different in terms of something. So if we
say that infinity
is not endlessness but rather is superlarge, we overcome that
obstacle.
Archimedes Plutonium
duality. What I wanted
to say about particle and wave duality as related to finite and
infinite is that
Physics demands duality for existence so that you have nothing along
with everything.
You have finite along with infinite. And we can sense that in particle
as a finite
object, and a wave as a sort of stretching across the entire Universe.
Energy is invisible
because it is in wave form and stretches the entire Universe.
So in this sense we can appreciate that we need both finite and
infinite, for they can
be substituted as particle and wave.
But here again when we equate Infinity with that of "endlessness" are
we making a mistake?
I believe so. I believe the correct view is to connect infinity as
wave as merely a superlarge
number and particle as finite as a large number but never as large as
a superlarge number.
So I think the concept of infinity of the past history with its
association to "endlessness" was
a big mistake and that Physics seems to point to the idea that
infinity is just merely a superlarge number compared to finite as a
large or small number.
Archimedes Plutonium
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > So, I can take this survey right to sci.math itself, and without
> > having to ask anyone who
> > posted to sci.math of its entire history of posts, I can conduct a
> > survey to see how many
> > people understand and accept the definition of "finite-number versus
> > infinite-number"
> > as being the definition: a finite-number repeats in an endless string
> > of zeroes leftward
> > so that 9999 is finite because it is .....00009999 whilst 9999....9999
> > is infinite. The immediate
> > contradiction of such a definition is that what about a number such as
> > 00009999....999
> > which is finite according to that definition.
>
> I'm curious about that survey you did in 1991. Did any of the
> professors mention a book or two where this definition of
> "infinite left strings of zero digits" can be found?
>
Every book that talks about the Reals in depth because the Reals
are defined as a finite string leftwards. A finite string leftwards
is the definition: a finite number is ending in zeroes leftwards so
that 94 is finite-number since it is .....000094
Wikipedia even mentions this leftward finite string by noting the
rightward
infinite string on Reals.
Number theory with Peano Axioms also assumes finite-number as
ending zeroes leftwards.
> I've read a small stack of math books, but I've never seen a
> definition like that in any of them.
>
That is because noone was bright enough to realize that everyone
assumed
the definition, not bright enough to formalize their hidden
assumption.
> Or perhaps there is a name associated with that definition,
> like there are names attached to almost every other theorem
> in math?
>
The name is "worst hidden assumption in math history."
> And for the record, it's pretty easy to provide an understandable
> definition of "finite number" without having to mention sets.
That should be laughable. And do I have to wait 6 months before you
deliver?
Not to spoil your effort, you must know by now that there are only two
promising
definitions of "finite-number" and only one of them is correct:
(i) finite number: repeating zeroes leftwards, such as 94 is finite
due to ....00094
(ii) select a large number like 10^500 and call all numbers beyond as
infinite.
Any other definitions of "finite number" are so far off that they are
in the twilight zone.
So I guess yours, if you ever deliver is a twilight zone definition of
finite-number.
Should be fun and laughable.
pnyikos
<plutonium.archime...@gmail.com> wrote:
> Peter, shame on you, you snuck in that sci.philosophy.tech.
Not really -- I did say the thread originated in sci.math for the
benefit of the new readers.
> Usenet does not like to have more than 3 newsgroups per post.
Google sets a limit of 5 and I'm happy to keep going with that.
> And I do not want to post in any philosophy newsgroup because
> philosophy
> is not science
But there is something called philosophy of science, and it is my
understanding that that is part of what sci.philosophy.tech is about.
Peter Nyikos
pnyikos
<plutonium.archime...@gmail.com> wrote:
> pnyikos wrote:
> > This thread originated on sci.math
>
> > On Jan 13, 11:28 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > > Archimedes Plutonium wrote:
>
> > > > But mathematics never faced up to any ironies, until now.
>
> > It faced up to Zeno's Paradoxes, much more deserving of the term
> > "ironies", centuries ago.
>
> defined
> Finite versus Infinite starting with the Calculus.
True, there was a lot of trouble with infinitesimals until Cauchy and
Bolzano came along. So I should have said "well over a century ago."
Thanks for reminding me of that.
[...]
> But I think the concept of infinity was never really tackled until the
> advent
> of the Calculus, and there it was of a micro-infinity. So Zeno's
> tackling was
> just a sign of things to come.
>
> Honestly, I think that the precision definition of Finite
There is none that corresponds to the concepts mathematicians work
with.
> had to wait
> until someone
> recognized that Physics is far above mathematics
Sorry, I just don't agree. Mathematics has been autonomous since
about a century ago.
[...]
> > > > In order to well-define Finite versus Infinite we have to
> > > > get help from Physics
>
> > Or from our own intuition, which tells us "I am, I exist" [a la
> > Descartes] and "Time is".
>
> Not so. For apparently your well-defining of Finite is the same old as
> the current establishment:
I didn't well-define it at all in a way intelligible to anyone other
than myself. It is something each of us intuits for himself, like we
intuit time.
> Finite : a number is finite if it repeats endlessly to the leftward
> string in
> zeroes such that 55 is finite because it is 0000....000055.
That is one way of looking at it, but there are nonstandard models of
the Peano axioms where the endless repetition comes only after an
infinite [in our INTUITIVE sense of the word] number of nonzero terms
to the left of the decimal point. THAT is why mathematicians cannot
well-define infinity in our intuitive sense of the word.
I did a letter to the editor of _Mathematical Intelligencer_ [there
may be a The in the title, I can't recall offhand] quite a number of
years back about this, and some sophomoric respondent said that second
order logic does the trick, not realizing that we cannot define second-
order logic in a way that unambiguously gets rid of the difficulty.
[...]
> > > > to tell us what number is large enough to
> > > > include all of Physics experimentation and that number is the largest
> > > > Planck Unit of 10^500. So beyond that number can be no physics since
> > > > we cannot measure beyond it. So we define in a well-defined definition
> > > > that Finite is 10^500 and below
> > > > (inverse for the microworld).
>
> > I suppose one could make a case for calling this "physics infinity"
> > but it just defines 10^500 (in the von Neumann convention, which
> > Archimedes apparently uses) as far as we mathematicians are concerned.
>
> You say "we mathematicians", but that must be a pitifully poor lot
> since you
> are not able to recognize when you have a "ill-definition of Finite"
I suppose most mathematicians are not, but I am, because I am
sophisticated in the foundations of mathematics.
>
> > Even that case is weak: if the "many worlds of quantum physics"
> > philosophical speculation becomes physically plausible, 10^500 is
> > dwarfed by the number of parallel universes that have branched off
> > from ours since ours began. I'm referring to the speculation that
> > says that every time an undetermined event takes place, universes fork
> > off from each other according to all possible outcomes of the event.
>
> Now you zoomed off into philosophy.
Lots of respected quantum physicists speculate that way.
But let me give you another example. The most respected cosmological
theory is that our universe will continue to expand forever, and that
forever allows for arbitrarily large units of time. So much for
10^500 or any other number you care to name.
> Ask yourself this question since you like philosophy more than
> anything.
Where do you get that idea?
> Since all the sciences such as chemistry, geology, sociology
> ultimately reduce
> to physics, then mathematics is also a subset of physics. Because math
> cannot
> exist without life
Neither can physics, if you want to play that game.
[...]
> > Peter Nyikos
> > Professor, Dept. of Mathematics
> > University of South Carolina
> >http://www.math.sc.edu/~nyikos/
[...] But I do appreciate you posting with a full name, and
> thank you
> for doing so,
You are welcome.
> since so much of the newsgroups are a wasteland of phony
> names and posts that are , thus, not worth reading. I strive to reply
> to every
> post that is from a true name and address.
>
> Archimedes Plutonium
Is that the name which you go by in your everyday interactions with
people?
Peter Nyikos
Archimedes Plutonium
Peano, Hensel,
Cohen. All concerning the definitions of finite-number versus infinite-
number.
My question is, can I get into trouble with Wikipedia over copyrights
and Fair Use
laws?
David Tribble or Peter Nyikos probably know something about copyright
infringement
on Wikipedia. From what I understand it is a free encyclopedia and so
they would
allow quoting, but how much quoting? I want to quote a paragraph of
each of those
mentioned names in one single post.
Has anyone gotten into trouble with quoting Wikipedia at length?
Anyone?
Archimedes Plutonium
pnyikos wrote:
> On Jan 15, 3:10 pm, Archimedes Plutonium
> <plutonium.archime...@gmail.com> wrote:
> > pnyikos wrote:
> > > This thread originated on sci.math
> >
> > > On Jan 13, 11:28 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > > > Archimedes Plutonium wrote:
> >
> > > > > But mathematics never faced up to any ironies, until now.
> >
> > > It faced up to Zeno's Paradoxes, much more deserving of the term
> > > "ironies", centuries ago.
> >
> > Not really. They should have wrestled with the definition of a well-
> > defined
> > Finite versus Infinite starting with the Calculus.
>
> True, there was a lot of trouble with infinitesimals until Cauchy and
> Bolzano came along. So I should have said "well over a century ago."
> Thanks for reminding me of that.
>
Yes, I would also like to link up or connect with Series and Sequence
definitions in
mathematics with the Successor function in the Peano axioms. This is
sorely needed
for the Natural Numbers, because the Hensel p-adics is a axiom system
which uses
the Series. But as far as I can make out, these three are really one
and the same
concepts, only in different contexts.
(i) Successor function
(ii) series such as 1 + 1 + 1 +....+ 1
(iii) sequence convergence {1,2,3,4, ....
The thing is, that those three end up equalling 9999....9999
> [...]
> > But I think the concept of infinity was never really tackled until the
> > advent
> > of the Calculus, and there it was of a micro-infinity. So Zeno's
> > tackling was
> > just a sign of things to come.
> >
> > Honestly, I think that the precision definition of Finite
>
> There is none that corresponds to the concepts mathematicians work
> with.
>
Thanks for being honest about that. I feel the time in history of ,
right now, is
a ripe time to thus make finite-number precision definition. What we
have
now that the pre-1990 folk never had were these ideas:
1) Physics is master and mathematics is subset
2) Numbers have a frontview with a backview and infinity fits into the
middle segment of the
number if the number is an infinite-number.
3) Physics has no infinites as realities. And Physics has a model for
which infinity is easy
to model thereof. The speed of light is a huge and large number and
for which there can be
no faster speed. So that Infinity, modeled from Physics, is not a
concept of "endlessness"
but a concept of "superlarge number", and this implies that Finite-
number is a selected special number tied to Physics which is a large
number, but not a superlarge number.
4) The Peano axioms are contradictory and inconsistent
5) Mathematics has a huge pile-up of unsolved and unsolvable problems,
not because these
problems are difficult but because mathematics is on whole
inconsistent and flawed since
noone bothered to precisely define "finite versus infinite". Once a
precision definition is given
then all those unsolved problems are solved almost overnight.
> > had to wait
> > until someone
> > recognized that Physics is far above mathematics
>
> Sorry, I just don't agree. Mathematics has been autonomous since
> about a century ago.
>
Take a look at Quantum Physics and Chemistry of the chemical element
plutonium. It has in rational number form, 22 subshells inside of 7
shells,
and it has only 19 of those subshells filled in 7 shells. Consider
that the
Big Bang is a fake theory but that the Plutonium Atom Totality is the
correct
theory of astronomy. So the Cosmic Atom has a belt length of 22/7 in
rational
form and this Cosmic Atom that we know as the Universe has only 19/7
filled in rational form.
So in other words Peter, numbers in mathematics are created and caused
by
Physics. If we lived in a Xenon Atom Totality, human civilization
would find
the value of pi and the value of "e" to be far different than 22/7 and
19/7.
The reason that mathematics even exists, is because atoms are numerous
giving
rise to arithmetic and atoms have shape and size giving rise to
geometry.
So, Peter, you can live and think as the pre1990 people do and have
done, but
the new world of understanding has arrived.
> [...]
> > > > > In order to well-define Finite versus Infinite we have to
> > > > > get help from Physics
> >
> > > Or from our own intuition, which tells us "I am, I exist" [a la
> > > Descartes] and "Time is".
> >
> > Not so. For apparently your well-defining of Finite is the same old as
> > the current establishment:
>
> I didn't well-define it at all in a way intelligible to anyone other
> than myself. It is something each of us intuits for himself, like we
> intuit time.
>
But you accepted the Peano Natural Numbers as all finite-numbers and
you
accepted the Real-Numbers with its finite-number portion to the left
of the decimal
point. Thus, whether you like it or not, by accepting the Peano
Natural Numbers
and the Real-Numbers, you have thus accepted this as the definition of
finite-number:
A finite-number is one in which repeating zeroes goes to infinity in
the leftward string
of the number, so that 94 is finite-number because it is ....000094
Cry and complain all you want Peter, since you accepted the Peano
Natural Numbers
and the Reals, you have tacitly accepted the hidden assumption of
finite-number as
ending in zeroes.
>
> > Finite : a number is finite if it repeats endlessly to the leftward
> > string in
> > zeroes such that 55 is finite because it is 0000....000055.
>
> That is one way of looking at it, but there are nonstandard models of
> the Peano axioms where the endless repetition comes only after an
> infinite [in our INTUITIVE sense of the word] number of nonzero terms
Don't you find that unreasonable, that you never defined finite and
here you
want to use infinity to get at defining finite. So if you never
defined finite
first, how in the world would you want to enlist infinity for any
help.
> to the left of the decimal point. THAT is why mathematicians cannot
> well-define infinity in our intuitive sense of the word.
>
They cannot define infinity, obviously, because you need to well
define
finite first.
Now, what is the most obvious way to define Finite? If really pressed,
what is the
most obvious way?
Clearly, we just jump to Physics and ask what is the largest number
where Physics
runs out of able to measure or observe or experiment? And we call that
largest
number the end of Finite.
So, Peter, that was not hard at all, was it.
> I did a letter to the editor of _Mathematical Intelligencer_ [there
> may be a The in the title, I can't recall offhand] quite a number of
> years back about this, and some sophomoric respondent said that second
> order logic does the trick, not realizing that we cannot define second-
> order logic in a way that unambiguously gets rid of the difficulty.
> [...]
Well, here again, Peter, I think you are bogged down with old-time
thoughts
of using philosophy, when you should be relying totally on Physics.
That we
have made advances in Physics that addresses the question of finite
versus
infinite. These four advances puts your struggle of finite versus
infinite into
perspective:
(1) A number should have both FrontView with BackView and that
infinity is tucked
into the middle. So that we can see that Peano Natural Numbers due to
the Successor
delivers this set {0, 1, 2, 3, . . . . 9999....9999} and no longer
delivers this set
{0, 1, 2, 3, . . . . }
(2) Physics is superior to math and math is a subset of physics. And
since there
are no Infinities in reality in physics, that we must use Large
numbers as the concept
of the boundary between finite and infinite. And use the concept of
the speed of light
as the concept of infinity as that of "superlarge number" not
endlessness as the concept.
(3) Peano axioms are contradictory with its Successor.
> > > > > to tell us what number is large enough to
> > > > > include all of Physics experimentation and that number is the largest
> > > > > Planck Unit of 10^500. So beyond that number can be no physics since
> > > > > we cannot measure beyond it. So we define in a well-defined definition
> > > > > that Finite is 10^500 and below
> > > > > (inverse for the microworld).
> >
> > > I suppose one could make a case for calling this "physics infinity"
> > > but it just defines 10^500 (in the von Neumann convention, which
> > > Archimedes apparently uses) as far as we mathematicians are concerned.
> >
> > You say "we mathematicians", but that must be a pitifully poor lot
> > since you
> > are not able to recognize when you have a "ill-definition of Finite"
>
> I suppose most mathematicians are not, but I am, because I am
> sophisticated in the foundations of mathematics.
>
Well you have admitted that math has never well-defined with precision
the
concept of finite-number versus infinite-number which is a first step
in the
right direction.
> >
> > > Even that case is weak: if the "many worlds of quantum physics"
> > > philosophical speculation becomes physically plausible, 10^500 is
> > > dwarfed by the number of parallel universes that have branched off
> > > from ours since ours began. I'm referring to the speculation that
> > > says that every time an undetermined event takes place, universes fork
> > > off from each other according to all possible outcomes of the event.
> >
> > Now you zoomed off into philosophy.
>
> Lots of respected quantum physicists speculate that way.
>
> But let me give you another example. The most respected cosmological
> theory is that our universe will continue to expand forever, and that
> forever allows for arbitrarily large units of time. So much for
> 10^500 or any other number you care to name.
>
So you bought into the Big Bang. That is like buying into the Static
Earth
when Wegener was actively pushing the Continental Drift.
Usually, Peter, first comers in science or mathematics build a topic
or
theory of science or math for which later people usually throw out
unto
the trashheap and replace it with a more "true theory". If you read
the history
of science you see where most established theories are trashed by
future
generation of scientists. Earth on the back of a elephant or tortoise
gave
way to Geocentric gave way to Ptolemy gave way to Helocentric and
Nebular Dust Cloud. But the Nebular Dust Cloud is giving way to Dirac
New Radioactivities.
This sequence of replacements of a theory of science holds true for
most
of mathematics also, only it was seldom recorded in history as to the
trashing of fake mathematics. In the future of mathematics, the Cantor
infinite sets and the Peano Natural Numbers and the continuum
hypothesis
will be nonexistant because they will have been thrown on the
trashheap
of fake theory.
> > Ask yourself this question since you like philosophy more than
> > anything.
>
> Where do you get that idea?
Why talk about philosophy at all when my thread is all about math and
physics.
>
> > Since all the sciences such as chemistry, geology, sociology
> > ultimately reduce
> > to physics, then mathematics is also a subset of physics. Because math
> > cannot
> > exist without life
>
> Neither can physics, if you want to play that game.
>
Let us play that game. So you are saying that in order for the Sun to
have been borne
and the planets and the galaxies that we needed to have human's living
in order for that
Physics to have taken place?
The comment I am making is that mathematics is dependent on Biology in
order for the science to exist in the first place. Physics and Biology
are not dependent on a biological
mind in order for physics and biology to take place.
>
> [...]
> > > Peter Nyikos
> > > Professor, Dept. of Mathematics
> > > University of South Carolina
> > >http://www.math.sc.edu/~nyikos/
>
> [...] But I do appreciate you posting with a full name, and
> > thank you
> > for doing so,
>
> You are welcome.
> > since so much of the newsgroups are a wasteland of phony
> > names and posts that are , thus, not worth reading. I strive to reply
> > to every
> > post that is from a true name and address.
> >
> > Archimedes Plutonium
>
> Is that the name which you go by in your everyday interactions with
> people?
>
> Peter Nyikos
I have two Wikipedia articles on me, which explains my name:
One of them is this: http://en.wikipedia.org/wiki/User:Likebox/Archimedes_Plutonium
pnyikos
<plutonium.archime...@gmail.com> wrote:
> pnyikos wrote:
> > This thread originated on sci.math
>
> > On Jan 13, 11:28 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > > Archimedes Plutonium wrote:
>
> > > > But mathematics never faced up to any ironies, until now.
>
> > It faced up to Zeno's Paradoxes, much more deserving of the term
> > "ironies", centuries ago.
>
> > > > get help from Physics
>
> > Or from our own intuition, which tells us "I am, I exist" [a la
> > Descartes] and "Time is".
>
I am not mocking you at all. You are aware of your own existence
through being conscious, and the existence of time with absolute
certainty, even without reference to science, are you not?
> Why not examine your own
> definition of Finite versus Infinite.
My own, inexpressible-to-others concept, has been under severe
examination since 1978, when I made a searching study of the
foundations of mathematics and realized that this concept is not
definable in terms of other concepts; it is a primitive concept, like
"time".
> Here I am in a gigantic thread of over 303 posts, examining Finite
> versus
> Infinite, and then there is Nam and Peter, mocking the examination.
Oh, I see now; my mocking (as you call it) was in what I said to Nam,
and had to do with the sheer arbitrariness of defining "finite" to
mean < or = 10^500. Why such a round number instead of one grounded in
an elaborate physical theory? a fondness for integer powers of the
number of digits on your two hands, with the exponent also easily
definable by the number of digits on one hand, the number of digits on
two hands, and the number of hands?
> So state your definition of Finite, Peter Nyikos and state your
> definition of
> Finite Nam Nguyen. Is it any different from what most mathematician
> use--
> a number is Finite if it ends in a endless string of zeroes leftward
> such as
> 89 is finite because it is 0000....00089.
Did you comprehend my explanation of why your definition is ambiguous
as far as mathematics is concerned?
And what does this have to do with your 10^500? Are you suggesting
that a number with 502 nonzero digits to the left of the decimal
point, and then followed by an endless string of zeros, is finite
after all?
> > > > to tell us what number is large enough to
> > > > include all of Physics experimentation and that number is the largest
> > > > Planck Unit of 10^500. So beyond that number can be no physics since
> > > > we cannot measure beyond it. So we define in a well-defined definition
> > > > that Finite is 10^500 and below
> > > > (inverse for the microworld).
>
> > I suppose one could make a case for calling this "physics infinity"
> > but it just defines 10^500 (in the von Neumann convention, which
> > Archimedes apparently uses) as far as we mathematicians are concerned.
>
> Peter says: "as far as we mathematicians are concerned"
>
> I do not think other mathematicians like it when Peter makes such
> comments.
Some mathematicians don't like the von Neumann convention, but it is a
concise way of stating the concept.
> Again, Peter, you entered a very long thread that dives into the
> precision definition
> of Finite versus Infinite. And for anyone with a milligram of logic
> would first off,
> offer their definition of what they thought and used as "finite".
I think I've made my meaning clear, second off.
> You see, Peter and Nam, I have given a precision definition of finite
> as 10^500.
You could also define the word "frolloog" to mean the same thing. The
trick is to make the definition have some relevance to mathematics, as
great a relevance as the distinction between finite and infinite in
common use by mathematicians.
> Peter and Nam, define what you mean by this ellipsis of the Peano
> Axiom set
> of Natural Numbers:
>
> {0, 1, 2, 3, 4, 5, 6, . . . . }
Well, the best I can do is that it is the set of numbers common to all
models of the Peano axioms. But it would take hours to explain that
adequately.
[...]
> So, Nam and Peter, why is it that you can accept this set as the
> Natural Numbers from
> the Successor Axiom
>
> {0, 1, 2, 3, ....}
>
> but why you cannot accept this set from the Successor Axiom:
>
> {0 , 1, 2, . . . . , 9999....998, 9999....999}
Because the last element has a successor, and so on *ad infinitum*.
Peter Nyikos
David R Tribble
>> Well, no. I was wondering how 999...999(10) and 111...111(2)
>> could both be evenly divisible by 9.
>
Archimedes Plutonium wrote:
> 1 11
> 1001 | 1111 111111.....
> 1001
> 1101
> 1001
> 1001
Continuing the binary division:
0001 1100 0111 000...
-------------------------
1001 ) 1111 1111 1111 1111 ...
1001
----
110 1 <A
100 1
-----
10 01
10 01
-----
00 001
0 000
-----
0 0011
0000
----
0011 1
000 0
-----
011 11
10 01
-----
01 10
00 00
-----
1 101 <B
1 001
-----
....
The division sequence repeats the group between A to B,
giving a quotient of 000 111 000 111 000 111 000...,
which at no point has a remainder of 0. So 1001 does
*not* divide 111...111 evenly.
Which leads us to conclude that 111...111(2) is *not* the
same as 999...999(10).
David R Tribble
>> I'm curious about that survey you did in 1991. Did any of the
>> professors mention a book or two where this definition of
>> "infinite left strings of zero digits" can be found?
>
Archimedes Plutonium wrote:
> Every book that talks about the Reals in depth because the Reals
> are defined as a finite string leftwards. A finite string leftwards
> is the definition: a finite number is ending in zeroes leftwards so
> that 94 is finite-number since it is .....000094
Really? Can you provide a citation to a math book that
defines real numbers that way? I've never seen one like that.
> Wikipedia even mentions this leftward finite string by noting the
> rightward infinite string on Reals.
To be precise, Wikipedia says that "a real number can be given
by an infinite decimal representation". It does *not* say that that
is the way real numbers are defined. In fact, it goes on to say
that "these descriptions of the real numbers, while intuitively
accessible, are not sufficiently rigorous for the purposes of
pure mathematics". The rest of the article then gives several
different rigorous formulations for them.
David R Tribble wrote:
>> And for the record, it's pretty easy to provide an understandable
>> definition of "finite number" without having to mention sets.
>
Archimedes Plutonium wrote:
> Not to spoil your effort, you must know by now that there are only two
> promising
> definitions of "finite-number" and only one of them is correct:
> (i) finite number: repeating zeroes leftwards, such as 94 is finite
> due to ....00094
> (ii) select a large number like 10^500 and call all numbers beyond as
> infinite.
And a handful of other, better definitions. You really should read the
articles you quote from.
spudnik
of "irony" is perhaps a definition of some other (English) word?...
I invite others to supply a better word to his putative de-finite-ion!
I would, again -- at the risk of contributing to any royalties
taht doctor Plutonium recieves for attending to his **** -- like
to refer to Ore's _Number Theory and Its History_
for a de-finite-ive account of Stevin's revolution
of _The Decimals_, and the reference to it in Munk's treatise,
published by a "vanity press," as he had used
during the Great Depression to publish the first "layman's" account
of aerodynamics. (am I recalling correctly,
taht this caused the Plutonium One to issue a threat
upon my life -- very scarey ?-)
> To be precise, Wikipedia says that "a real number can be given
> by an infinite decimal representation". It does *not* say that that
> is the way real numbers are defined. In fact, it goes on to say
> that "these descriptions of the real numbers, while intuitively
> accessible, are not sufficiently rigorous for the purposes of
> pure mathematics". The rest of the article then gives several
> different rigorous formulations for them.
> And a handful of other, better definitions. You really should read the
> articles you quote from.
thus:
(1) is not self-consistent?
how does it materially differ from Gauss's arithmetical series,
prime_number + 2n, or what ever?
> k^2 = [X + (2^n)T^2]/M (1)
> Conditions: All the variables are odd integers, relatively prime, each
> > 1, prime k > 3
> Claim: (1) is not consistent
--l'OEuvre!
http://wlym.com
spudnik
> The division sequence repeats the group between A to B,
> giving a quotient of 000 111 000 111 000 111 000...,
> which at no point has a remainder of 0. So 1001 does
> *not* divide 111...111 evenly.
>
> Which leads us to conclude that 111...111(2) is *not* the
> same as 999...999(10).
thus:
first of all, did anyone point out that the Archimedean valuation
of "irony" is perhaps a definition of some other (English) word?...
I invite others to supply a better word to his putative de-finite-ion!
I would, again -- at the risk of contributing to any royalties
that AP gets for any one attending to his **** -- like,
to refer to Ore's _Number Theory and Its History_
for a de-finite-ive account of Stevin's revolution
of _The Decimals_, and the reference to it in Munk's treatise
(published by a "vanity press," as he had used
during the Great Depression to publish the first "layman's" account
of aerodynamics.) [am I recalling correctly,
taht this caused the Plutonium One to issue a threat
upon my life -- very scarey ?-]
> You really should read the articles you quote from.
thus:
(1) is not self-consistent?
how does it materially differ from Gauss's arithmetical series,
prime_number + 2n, or what ever?
> k^2 = [X + (2^n)T^2]/M (1)
> Conditions: All the variables are odd integers, relatively prime,
> each > 1, prime k > 3
--l'OEuvre!
http://wlym.com
Archimedes Plutonium
1001 <C
1001
David R Tribble
>> but why you cannot accept this set from the Successor Axiom:
>> {0 , 1, 2, . . . . , 9999....998, 9999....999}
>
Peter Nyikos wrote:
> Because the last element has a successor, and so on *ad infinitum*.
Indeed so. If the set built from the Successor Axiom contains an
element that does not have a successor, then we'd be forced to
conclude that something was wrong with either the Axiom or
the construction of the set.
If the Axiom is wrong, then it is not really an axiom after all.
On the other hand, the Axiom dictates that every element of
the set has a successor, so logically there cannot be a "last"
element of the set.
So the reasonable conclusion is that either AP rejects the
Axiom as not really being an axiom, or that he is not using
the axiom to properly construct the complete set.
The third possibility is that he's sneaking in a hidden rule or
two about how the set is constructed, instead of using the
Axiom alone to do so.
spudnik
> The third possibility is that he's sneaking in a hidden rule or
> two about how the set is constructed, instead of using the
> Axiom alone to do so.
thus:
several have posted references to the non-null results
of the Michelson-Morely Xprmnt; Cahill (?) sites a paper
that gives a l o v e l y graphical comparison
of the "nulls" of M&M and successors (D.C.Miller e.g.);
read it and freak.
thus:
cool; would some one provide a tutorial?
> > the number of self-conjugate partitions of N is the same as the number of
> > partitions of N into distinct odd parts. Is there a way to determine the
> > number S(N;n) of self-conjugate paritions of N given that they all
> > must contain a largest element n?
> It seems to me that S(N; n) = S(N - 2 n + 1).
thus:
I agree with the above pundits;
all you have to do is actually create such a proof, or
you can just "work-through" any that were done,
such as Fermat's -- http://wlym.com --
the creator of the modern theory of numbers,
of which Godel was a rather crude arithmetical usage
-- totally elementary, but rather laborious --
I think. in particular,
Fermat's "reconstruction" of Euclid's "porisms" is supposed
to be exmplary, for a cannonical geometrical proof.
thus:
first of all, did anyone point out that the Archimedean valuation
of "irony" is perhaps a definition of some other (English) word?...
I invite others to supply a better word to his putative de-finite-ion!
I would, again -- at the risk of contributing to any royalties
that AP gets for any one attending to his **** -- like,
to refer to Ore's _Number Theory and Its History_
for a de-finite-ive account of Stevin's revolution
of _The Decimals_, and the reference to it in Munk's treatise
(published by a "vanity press," as he had used
during the Great Depression to publish the first "layman's" account
of aerodynamics.) [am I recalling correctly,
taht this caused the Plutonium One to issue a threat
upon my life -- very scarey ?-]
--l'OEuvre!
http://wlym.com
Archimedes Plutonium
pnyikos wrote:
> On Jan 16, 11:38 am, Archimedes Plutonium
(snipped)
> > Peter and Nam, define what you mean by this ellipsis of the Peano
> > Axiom set
> > of Natural Numbers:
> >
> > {0, 1, 2, 3, 4, 5, 6, . . . . }
>
> Well, the best I can do is that it is the set of numbers common to all
> models of the Peano axioms. But it would take hours to explain that
> adequately.
> [...]
And I suspect you think that is a reasonable answer. Where one can
explain
the entire Peano Axioms in one hour but that Peter has to take hours
just explaining
four-dot ellipsis. What I think is the case here with the ellipsis, is
that modern
day math does not want to admit the Peano Natural Numbers are a flawed
axiom system and that they use the ellipsis as a bastion of under the
rug
sweep away of the flaws.
Failure to see that Successor axiom contradicts that every Peano
Natural Number
is a finite-number. And that the ellipsis is also the "under the rug
sweeper" for
the lack of precision defining a finite-number.
Now in an earlier post, Peter, you said that precise definition of
finite-number
is akin to intuition and not defining. That finite is like time or
space, in that
we all intuit. But then, Peter, is not that just a big excuse. Is not
that a way
to want to weasel out of the job that math is supposed to be all
about--
precision.
And how is it, Peter, that you want to not well-define Finite-Number
but leave
it as "intuit" when nearly half of all of mathematics depends on
knowing what
is a finite-number from an infinite-number.
I mean, talk about status quo, ultraconservative, and never wanting to
change
anything.
So, Peter, is your concept of mathematics of a system of intuits, then
a system of
axioms, and then rest of the full bodied math.
It only takes me two seconds to type 1 + 1 + 1 + . . . . + 1 to
indicate
9999....9999 yet you need hours, Peter to explain the ellipsis. Don't
you see
some imbalance there?
>
> > So, Nam and Peter, why is it that you can accept this set as the
> > Natural Numbers from
> > the Successor Axiom
> >
> > {0, 1, 2, 3, ....}
> >
> > but why you cannot accept this set from the Successor Axiom:
> >
> > {0 , 1, 2, . . . . , 9999....998, 9999....999}
>
> Because the last element has a successor, and so on *ad infinitum*.
>
> Peter Nyikos
Ah, Peter, you never really thought about finite and infinite well
enough.
There is no successor to 9999....9999, unless you want an imaginary
one.
You are at "infinitum". Add 1 to 9999....9999, Peter, and what do you
get?
You get 10000....00000, but you already have 1000....00000 much
earlier
in the series 1 +1+ 1 + ....+1 as 10% of 9999....9999+1. Your mistake
Peter
is that your mind is not focusing on the infinite territory but is
still back on
finite shores, your intuited finite.
If I were to ask you, Peter to walk through finite into infinity,
judging from this
post you would be saying that you walk or run through the successor
axiom
of 9999 to 10000 to 10001 and much later from 10^500 to 10^500 +1 and
that you,
Peter, never gets to any number such as 1111....11111 or
2222....22222.
Judging from your remarks, your idea as you walk or run through the
Peano Natural
Numbers that none of them turn up as infinite-numbers. But you are
sadly mistaken.
On the one hand, Peter, you want to cling to the idea that every Peano
Natural
Number is a finite number, and on the other hand you want to say that
those numbers
form an infinite set. But you do not see the contradiction of that
desire, because
nothing can stop or is stopping that Successor from adding 1 ad
infinitum. And you cover
it up with the four dot ellipsis, saying that it would take hours to
explain. I think
what you meant to say was that it would take hours for an "excuse" not
an "explain".
The number 9999.....99999 has no successor, because adding anything to
it is a number
already in that list or set of all those numbers generated by the
Successor. The Successor
generates a list of numbers that are All Possible Digit Arrangements.
Add 1 to
9999....99999 and we already return to the number 10000.....0000 which
is 10%
of 999....9999 +1 and the number 10000.....0000 has the predecessor of
0999....9999
and the successor of 10000....00001.
So many people use the phrase "ad infinitum" not as a badge of
understanding but
more of a signal to shut-up and go away. There is a concept of All
Possible Digit Arrangements,
and it comes from a subject of mathematics that is above Algebra, and
is called
Probability Theory. Yes, it is more important and deeper than Algebra.
In Probability
theory there is the Universal Space of all possible outcomes. So the
Peano Natural
Numbers has a Universal Space and it is this All-Possible-Digit-
Arrangements.
There is only one number in that space that is 10000.....00000 and it
is not
999....9999+1 but rather the 10% below. So the Universal Space of all
the Natural
Numbers has 9999....99999 and its predecessor was 9999....99998.
But getting back to your walking or running through the Peano Natural
Numbers and your
acceptance that they are this set {0, 1, 2, 3, 4, . . . .} and your
acceptance that the Successor
axiom is the endless adding of 1 to generate the successors. You do
not believe any
infinite-numbers are in that set, yet you accept the successor as an
infinite adding of 1.
So how in the world, Peter, can you accept both of those ideas, and
yet never understand
that you committed a contradiction, and that your axioms are
inconsistent. Is it because you think everyone else will come to your
rescue and turn against me and what I am saying. And is it because you
feel everyone else will confederate around you, and will turn against
the "new thinker".
You say you are a mathematician, but when asked to define precisely
"finite-number" the
best you can do is talk about intuit to time and space. That is an
awfully poor job, would you
not say Peter. Of course, you can always do what Nam does, run away
and hide.
Peter Webb
it is a number
already in that list or set of all those numbers generated by the
Successor.
___________________________
In the context in which we are talking (Peano's axioms) 999...999 is not a
natural number. It couldn't possibly be, because it has no successor (as you
yourself pointed out), so its not part of the set of numbers defined by
Peano's axioms.
Similarly, 1/3 is not a natural number, and nor is the square root of 2,
or -1 for that matter. You don't need p-adics to find examples of numbers
that aren't Natural numbers and hence are not part of the set of numbers
generated by Peano's axioms. Any non-integral Real will do.
Archimedes Plutonium
David R Tribble wrote:
> Archimedes Plutonium
> >> but why you cannot accept this set from the Successor Axiom:
> >> {0 , 1, 2, . . . . , 9999....998, 9999....999}
> >
>
> Peter Nyikos wrote:
> > Because the last element has a successor, and so on *ad infinitum*.
>
> Indeed so. If the set built from the Successor Axiom contains an
> element that does not have a successor, then we'd be forced to
> conclude that something was wrong with either the Axiom or
> the construction of the set.
>
Granted, if you are given two axioms
(1) existence of 0 and 1, thus creating a metric unit distance
(2) Successor function using that unit distance and thereby creating
all the other Natural Numbers by an endless adding of 1.
So every number from 0 to 9999....9999 is created from that
Successor. But now apply the Successor to 9999....9999 and it fails
to work since 10000....00000 was already created far below as
10% of 9999....9999+1.
Correct me if wrong but there is another Peano Axiom that says
0 has no predecessor. If memory is wrong, well, refresh it.
So I think that we need another axiom that says 9999....9999 has
no successor. But to me, these sort of axioms are cosmetics, or
unimportant irrelevant add ons. I think these are not really axioms
independent and should somehow be incorporated within the
"driving axioms". So that when we list the axiom of the
Successor, we should include within that axiom itself the datum
that 0 has no predecessor and that 9999.....9999 has no successor.
That it is a mistake to list as a separate axiom, a piddly paddly
axiom that 0 has no predecessor and 9999...9999 no successor.
Axioms should be streamlined and not made petty with petty axioms.
> If the Axiom is wrong, then it is not really an axiom after all.
>
The Successor is the main axiom of the Natural Numbers. As we
can see it is the mirror reflection of the Hensel axioms for the p-
adics
only there it is called a series axiom. Now I am not aware of how
Hensel dealt with .....99999r and with 0 as it relates to successor
and
predecessor.
But it shows how overinflated the Peano Axioms really are, and that
there never was a need to have Math Induction as an axiom, because
Hensel p-adics built a larger set out of fewer axioms, both using of
course
a main-engine axiom of Series and Successor.
> On the other hand, the Axiom dictates that every element of
> the set has a successor, so logically there cannot be a "last"
> element of the set.
>
Not so, the trouble is that Peano and Dedekind and others when
crafting the Axioms never had the technological advance of the concept
of FrontView with BackView and the idea that you put the infinity into
the middle
segment of the number. Last night I watched a history documentary on
the
Influenza of 1918 with the song ditty:
I had a bird named Enza
I opened up the window
And in Flu Enza
And where the scientists at the time had microscopes that could only
see bacteria,
but not viruses. The electron-microscope that can see viruses took
many decades
later.
Likewise, if Peano-Dedekind had had FrontView with BackView with the
idea that
Infinity is nestled into the midsection of a number, then math history
would have
been vastly improved and without the piles of fakery that much of the
20th century
math turned out.
Dedekind and Peano, never had the technological advancement of seeing
a number
from its Front and from its Back with infinity tucked into its
midsection. So they
could not have foreseen that 999....999 has no successor, but were
able to spot that
0 had no predecessor.
> So the reasonable conclusion is that either AP rejects the
> Axiom as not really being an axiom, or that he is not using
> the axiom to properly construct the complete set.
The Successor axiom is the most important axiom of the Natural
Numbers,
it was just unfortunate that FrontView and BackView with infinity in
the middle
was unknown at that time.
>
> The third possibility is that he's sneaking in a hidden rule or
> two about how the set is constructed, instead of using the
> Axiom alone to do so.
Tribble, you should be more flexible, since you are so much younger
than
Peter Nyikos. The Peano Axioms are flawed and inconsistent because
they require a Successor Axiom which builds this set
{0, 1, 2, 3, 4, 5, 6, . . . . , 9999....9999}
And that means the creation axiom of 0 must include 1. And thence the
Successor
Axiom is the same only it states also that 0 has no predecessor and
9999....9999
has no successor. Sort of a symmetry. Or one could say that the
successor of
9999....9999 is 0 and predecessor of 0.
So I am not sneaking anything but only correcting the contradictions
of Peano
axioms.
The Peano axioms of old were like the scientists that had only
microscopes to see
bacteria in 1918, but with FrontView and BackView and infinity in the
middle, then
we have new technology of the electron-microscope to actually confirm
the virus.
Archimedes Plutonium
pnyikos wrote:
> On Jan 16, 11:38 am, Archimedes Plutonium
(snipped)
> > Why not examine your own
> > definition of Finite versus Infinite.
>
> My own, inexpressible-to-others concept, has been under severe
> examination since 1978, when I made a searching study of the
> foundations of mathematics and realized that this concept is not
> definable in terms of other concepts; it is a primitive concept, like
> "time".
>
Peter, lets raise the stakes here. Instead of the physics "time or
space"
as intuition. Let us talk about Primitive Concepts and the two most
famous
in mathematics are in geometry, not numbers, and are the primitive
concepts
of "point and line" in geometry.
Now let us ask a question. Are these the only two primitive concepts
in all of mathematics?
I would say no, from a physics standpoint of duality. That Geometry is
the dual of Numbers,
and since Geometry gets by with having two primitive concepts, it is
likely that Numbers
must have two and only two primitive concepts that relates to "points
and lines".
So are there any primitive concepts in Numbers? I can think of two
primitive concepts,
set and membership.
So I think that in the whole of mathematics, there are two and only
two primitive concepts
and all the other things have to be either axiomatized or defined.
Now looking at "point and line" as primitive concepts of Geometry, do
they reflect in some
meaningful way "membership and set" for Numbers? Is not a line a set
of points and is not
a point a member of a line? Likewise, the reverse, is not a set a line
of members and is not
a member a point of a set?
So I think where Peter is trying to escape from having to well-define
or precisely define a
Finite-number versus an Infinite-number by shrugging the task off as a
intuit or primitive concept, is not allowed. That mathematics already
has all the allowable primitive concepts and that everything else in
mathematics has to be precision defined. If Geometry can have
only two primitive concepts and all the rest are precision defined
then Numbers have to
have a precision defined Finite-number and Infinite-number.
Archimedes Plutonium
Archimedes Plutonium wrote:
> pnyikos wrote:
(snipped)
> >
> > > Since all the sciences such as chemistry, geology, sociology
> > > ultimately reduce
> > > to physics, then mathematics is also a subset of physics. Because math
> > > cannot
> > > exist without life
> >
> > Neither can physics, if you want to play that game.
> >
>
> Let us play that game. So you are saying that in order for the Sun to
> have been borne
> and the planets and the galaxies that we needed to have human's living
> in order for that
> Physics to have taken place?
>
> The comment I am making is that mathematics is dependent on Biology in
> order for the science to exist in the first place. Physics and Biology
> are not dependent on a biological
> mind in order for physics and biology to take place.
>
So far I prove that Mathematics is a subset of Physics due to the fact
that
"pi and e" have the values they have because our Universe is an Atom
Totality
where it has 22 subshells inside of 7 shells and only 19 occupied
subshells
giving 22/7 and 19/7 in rational number form. And that geometry exists
because
atoms have shape and size and arithmetic exists because atoms are
numerous.
But I found a second proof that is more obvious. The above, where the
only place
that mathematics exists is in the minds of intelligent life. But that
Physics exists
regardless of whether there is intelligent life or not. That Physics
and chemistry
and biology and geology can exist and carry on regardless of whether
the Universe
has or has no intelligent life. But the science of mathematics is
dependent on a
creature with a mind that can do mathematics. So mathematics is closer
to
psychology than it is to physics. And we all know that chemistry is a
subset of
physics, and biology a subset of chemistry. So, surely, since
mathematics is so
far removed by its dependency of the presence of intelligent life that
it is a subset
of Physics.
Our Solar System experienced Physics for at least 4.5 billion years of
ongoing
Physics carried out, but only in the last hundred thousand years was
there a animal
mind capable of doing mathematics. So as a subject that existed in
reality, Physics
has been around for as old as the Universe itself, but mathematics has
been around
only for the past 100,000 years.
Anybody else want to raise the arrogant thought that mathematics is so
big and important
when in fact it probably ranks as one of the lowiest of the sciences
in importance. And it
is probably more important to make progress in the science of
psychology than to make
progress in mathematics.
Our entire education system should put mathematics to a lesser
priority and instead of
teaching English and Math as the two heavyweights should replace the
Math with either
Physics or Chemistry or Biology, or those three in combination.
The education in mathematics should be slimmed down to those ideas
that can handle
money calculations. We spend too much time on mathematics in our
education system
which is a shame, and not enough time on the important sciences of
physics, biology,
chemistry.
A
Great, then you want a definition of the natural numbers which takes
SETS AND MEMBERSHIP as logical primitives. Then why did you complain
so much when I provided a definition of the natural numbers which did
EXACTLY THAT--derived the natural numbers from counting elements of
sets??
A
Archimedes Plutonium
Alright, according to Wikipedia the 7th axiom of Peano reads thus:
--- quoting Wikipedia ---
For every natural number n, S(n) ≠ 0. That is, there is no natural
number whose successor is 0.
--- end quoting ---
Alright, let us as Physics what that axiom should be. When I did the
AP-adics infinite-integers
I restricted them to a hemisphere and the not full sphere. But I never
asked Physics what
was the correct idea. I restricted them because I felt that 0 should
be the dual of the largest
number 9999....9999+1 = imaginary number = infinity. So that the North
Pole= 0 is dual
to the South Pole = infinity.
But I never allowed Physics to weigh in on this decision.
Physics is Quantum Logic and that means duality. So Aristotelian Logic
which is common everyday mathematics of algebra with add, multiply and
operators only is upheld and supported in the Finite and Incognitum
regions but not in the Infinity regions. So we lose
algebra in the infinity territory. And Duality is a circular logic, a
sort of what philosophy calls
yin and yang, much like magnets of positive and negative charge.
So the question put to Physics is it better to have 9999.....99999
have no successor or is it
better to have 9999....9999 be the predecessor of 0?
I think the answer that Physics provides with its Duality and Quantum
Logic is that 999....9999 is the predecessor of 0 and that 0 is the
successor of 9999....99999.
It is what gives mathematics its cyclicity.
And it is what makes Elliptic Geometry.
Also, I should have used binary system, in that the South Pole now
becomes
10000.....000000 and the predecessor of 0 is 11111....111111
I should also note that Hensel P-adics confirms this in the fact
that ....99999r
is a substitute for -1 and that ....99998r is a substitute for -2 so
all the negative
numbers exist as this sequence ...., ....9998, ....9999, 0, 1, 2,
3, ....
And that makes better sense in an Atom Totality theory, where the
universe cycles back and
forth between becoming Cosmic Atoms and then back to the starting
point of becoming
a Neutron or a Hydrogen Atom Totality and then recycling upwards to a
Plutonium Atom
Totality and beyond only to cycle back down to a Neutron Totality.
Also, notice that in Binary representation of All-Possible-Digit-
Arrangements we
do achieve a symmetry since 1111....1111 is a reverse of 0000....0000
by substituting
1s for 0s and in this manner we see that 1/2 of all the numbers are a
reverse of the
other half. So that we can say that every point of a hemisphere
longitude has a symmetric
reverse on the other side of that same longitude.
So I definitely think that as Physics weighs in on the question of 0
having no predecessor
and 1111....1111 binary having no successor, is answered by saying
that Physics
knowledge would say that 0 was the successor of 1111....1111 and
1111....1111 the
Archimedes Plutonium
just correcting the Peano
Axioms.
And whether I do a solo book on Peano Axioms or include it in this big
book on overall
Correcting Math, I should start the discussion of Peano Axioms by
travelling through the
mind of someone who just learns the Peano Axioms and is thus
brainwashed and blinded
by not comprehending all the flaws and fakeries of what he/she just
thought they had
learned and understood. So let me travel through the mind of a typical
person having
learned the Peano Axioms and the Natural Numbers thereof.
Let me also remark about a poster I once saw. I think I saw it in the
University in the
1970s of a poster where the student goes to College and is met with by
a team of
surgeons that carefully saw into his cranium and so to speak "lift the
lid open" as seen
in the next frame. And then a frame in which a bunch of assistants
carrying some huge
pot of liquid and then pouring it into the student's cranium. Then the
student is sewn
up and graduates with a College degree. Now, many would be cynical and
adverse to this
poster but the poster rings with alot of truth to it. Especially in
the sciences, where more
often than not, that these educated people take their education so
inflexible that they
are never able to question what was poured into them as "education and
knowledge."
Especially mathematics in that if a student sees something in one book
and two or
three books then for the remainder of their lives, no matter how wrong
those tidbits are,
can never seem to examine or question those tidbits. I call it the
dogma of education
or the turning of science into a religion.
But now let me travel through the mind of some individual student who
has just learned
the Peano Axioms and the Natural Numbers they represent. So they learn
about a Successor axiom and an axiom that creates 0 (should create
both 0 and 1 together) but we will
not labor that point. So the student sees that we have a 0 and this
Successor that adds 1 to
0 and delivers a new number of 1. And continuing it delivers another
new number by adding
1 of the Successor function and arriving at 2. Now the student
realizes that this is a continuing
adding of 1, and endless adding of 1, delivering the Counting Numbers.
And those numbers never stop because if you stop, you can add another
1 to it and have a new number.
Now we reach a point in the mind of this student that the student
realizes since we never stop
adding 1 that these numbers go on endlessly and go to infinity. And
this is what the brain and
mind of every student sees in learning the Peano Axioms. That these
numbers are unstoppable since you endlessly keep adding 1 more to the
previous. And so this student
has a picture of the Peano Natural Numbers as numbers that start with
0 then 1 then 2
and go to infinity. Now most students never bother about asking
anything about whether each of those numbers is a finite-number,
because, well Peano never raises that issue of
finite-number and it is not written in any of his axioms. So basically
travelling through the mind
of a typical student who learns the Peano axioms of the Natural
Numbers comes away with a
picture that these are the counting numbers and that they occur by
adding 1 endlessly to produce a infinite set.
But later on, this student begins to learn that there are two types of
numbers, one called
finite-number and another called an infinite-number and where they
pick up this new found
knowledge is usually those inquisitive when they learn what a Real
Number is. They
learn that a Real Number has a unique decimal representation and that
consists of a
finite-string leftwards of the decimal point and a infinite-string
rightwards of the decimal
point. And they associate that finite-string as a Peano Natural-
Number. Usually the student
never examines this Real Number finite string with the Peano Natural
Number and asks
a relevant question. Since the Successor is an endless adding of 1,
well, will that endless
adding trespass into infinite-strings leftward and not just sticking
to finite-strings.
Usually, no student of mathematics ever gets to be that curious. Most
everyone
drop the issue of why does not the Successor produce infinite-numbers,
and not
just so called finite-numbers. Most students are not bright enough to
question
what the textbooks tell them.
Or when this student learns about the Hensel p-adics. And the
student learns that the Hensel p-adics have the same Successor
function only it
is in the form of a Series addition of endless adding 1. But the
bizarre thing is that
the student learns that the p-adics starting with 0 then 1 then 2,
goes to ....99999
which they are taught is the same as -1. But does the student begin to
be curious
as to why in the Peano axioms the Successor is supposed to create only
so called
finite-strings of numbers and is not supposed to create a infinite-
string. But in
Hensel p-adics that very same endless adding of 1 creates 2222....2222
or
9999....9998 or 9999....9999.
Now, let us say that Student A is super curious and smart and asks
questions as
he/she is learning the Peano Axioms, and says, well, since the
Successor is endless
adding 1, that the numbers that are spit out of the machine that
creates them from
the Successor that those numbers are going to be both finite and
infinite strings, because
the Successor cannot help but produce or create infinite-numbers since
it is an infinite
adding.
This smart student also knows that the Series 1 + 1 + 1 + . . . .+1
although you stop
it at intervals and can create 0 , 1, 2, 3 but since it is infinite
means it also creates
4444....44444 or 8888....812345 or 9999....99999.
Now if Peano did not want any infinite-numbers in his Natural Numbers,
he should have
defined where he would like for the Successor to stop, so that the
numbers like
9999....9999 were not part of his Natural Numbers.
And this is where it is extremely difficult for students to tackle or
wrestle with. There is no
stopping point or juncture where you can say from this point onwards
are only infinite-numbers
and below are only finite-numbers. The only way to do that is to
define Finite-number as
a specially selected number such as 10^500.
Now the above is only my first attempt of explaining what goes on in a
typical mind of a
student when he/she learns the Peano Axioms, and thinks he/she
understands them
and accepts them. The reason I want to perfect this post by editing
and re-editing in the
future is because it gets at the heart of the problem of lack of
comprehension on the part
of students of math and especially on the part of instructors of math
who have been
brainwashed worse than the students, and like the cutting open of the
cranium and pouring
in the broth of a pot and calling that a degree in mathematics from a
College.
I mean, in a sentence, how much more stupid can one be, to think that
if you have
an endless adding of 1, that all your numbers are going to be finite-
numbers, finite-strings,
when a reasoned person can easily grasp that the endless adding of 1,
whether Successor
or Series, will fetch infinite-strings or infinite-numbers. It is
fascinating, that virtually every
mathematician today, was brainwashed and blinded into not realizing
that a Successor
function must yield infinite-strings; must yield infinite-numbers.
So I have my work cut out for me to get this post edited and re-
edited, where I travel
down the brain and mind of a student learning the Peano axioms and
discussing in
detail, how easy it is to get brainwashed and blinded.
P.S. the next time any person tries to persuade you that the Successor
cannot create
an infinite-number. Then just turn the tables on him by saying " so
you think that the
series 1 + 1 + ....+ 1 does not go to infinity but is equal to a
finite-number.
Archimedes Plutonium
Archimedes Plutonium wrote:
>
> P.S. the next time any person tries to persuade you that the Successor
> cannot create
> an infinite-number. Then just turn the tables on him by saying " so
> you think that the
> series 1 + 1 + ....+ 1 does not go to infinity but is equal to a
> finite-number.
>
Alright the last post talking about how students learn the Peano
Axioms with its
Successor Axiom and how they are beguiled or mislead into a lifetime's
worth of
fakery. How they are mislead into thinking they have an infinite set,
whose individual
members are all finite-specimens.
And the trouble seems to lie with the fact that Peano did not use
Cauchy and the long
history before the Peano axioms concerning the theory of Series. That
if Peano had
simply used the theory of Series, he could not have thus clouded the
mathematics
world with the fakery of Successor as delivering only finite-
specimens. Now it is interesting
that a contemporary of Peano did not miss the opportunity to create
another system of
numbers whose axioms use the Cauchy Series, rather than some
convoluted Successor
function and that was Hensel with the p-adics.
So I am going to search through Cajori's book "History of Mathematics"
1991, to try to assemble the dates, rather than using Wikipedia which
would have few dates and be more gap
ridden since not one author wrote the entries. Cajori writes on page
373 " The faults of his time found their culmination in the
Combinatorial School in Germany, which has now passed into
oblivion."
What I am going to spy into is the idea that if Dedekind and Peano and
others had fixated on a
Series rather than a Successor function for the Natural Numbers, that
the Natural Numbers would be on the path of truth instead of their
path of disgrace and that the Cantor era going into
the Cohen Continuum farce would have never happened. As Cajori notes
of one case of
oblivion in the 1800s, it is my belief that if the Natural Numbers
Axioms had been based
strictly on Series would have caused Dedekind and Peano to also have
to define precisely
finite-number along with infinite-number and thus mathematics by the
1900s to 1990 would
have been a far more "cleaner and logical mathematics" rather than its
present day murky
and filled with fakery.
So let me follow some of that history, because the trouble I am seeing
is that for some reason,
Dedekind and Peano and others abandoned the Series and came up with a
screwy and wretched function of Successor, when their best path was to
have taken the Series and
made that the axiom of the Natural Numbers.
Also, one can argue that when Peano used the Successor as a function
and not as a purely
Series concept, that not only would that cover up the needed outcome
that the Natural Numbers have both finite-numbers and infinite-numbers
but that Peano, by using the Successor function is using the very idea
of Natural Numbers as counting, so that it is a
circular definition or circular axiom. In other words, Peano
incorporated into his Successor Axiom what the Natural Numbers are,
when it is the axiom that is supposed to create the
Natural Numbers. You see what I am saying, is that Peano had a
preconceived notion of
Natural Numbers and incorporated his preconception into the Successor
Axiom which
is verboten. A modern day analogy is that Svante Paabo is trying to
sequence the DNA of
Neanderthal and he is verboten to have impurity contamination into his
extracted Neanderthal
samples, such as the DNA from the modern humans of their hair or skin
falling into the
test samples of Neanderthal. So contamination is a vital concern in
Neanderthal DNA
sequencing. In like manner, when Peano devised a Successor function
for the axiom to
build the Natural Numbers, rather than just purely using Series of
endless adding of 1, that
Peano had contaminated the axioms with a preconceived notion of what
the Natural Numbers
already were, before creating the Natural Numbers. The Successor
function of Peano, has
the Natural Numbers already existing, and thus a erroneous set of
axioms.
Archimedes Plutonium
Now Wikipedia is good for an overall perspective of Series in
mathematics, but not good for
tracking down some dates and those mathematicians who contributed to
the progress of
Series as a subject of mathematics.
--- quoting Wikipedia on some pitfalls of Series ---
Potential confusion
When talking about series, one can refer either to the sequence { SN }
of the partial sums, or to the sum of the series,
i.e., the limit of the sequence of partial sums (see the formal
definition in the next section) – it is clear which one is meant from
context. To make a distinction between these two completely different
objects (sequence vs. summed value), one sometimes omits the limits
(atop and below the sum's symbol), as in
∑
an
n
in order to refer to the formal series, that may or may not have a
definite sum.
--- end quoting Wikipedia on Series ----
Now it is not surprizing that Series history goes back to Ancient
Greece especially
with Archimedes of Syracuse with his method of exhaustion which is the
preCalculus.
And not surprizing that Series is a big part of mathematics since it
is the trigonometry
and Calculus. Series is a part of what is known in mathematics as
Analysis.
So it is rather surprizing and rather silly and ridiculous that when
Dedekind and Peano
set about to formalize the Natural Numbers that they would not use
Series as the
foundation of their axiom system but rather invent a newer concept of
Successor, for
which mathematics never had a "successor concept" before, and
beguiling in the fact that
"successor" has already incorporated the concept of counting numbers,
when it is the job and
duty of the axioms to create the counting numbers and not have them
incorporated within
the axioms themselves. So that the Peano Successor axiom is a circular
axiom and not a
legitimate axiom that creates the Natural Numbers. If Peano had used a
Series Axiom, which
does not have the preconceived idea of counting already incorporated,
then Peano would have
a far cleaner and logical axiom set. And by using a Series Axiom
rather than a Successor
Axiom, there would have been no hiding or escape from the fact that
some of the Natural Numbers are finite-numbers whilst others are
infinite-numbers. And so, Peano and Dedekind
would have been faced with providing a precision definition of finite-
number.
I contend that it would have been almost impossible for the technology
of mathematics during the time of Peano and Dedekind or even including
the time period of 1900 to 1990 for mathematics to have ever been able
to define finite-number, since Physics was not mature
until far after 1930s.
So as much as I begrudge Peano, Cantor, Dedekind, Cohen, Godel, and
many others
for failing to precisely define "finite-number" one can easily see
that since the times
in which they lived, they were technologically infeeble to define
finite-number. They did not have these items:
(1) Physics is master and math is subset, especially having Quantum
Mechanics
(2) they did not have frontview with backview of a number where
infinity fits into the middle.
But they did have Series and we can easily see that Hensel made use of
Series to create
the p-adics which have infinite-numbers. And we see that the Hensel p-
adics are what the
Natural Numbers should be like.
Archimedes Plutonium
proffer my own
speculation as to why Peano went off track, off course with a
Successor Function axiom
rather than to tie into the bulwark of history by giving this axiom a
Series definition. It would
be helpful if Peano left behind alot of notes, personal notes and
whether he gave any
autobiography or notes as to why he constructed a Successor Axiom
rather than a
Series Axiom.
--- quoting by typing from various pages: Cajori's book "History of
Mathematics" 1991 ---
More fortunate in reaching the public was A. L. Cauchy, whose Analyse
Algebrique of
1821 contains a rigorous treatment of series. (page 373)
G. Cantor began his publications in 1870; in 1883 he published his
Grundlagen einer allgemeinen Mannichfaltigkeitslehre. In 1895 and 1897
appeared in Mathematische Annalen his Beitrage zur Begrundung der
transfiniten Mengenlehre.2 (page 400)
In 1891 appeared under G. Peano's editorship, the first volume of the
Rivista di Matematica
which contains articles on mathematical logic and its applications,
but this kind of work was
carried on more fully in the Formulaire de mathematiques of which the
first volume was published in 1895. (page 408)
A new and powerful method of attacking questions on the theory of
algebraic numbers was
advanced by Kurt Hensel of Konigsberg in his Theorie der algebraischen
Zahlen, 1908, and in his Zahlentheorie, 1913. (page 445)
--- end quoting (by my own typing) from Cajori, on various pages the
dates of Cauchy, Cantor, Peano, and Hensel ---
Now the purpose of this outline of math history is that I suspect I
know why Peano and others
veered off course by using a Successor Function for the axioms of the
Natural Numbers
rather than use the more logical choice of the Series, for which the
Series was designed
to be the axiom that creates the Natural Numbers.
My speculation is that, whether we like it or not in science, that
humanity as a whole
is too spiritual, or, over-spiritual, and when it comes to a choice
between two theories
of science as to which is accepted first by the human civilization
that this spiritual or religious component of the general public or
the individual
scientist making the new theory, that the choices selected are those
that have more
"spiritual or religious connotations as the first endorsed theory of
that particular subject."
So for example, the first selected theory of creation was of course
Bible-religion creation
theory. The first selected theory of astronomy was not the
heliocentric theory even though
Ancient Greeks discovered and proved it to be true, but rather the
spiritual-religion theory
of geocentric. The first geology theory of continents was a static
continent theory much like
the static geocentric of astronomy because it is more conforming to
the prevailing social
religion and spiritualism, rather than the alternative of a
Continental drift theory.
And the first cosmology theory to be accepted would of course be the
Big Bang rather than
the Atom Totality which there is plenty of room to fit a religion god
into the Big Bang but in the
Atom Totality, god is the Atom Totality.
That is my hunch as to why mathematics veered off course from Cauchy
to Cantor to Peano
to Hensel. That the first time in math history we examine "infinity"
we of course, laden with
too much spiritualism and religion in the society as a whole would
come down onto acceptance of "infinity" as per Cantor that we lose
sight of truth and reality. That comes
Peano with a choice to use the Series for the Successor and the Series
thus shows us
that the Natural Numbers are a mix of finite-numbers and infinite-
numbers and that would be
rather-- anti-religious or anti-spiritual, for the finite is of
humanity and god is of the infinite.
And really ironic and funny as that the history of mathematics, by the
20th century, had come
to the silly situation that mathematics knew more about "infinity"
with Cantor and then Godel
and Cohen on continuum, that math espoused to have more knowledge and
insights into
infinity than mathematics knew about the lowly "finite", because,
well, anyone wanting a precision definition of finite-number, was out
of luck, because noone was about to offer what
finite meant, yet we have tomes of books waiting at every university
to saturate the student
with infinity.
Archimedes Plutonium
Up until the last several posts, I thought the Successor Axiom was
going to survive the
changes, but it looks as though it needs refurbishing also. It looks
as though Peano presumed the Natural Numbers as Counting Numbers
within the Successor function, whereas the Series
was the better means of defining succession. It is the Series that
makes no distinction as to
finite and infinite numbers and a Successor Function prejudices or
biases what the number
is. Hensel shows us that with the p-adic axiomatics using a Series
that the numbers are both
finite numbers and infinite numbers and that is how the Natural
Numbers should be also.
So here is a quick summary of the changes needed to make the Peano
Axioms a consistent
set.
(1) The creation of two numbers of both 0 and 1 and not just a solo 0.
This is needed since we
need a metric distance of 1 for the Series (successor axiom).
(2) Since the p-adics shows us that going to infinity is like that of
circling around a globe. Another way to think of this is that in
Physics, the Atom Totality does not go to infinity but
is in a elliptic geometry shape and that means if we go far enough we
end up coming to the
starting point. So that axiom of Peano that says 0 has no predecessor
is wrong. For that
the largest possible Natural Number of 9999....9999 when adding 1
becomes 0 again.
So that the successor of 9999....9999 is 0.
(3) Again, the Successor function needs to be replaced with Series so
that
1 + 1 + . . . . + 1 represents all the Natural Numbers beyond 0 of
finite stopping and that
this Series also included infinite-numbers.
(4) Define Finite-number as equal to or less than 10^500, anything
beyond is an infinite-number. Caveat: we can include a Incognitum as a
region between finite and infinite and
we can well-define the Incognitum as a territory in which Algebra is
still obeyed of its
operations but beyond the Incognitum, algebra is untrustworthy. We use
a 100-Model
where multiplication ceases at 11x9 since 12x9 is larger than the
largest number of 99.
So the Incognitum is about 10% of all the numbers of all-possible-
digit-arrangements
where 9999.....99999 is the last and largest Natural Number.
(5) Mathematical Induction is superfluous and can be derived as a
theorem from the
other axioms.
Now I need to talk about what the above does to my earlier book of AP-
adics where
I was trying to synthesis this geometry theory:
Eucl geom. = Elliptic geom unioned to Hyperbolic geom.
and where I said that was equivalent to the Number systems
of Reals, AP-adics, and Doubly Infinites.
However, I can see alot of changes to that theory, not the geometry
aspect
but to the Number Systems aspect. Because I have turned into a
*finitist*
and the Doubly Infinites or Reals or AP-adics no longer have much
meaning
since mathematics is confined to only 10^500 for meaning.
Now this helps that other book of mine because there is no longer any
need
for me to wrestle with Algebra because Algebra decays after it
reaches
10^500. It is meaningless to consider what is 5000....0000 x
6666....6666.
So Algebra is no longer a problem. But there is a further help in the
fact that
geometry is also confined to the finite region.
So in the equation above of Eucl = Ellipt unioned Hyperbolic we draw
a
small triangle on a sphere surface (which is elliptic triangle since
the sides
are concave outward) and on the underside of the sphere we draw the
corresponding hyperbolic triangle whose sides are concave inward and
so if we were to place
these two triangles together (unioned) they cancel their concavity and
yield a
Euclidean triangle. So this facet is made easier with the finite
restriction.
Nam Nguyen
> Nam Nguyen wrote:
>> Marshall wrote:
>>> On Jan 15, 11:43 pm, Archimedes Plutonium
>>> <plutonium.archime...@gmail.com> wrote:
>>>> I want to prove that the only way to well-define or
>>>> precisely define Finite is to pick a large number and
>>>> say that is the end of Finite.
>>>
>>> Anyone can define anything to be anything. The idea that
>>> there is only one right definition of something is a failure
>>> to understand what definitions are.
>>
>> Totally agreed with you on this. (Not that AP's "precise" definition
>> of "Finite" would make a lot of mathematical sense anyway).
>>
>> So are you with me that the currently widely accepted definition of
>> the "natural numbers" is *not* the only right definition?
>>
>> For instance, the following 2 definitions would be equally the right
>> ones (as well as the current one):
>>
>> Let F be the formula "There are infinite counter examples of GC"
>>
>> Def 1: The natural numbers = the current definition + that F is true.
>> Def 2: The natural numbers = the current definition + that F is false.
>>
>> Right?
>
> I can be said that the key shortcoming of Godel's work is that it doesn't
> recognize the truth of the meta statement:
>
> (I) For any concept as strong as that of the natural numbers, there are
> concepts such as F = "There are infinite [number of] counter examples
> of GC" that would be independent from [the original concept].
>
> It's truly an irony that this statement rhythms (i.e. sounds like) a
> statement in his Incompleteness work, when he tried to point out the
> weakness of relying on one "giant" formal system for proving _all useful_
> mathematical/arithmetical properties and relations.
>
> In other words, in pointing out the Incompleteness of mathematical
> provability of any one "giant" formal system, Godel ignored the
> Incompleteness of knowledge of any "giant" definition of "The Natural Numbers".
For lack of a better name, let's call such F a G2 (Godel-Goldbach) sentence,
reflecting a concept in L(PA). Similarly (I) would be called here G2IT
(Godel-Goldbach Incompleteness Theorem [of Knowledge]).
Certainly G2IT wouldn't rhyme well with GIT unless for *any* current
concept A of arithmetics (or "natural numbers"), we can demonstrate an
existence of an independent concept, say, F = G2(A) such that A + G2(A) and
A + ~G2(A) are both equally extended concepts of the natural numbers, which
one could choose as a new "the standard model" of say Q (i.e. a new "the
natural numbers"). Furthermore, for any new arithmetic A' = A + G2(A), or
A' = A + ~G2(A), there would have to be a new G2(A') similarly.
We'll demonstrate that for any concept A of "the natural numbers" such a G2(A)
would exist in subsequent posts. But here, as an introduction to G2IT, let's
briefly touch base on what we'd mean by a concept (e.g. G2) _independent_ of
an underlying arithmetic A.
***
Since Godel, we know that an F is undecidable(F and ~F are independent) in T
means provability-wise we can't know the fact, if F is genuinely so. The matter
would be resolved only through (subjective) interpretation known as model,
where we'd find 2 models of T opposing the truth of F. In other words, model
is a kind of "the-buck-stops-here" stigma in mathematical truth-knowledge, beyond
which certain truths are impossible. So then, how could we _reliably_ demonstrate
an concept (e.g.G2) independent from a foundational arithmetic A which is
purported to be "the standard" "model" of important arithmetic formal systems
such as Q or PA?
The answer, imho, would be *not* using Induction. L(PA) is L(0,S,+,*,<) and
'S', '+' are well known for being instrumental in carrying out concepts that
would depend on Induction. Then, if we have 2 formulas F and F' in which
F includes 'S' or '+' while F' doesn't and (concept wise) F' is a semantically
translated version of F, F could be a candidate for being independent from A.
For instance,
- instead of defining One as S0 we could have One = "the unique minimum number
that 0 is less than",
- instead of defining Two as SS0 we could have Two = "the unique minimum number
that One is less than",
- we could then define an even number e as Two*x for some x, ....
Ultimately we'd translate certain concepts related to GC into formulas that
are independent of Induction, in the sense that the translated formulas are
free of 'S' and '+'.
[To be continued...]
David R Tribble
>> Well, no. I was wondering how 999...999(10) and 111...111(2)
>> could both be evenly divisible by 9.
>
Archimedes Plutonium wrote:
>> 1 11
>> 1001 | 1111 111111.....
>> 1001
>> 1101
>> 1001
>> 1001
>
Archimedes Plutonium wrote:
> 1001 <C
> 1001
I don't see what you're driving at. The next term in the
long division sequence is 1001, just like at <A>, which
gives us another '1' digit in the quotient, which is then
followed by the next term of 0001, and so on.
At no point do we ever reach a term that gives us a
zero remainder. Which should be obvious, since we
never run out of '1' digits to tack on to the next term
we're dividing. (Check it out for yourself, but do please
try to go farther than one step.)
So the quotient is 000 111 000 111 000 111..., repeating
without end. Which proves that 111...111(2) is not evenly
divisible by 9, and thus cannot be the same as 999...999(10).
David R Tribble
> The Peano Axioms are flawed and inconsistent because
> they require a Successor Axiom which builds this set
> {0, 1, 2, 3, 4, 5, 6, . . . . , 9999....9999}
Unfortunately, you have never demonstrated how this can
be so. Specifically, you've never explained what happens
in the ". . . ." ellipses following the '6'.
You have also never given a good explanation of what the
"..." in the last number is supposed to mean. Is it 500
'9' digits in a row? Or 10^500 digits? Or an unending sequence
of digits? Or what?
Archimedes Plutonium
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > The Peano Axioms are flawed and inconsistent because
> > they require a Successor Axiom which builds this set
> > {0, 1, 2, 3, 4, 5, 6, . . . . , 9999....9999}
>
> Unfortunately, you have never demonstrated how this can
> be so. Specifically, you've never explained what happens
> in the ". . . ." ellipses following the '6'.
>
Your juvenile mind has accepted 1 + 1 + 1+ . . . + 1 diverging to
infinity
so that means it is not a finite-number, yet simultaneously
your juvenile mind thinks {0, 1, 2, 3, . . . . } are all finite
numbers.
How come you hold such simultaneous contradictory beliefs?
> You have also never given a good explanation of what the
> "..." in the last number is supposed to mean. Is it 500
> '9' digits in a row? Or 10^500 digits? Or an unending sequence
> of digits? Or what?
You have never given any definition of finite-number versus infinite
number? You said you would, but apparently you fail at this also.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped in many spots)
>
> --- quoting by typing from various pages: Cajori's book "History of
> Mathematics" 1991 ---
>
Let me quote two more sentences out of Cajori "History of Mathematics"
1991. The reason I choice Cajori is because he offers us the mindset
of the
mathematicians as history unfolded. So often history of math books are
in a lecture mood of the contents of what happened. I want a history
more
of what compelled mathematicians of their own time period to think
they were
on a correct path. I want more of a social environment that they lived
in and
what motivated them and their mindset. And Cajori is often good at
revealing
such environmental conditions of these past mathematicians.
--- quoting Cajori ---
" In some of the textbooks in common use in this country, the symbol
infinity is still
used as if it denoted a number, and one in all respects on a par with
the finite numbers.
(page 398)
When G. F. B. Riemann gave an example of a function expressed
analytically which was
discontinuous at each rational point, the need of a more comprehensive
theory became
evident. (page 400)
--- end quoting Cajori ---
> My speculation is that, whether we like it or not in science, that
> humanity as a whole
> is too spiritual, or, over-spiritual, and when it comes to a choice
> between two theories
> of science as to which is accepted first by the human civilization
> that this spiritual or religious component of the general public or
> the individual
> scientist making the new theory, that the choices selected are those
> that have more
> "spiritual or religious connotations as the first endorsed theory of
> that particular subject."
>
Let me expand on that conjecture of what mathematics becomes socially
accepted
by the math community and the human community at large. The reason
that
Peano, in my opinion did not use the Series definition for the
Successor axiom
is because it would immediately cause him hardship because of the
questions of
why exclude infinite-numbers and have his set only be finite-numbers.
It is not because
Peano, in my opinion, was scared of fighting or causing a fight, but
that rather,
why cause a fight and have your chances of acceptance of the axioms in
doubt.
It is my hunch that at the time of Peano with his axioms of Natural
Numbers, someone
else during that time period offered up a axiomatics which included a
Series definition
rather than a Successor definition. And because of my conjecture that
the theory which
is in most accord or harmony of the existing religion and spiritual
order of that time
period is the theory that is going to be printed and published as the
truth of that time
period and all other contenders are going to be waylaid or ignored.
Also, above I quoted a sentence from Cajori of probably what sent
Cantor off on chasing
after infinite sets. The Riemann function that was discontinuous
everywhere is likely to
have been Cantor's model for his erecting hierarchies of infinity.
Now, transporting into the
future with AP, what sent me off onto finitism and that finite be
defined as 10^500 is Physics,
the entire subject of Physics is deplete of any actual-infinity.
Physics even goes out of its
way to renormalize away any infinity. There is no infinity in physics
nor any other science.
It is only in mathematics, a subset of physics, that we encounter
infinity as if it really existed.
I believe infinity exists as a potential and is a dual of finite, but
I think that all of Cantor's
work on infinity is going to go into the dump trash.
I also quoted a sentence from Cajori that shows that mathematicians
have thought of defining
Finite-number. But I wonder if Cajori ever came across any
mathematician throughout the history of mathematics that sat down and
gave a precision definition of finite-number. I doubt
it. I doubt that Cajori ever came across a precision definition of
finite-number. Even though
Cajori mentions "finite number" on page 398. Certainly Peano never
defined what is a finite-number, yet assumed all his Natural Numbers
were finite-numbers. I do not know if Cajori is
still alive and able to answer that question.
Archimedes Plutonium
Well, part of the above is worth saving to show readers who never
really
think about what they believe in and who are quick to raise silly
stupid questions.
Who have an education that is mostly rote memory and what they think
others
will say about a topic. Rather than search through their beliefs to
see whether they
really stand up. As Socrates said, the unexamined life is not worth
living. And apparently
Tribble is more apt to raise silly questions and never examine his own
contradictory beliefs.
David R Tribble
>> The Peano Axioms are flawed and inconsistent because
>> they require a Successor Axiom which builds this set
>> {0, 1, 2, 3, 4, 5, 6, . . . . , 9999....9999}
>
David R Tribble wrote:
>> Unfortunately, you have never demonstrated how this can
>> be so. Specifically, you've never explained what happens
>> in the ". . . ." ellipses following the '6'.
>
Archimedes Plutonium wrote:
> Your [juvenile insult] mind has accepted 1 + 1 + 1+ . . . + 1 diverging to
> infinity so that means it is not a finite-number, yet simultaneously
> your juvenile mind thinks {0, 1, 2, 3, . . . . } are all finite numbers.
> How come you hold such simultaneous contradictory beliefs?
What contradiction?
1 = 1
2 = 1 + 1
3 = 1 + 1 + 1
...
n = 1 + 1 + 1 + ... + 1 (n times)
Every number in the set {0, 1, 2, 3, ...} is the sum of a fixed
number of '1' terms. Thus every number in the set is a finite
sum.
At no point in the set is there ever a number or sum of an
unending number of '1' terms. Yet the sum 1+1+1+... is a sum
of unending terms, therefore it does not exist in the set.
Furthermore, it cannot be a finite number, otherwise it would
be a member of the set.
So where is the contradiction?
David R Tribble
>> You have also never given a good explanation of what the
>> "..." in the last number is supposed to mean. Is it 500
>> '9' digits in a row? Or 10^500 digits? Or an unending sequence
>> of digits? Or what?
>
Archimedes Plutonium wrote:
> You have never given any definition of finite-number versus infinite
> number? You said you would, but apparently you fail at this also.
So you're saying that you can't explain what your new notation
is until I explain to you some terms that have been written
down for 150 years?
Archimedes Plutonium
David R Tribble wrote:
> Archimedes Plutonium wrote:
> >> The Peano Axioms are flawed and inconsistent because
> >> they require a Successor Axiom which builds this set
> >> {0, 1, 2, 3, 4, 5, 6, . . . . , 9999....9999}
> >
>
> David R Tribble wrote:
> >> Unfortunately, you have never demonstrated how this can
> >> be so. Specifically, you've never explained what happens
> >> in the ". . . ." ellipses following the '6'.
> >
>
> Archimedes Plutonium wrote:
> > Your [juvenile insult] mind has accepted 1 + 1 + 1+ . . . + 1 diverging to
> > infinity so that means it is not a finite-number, yet simultaneously
> > your juvenile mind thinks {0, 1, 2, 3, . . . . } are all finite numbers.
> > How come you hold such simultaneous contradictory beliefs?
>
> What contradiction?
>
> 1 = 1
> 2 = 1 + 1
> 3 = 1 + 1 + 1
> ...
> n = 1 + 1 + 1 + ... + 1 (n times)
>
Your "n" is wrong for you use the three-dot ellipsis saying it is
infinite
yet you say it is "n times". So that is contradictory, but you
probably
never perceived that contradiction.
If you had studied Series you would know that "n times" looks like
this
n = 1 + 1 + 1+ . . +1_n
Notice the two-dot ellipsis.
The above is
infinity = 1 + 1 + ...+ 1
> Every number in the set {0, 1, 2, 3, ...} is the sum of a fixed
> number of '1' terms. Thus every number in the set is a finite
> sum.
So your saying Peano Axioms go to "n" and not to infinity.
It would help you if you ever defined finite from infinite with
a precision definition. Something you have been unable to do
in all your postings. And why have you been unable? Because
all of mathematics has never precisely defined finite.
>
> At no point in the set is there ever a number or sum of an
> unending number of '1' terms. Yet the sum 1+1+1+... is a sum
> of unending terms, therefore it does not exist in the set.
> Furthermore, it cannot be a finite number, otherwise it would
> be a member of the set.
>
Again, you are saying the Peano axioms go to "n" and never to
infinity.
> So where is the contradiction?
So where have you provided any truthful information, other then your
gaggle
of false beliefs. You have never provided enough truthful information
to warrant
a conversation with you. I get tired of having to correct every
sentence of yours.
This does not make for a dialogue or conversation.
David R Tribble
> Wikipedia writes:
> In mathematics, a set A is Dedekind-infinite if some proper subset B
> of A is equinumerous to A. Explicitly, this means that there is a
> bijective function from A onto some proper subset B of A. A set is
> Dedekind-finite if it is not Dedekind-infinite.
> --- end quoting Wikipedia ---
>
> Also, looking at Wikipedia defining finite-number is the pitiful
> definition of "not infinite."
>
> This is probably the worst cesspool in mathematics where definitions
> are given which are void of meaning--
>
> Transcendental-number -- not algebraic
> Finite-number -- not infinite
>
> When mathematics does something like this, you wonder if everyone went
> on vacation
> and left the shop to be attended by goats and dogs.
I'm curious as to why you're having so much trouble understanding
those
definitions.
"A Dedekind-infinite set is a set that can be placed in a one-to-one
correspondence with some proper subset of itself". It seems that you
don't have any problem with that part.
"A Dedekind-finite set is a set that is not Dedekind-infinite". That
seems to be where your difficulty in understanding lies. It seems
that you're having trouble seeing what it means when a set is "not
Dedekind-infinite". But it's fairly easy to tease out the meaning.
If a Dedekind-infinite set has a proper subset that it can be placed
in bijection with, then obviously a set that does *not* have a subset
that it can be bijected with must *not* be a Dedekind-infinite set.
So clearly a set that is Dedekind-finite is *not* Dedekind-infinite,
which means that it does not have any subsets that it can be bijected
with. That seems pretty simple.
Some obvious examples come to mind, e.g., any finite set of counting
numbers, such as {1, 2, 3}. That particular set has 3 members, and
2^3 = 8 subsets, 7 of which are proper subsets of it. None of those
proper subsets has 3 members, though, so none of them can be bijected
with the original set. So we conclude that the original set is not
Dedekind-infinite, so therefore it must be Dedekind-finite.
Perhaps you could point out where your problems with the Dedekind
definitions lie in the context of the example above?
David R Tribble
>> Your [juvenile insult] mind has accepted 1 + 1 + 1+ . . . + 1 diverging to
>> infinity so that means it is not a finite-number, yet simultaneously
>> your juvenile mind thinks {0, 1, 2, 3, . . . . } are all finite numbers.
>> How come you hold such simultaneous contradictory beliefs?
>
David R Tribble wrote:
>> What contradiction?
>>
>> 1 = 1
>> 2 = 1 + 1
>> 3 = 1 + 1 + 1
>> ...
>> n = 1 + 1 + 1 + ... + 1 (n times)
>>
>> Every number in the set {0, 1, 2, 3, ...} is the sum of a fixed
>> number of '1' terms. Thus every number in the set is a finite
>> sum.
>
Archimedes Plutonium wrote:
> So your [sic] saying Peano Axioms go to "n" and not to infinity.
Um, I'm saying that the set of natural numbers contains only
natural numbers, and does not contain any infinite numbers.
For any given natual number n which is in the set, its
successor S(n) = n+1 is also in the set.
From that we can conclude several things:
that the set contains only natural numbers;
that every successor of a natural number is also a natural number;
that there is no "last" or "largest" natural number, because such
a thing would not have a successor and so it could not be a
natural number.
And so on.
> It would help you if you ever defined finite from infinite with
> a precision definition. Something you have been unable to do
> in all your postings. And why have you been unable? Because
> all of mathematics has never precisely defined finite.
Non sequitur. You say I can't define "finite" because all of
mathematics has never defined it. If that's true, then by the
same logic, you can't define it, either.
David R Tribble wrote:
>> At no point in the set is there ever a number or sum of an
>> unending number of '1' terms. Yet the sum 1+1+1+... is a sum
>> of unending terms, therefore it does not exist in the set.
>> Furthermore, it cannot be a finite number, otherwise it would
>> be a member of the set.
>
Archimedes Plutonium wrote:
> Again, you are saying the Peano axioms go to "n" and never to
> infinity.
If you mean the Peano set is not an infinite set, then, no,
because the Peano set is indeed infinite.
If you mean that the Peano set does not contain any infinite
numbers, then, yes.
Since you apparently believe the opposite is true, and since
it is not very obvious how the Peano axioms produce such a
thing as an "infinite number", then it is incumbent upon you to
demonstrate how they do, in fact, result in a member of the
natural numbers that is unlike all the other naturals, i.e., not
finite.
If you simply mean that there are naturals larger than 10^500
(your definition of an "infinite number"), then yes, obviously,
that's true. If you mean something else, then you'll have to
show that can be.
David R Tribble wrote:
>> So where is the contradiction?
>
Archimedes Plutonium wrote:
> So where have you provided any truthful information, other then your
> gaggle
> of false beliefs. You have never provided enough truthful information
> to warrant
> a conversation with you. I get tired of having to correct every
> sentence of yours.
> This does not make for a dialogue or conversation.
Well, that certainly makes it easier for you to not provide any
meaningful answers to our questions, doesn't it?
Archimedes Plutonium
Why resort to liaring in sci.math? Finite-number has never been
precisely
defined in mathematics until I said it was the largest Planck Unit at
10^500 and above that is Infinity where we can sandwich in between
a Incognitum.
So, never before in the history of mathematics has anyone given a
precision definition of Finite number versus Infinite number until AP
did so.
David, do you remember when I precisely defined Finite-number?
Probably not.
But I defined it due to a conversation with a bozo named David
Tribble, and I remember
where Lwalk burst into the conversation with panic and fear that I had
"truly meant it."
I guess Lwalk at that time was scared that I would abandon the
"infinite" and which
his fear would be confirmed, only it took me some years after the fact
to abandon infinity.
And Lwalk was probably scared that -- I believed it myself, that I had
picked a large number and said that is the end of finite. I am too
lazy to search back for those posts. Probably in 2007 or 2008 or 2009
when this had occurred in sci.math. There is a written record of these
posts, so it is easy to confirm. So it was a bozo like you, David,
that
propelled me to discover a precision definition of "finite number" and
it is exactly for
that reason that I continue to talk to you even though your posts are
mostly dust
binnish. It is for those rare moments such as the discovery of the
true and precise
definition of finite, that I talk to you.
Now let us return to your liaring or confusion above of "150" years.
You said in some post about a week ago that you can precisely define
finite-number without set theory.
Yet you continue to not reveal that definition. The minimum
requirement for anyone to
be in this thread is that they be able to offer forth their precision
definition of "finite number"
just as the minimum requirement for entering the proof of Euclid's
infinitude of primes proof
when I was doing that, was to be able to offer their own version.
So far, David Tribble, you have not been able to meet the minimum
requirement of engaging
in a conversation in this thread, you duck and hem and haw about
defining "finite number"
but fail to do so.
Let us compare you David with Peter Nyikos in this same thread
concerning the definition
of finite-number. Now Peter is a working mathematician and you David
is a bloke who took
some courses in math in school. And Peter says the best he can do for
the time being
here is that finite-number is
refer to "intuit" like a primitive concept where "finite number" is
like "time or space" intuit.
Now David on the other hand, claims he can post a precision definition
of "finite-number"
without using set theory, but has never provided. And now David says
this precision definition
has been around for 150 years.
So who is a reader to believe, the working mathematician of Peter
Nyikos who talks about
"intuit and time" or the bloke David Tribble who dabbles in math and
claims a precision definition of finite-number is 150 years old.
Maybe you gloat in looking so silly in sci.math, David, but I already
told you why I keep
talking to you when possible. Because your gaggle of nonsense,
sometimes, although rare, propells me into new directions.
Archimedes Plutonium
David R Tribble wrote:
(snipped the garbled nonsense)
>
> Archimedes Plutonium wrote:
> > So your [sic] saying Peano Axioms go to "n" and not to infinity.
>
> Um, I'm saying that the set of natural numbers contains only
> natural numbers, and does not contain any infinite numbers.
>
> For any given natual number n which is in the set, its
> successor S(n) = n+1 is also in the set.
>
The number 0000....99998 has a n+1 and it is 0000....99999 which has a
n+1
and it is 0000....00010000....0000 and it has a n+1 and it is
0000....00001000....0001
I can work backwards with 0000.....99998 for it has a predecessor of
0000....99997
which in turn has a predecessor of 0000....99996, all going down to
the number
6, then 5, then 4 then 3 then 2 then 1 then 0.
I could go on forever. But the dullard David Tribble is too blind to
realize that since
he never gave a definition of finite-number that would eliminate
0000....999 as a
Peano Natural Number. Since David never defined finite-number from
infinite-number
that David can only say that the Peano Axioms are an inconsistent set,
a fakery set.
I deleted the rest of David's waffling on. Until David posts a
precision definition of
finite-number there is no point in answering any more of his crank
nonsense.
Archimedes Plutonium
On Jan 19, 2010 11:45 PM
David R Tribble wrote:
>> And for the record, it's pretty easy to provide an understandable
>> definition of "finite number" without having to mention sets.
>
So what is up today David, are you in a liaring mode or are you super
confused and dazed.
You keep saying that a precision definition of finite-number exists
for 150 years and
that you can provide it without mentioning "sets". But apparently you
cannot.
As I said, the minimum requirement to engage in this thread is to
provide a precision
definition of finite-number. And you failed not once, not twice but
three times now.
So are you trying to liar to get by or just so rattled confused and
dazed?
So you need to exit this thread as a failure. You cannot learn from me
because you
hate my guts too much, but you can learn from Peter Nyikos who earlier
in this thread
teaches you-- David Tribble, that the best he can do for defining
finite-number is an
intuit of analogous to "time". Believe me, David that if Peter, a
working mathematician
can do only "intuit" that your silly search for finite-number is sure
to fail.
Brian
the number of ways of arranging these arrangements. Continue long
enough and you'll eventually surpass the "largest number" in finitely
many steps.
This might prove useful to your investigation :
http://en.wikipedia.org/wiki/Finite_set
Descriptors in mathematics may not be how those descriptors are used
outside of mathematics. As an example, a number is called "rational"
iff certain criteria are met. However, outside of mathematics, if
someone is irrational, then they are in some way psychologically
unstable. Assuming that numbers don't possess minds of their own, a
rational number does not literally mean it is psychologically stable/
healthy.
Mathematical descriptors are not compelled to coincide with either
intuition or how those descriptors are used in the natural language.
The example at hand is the descriptor 'finite'.
The intuition on the word is that if you have a group of marbles and
if there is a basket big enough to hold them, then that group is
finite.
We might as well classify numbers as Type I and Type II where Type II
would be the property "not Type I."
Using these generic descriptors, we can move past the baggage
associated the word 'finite'
Nam Nguyen
You've challenged few of us to come up with the definition of "finite-
number" or else you'd go on with your ignorant babbling. So here they are,
the definition of properties Finite(x) and Infinite(x):
P(x) <-> Ey[y <= x)
(*)P(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)]
Finite(x) <-> ~(*)P(x)
Infinite(x) <-> ~Finite(x)
Can you quit babbling now?
Nam Nguyen
Exactly. Apparently AP doesn't know what "contradiction" means, in mathematical
contexts.
David R Tribble
>> You have never given any definition of finite-number versus infinite
>> number? You said you would, but apparently you fail at this also.
>
David R Tribble wrote:
>> So you're saying that you can't explain what your new notation
>> is until I explain to you some terms that have been written
>> down for 150 years?
>
Archimedes Plutonium wrote:
> Why resort to liaring [sic] in sci.math? Finite-number has never been
> precisely
> defined in mathematics until I said it was the largest Planck Unit at
> 10^500 and above that is Infinity where we can sandwich in between
> a Incognitum.
>
> So, never before in the history of mathematics has anyone given a
> precision definition of Finite number versus Infinite number until AP
> did so.
Wow. That must come as a shock to everyone who's been using
the terms that were rigidly codified back in the 1880s.
> So far, David Tribble, you have not been able to meet the minimum
> requirement of engaging
> in a conversation in this thread, you duck and hem and haw about
> defining "finite number" but fail to do so.
Your standard response to simple questions is juvenile insults.
Providing any more grist for your illogical, useless, and boring
personal attacks would seem to most sane people to be a waste
of time.
David R Tribble
> [...] I could go on forever.
Apparently not, at least according to your own theories of infinity.
Archimedes Plutonium
Brian wrote:
> You have the number of ways q quanta can be arranged. Next you have
> the number of ways of arranging these arrangements. Continue long
> enough and you'll eventually surpass the "largest number" in finitely
> many steps.
>
Well, Brian, mathematics first job is to define precisely what finite-
number is.
Math has failed to do that and as a consequence we have thousands of
unsolved
and unsolvable problems in mathematics number theory. Not because they
are
difficult problems, but only because the concept of finite-number is
crucial to those
problems and yet finite-number is ill-conceived.
Example: in geometry we define the rectangle with precision for
Euclidean geometry,
and as a consequence there is no backlog of unsolved problems
concerning rectangles
in Euclidean geometry. Why is that? Because we have a precision
definition.
Now look at mathematics Number theory and there sits the worlds oldest
unsolved
problem going back to Ancient ancient Greeks of the perfect numbers
conjecture. It
is a simple to understand problem and there is nothing in any of the
other sciences that
even remotely resembles the unsolvability of Perfect Numbers. Now most
everyone in
the world including you, probably, thinks that Perfect Numbers is just
a tremendously
hard problem. But I think the truth of the matter is that because
finite-number was
never well-defined, never given a precision definition, that Perfect
Numbers could never
and will never be proven.
Now, let us say tomorrow we define Finite Number precisely as 10^500
or below
and beyond is Infinite-numbers. Then in a weeks time, we can check
through all those
numbers and announce that Perfect Numbers was proven true.
You see, when math has trainloads of murky defined concepts gives rise
to unsolved
and unsolvable problems. Finite Number is math's most horribly defined
concept and that
is why math has to reach back to Ancient Greeks with still Perfect
Numbers open. Physics
nor chemistry, nor biology, nor geology, nor astronomy nor medicine
nor any other science
has to reach back to Ancient Greeks for solving some problem started
there. Because all of those sciences have made more precise
definitions of their concepts than the Ancient Greeks
ever dreamed about. However, math, the only science at present day,
still has to go back to
Ancient Greeks to try to solve the Perfect Numbers.
So, Brian, do you really think that the concept of finite-number was
ever given a precision
definition?
> This might prove useful to your investigation :
> http://en.wikipedia.org/wiki/Finite_set
>
Finite-number, not finite set. Numbers and sets are independent in
this topic
just as if I were talking about a precision definition of finite-line,
I would not
look for set theory to render a definition of finite-line. To define a
finite-line from
an infinite-line I would do the same as with numbers. I would look to
Physics
for the largest number possible in Physics which is the Couloumb
Interactions
of element 109 of a number about 10^500. So I would define a finite-
line as
all lines of that number or less and any above that number are
infinite lines.
I could have a Incognitum region of lines sandwiched in between finite
and infinite.
Now whether you like that or not Brian that is a **precision**
definition for there
are never any doubt as to whether a line is finite or infinite. Now
you may ask is that
10^500 cm or meters or kilometers or what. And I would answer you that
in Physics
there is no need for any number beyond 10^500 in any physical physics
measurement
or experiment so it does not matter whether you want kilometers or
millimeters. If I
remember correctly the Planck Unit for distance is very far below
10^500.
The point is Brian-- precision precision precision and when you say
10^500 you have
fulfilled the precision task.
> Descriptors in mathematics may not be how those descriptors are used
> outside of mathematics. As an example, a number is called "rational"
> iff certain criteria are met. However, outside of mathematics, if
> someone is irrational, then they are in some way psychologically
> unstable. Assuming that numbers don't possess minds of their own, a
> rational number does not literally mean it is psychologically stable/
> healthy.
>
Precision definition is far more of a task than your mere ideas of
descriptions.
Precision definition eliminates ambiguity and confusion.
Is the number 0000.....99999 a finite-number or a infinite-number?
According to
present day mathematics, noone knows. Noone in the math community has
a definition
of finite-number to decide, except me, who says that 10^500 is finite
and that number
is thus infinite-number. But everyone else in the math community must
admit that
0000....99999 is a finite number since it ends in zeroes to the
leftward of the string
just as 99 is finite because it is 000000.....0099
So we are talking about precision, not loose descriptions.
> Mathematical descriptors are not compelled to coincide with either
> intuition or how those descriptors are used in the natural language.
>
> The example at hand is the descriptor 'finite'.
>
> The intuition on the word is that if you have a group of marbles and
> if there is a basket big enough to hold them, then that group is
> finite.
>
> We might as well classify numbers as Type I and Type II where Type II
> would be the property "not Type I."
>
> Using these generic descriptors, we can move past the baggage
> associated the word 'finite'
Oh, so you are saying much of what Peter Nyikos says that finite-
number
is a intuit like that of "time or space".
Well, math primary job is precision and not your foggy intuition or
descriptions.
Math did a precision on defining rectangle or square or circle or
tangent or
prime number or derivative or integral.
So why become derelict and lazy about a precision definition of finite-
number.
Nam Nguyen
>
> Brian wrote:
>> You have the number of ways q quanta can be arranged. Next you have
>> the number of ways of arranging these arrangements. Continue long
>> enough and you'll eventually surpass the "largest number" in finitely
>> many steps.
>>
>
> Well, Brian, mathematics first job is to define precisely what finite-
> number is.
Well, AP, the last time one look a (making-sense) definition has existed.
> Well, math primary job is precision and not your foggy intuition or
> descriptions.
No kidding. But you forgot that a a mathematical definition *must* also
make mathematical sense. If I define a finite number as one no greater than
1, would that make sense.
Similarly, your definition of a finite number as one no greater than 10^500
doesn't mathematically make sense.
> Math did a precision on defining rectangle or square or circle or
> tangent or prime number or derivative or integral.
OK.
>
> So why become derelict and lazy about a precision definition of finite-
> number.
It's you who's derelict and lazy to learn that one can easily come up
with a sound definition of a finite number, and that there have *actually*
been more than 1 or 2 ways of defining it!
Archimedes Plutonium
Nam Nguyen wrote:
>
> You've challenged few of us to come up with the definition of "finite-
> number" or else you'd go on with your ignorant babbling. So here they are,
> the definition of properties Finite(x) and Infinite(x):
>
> P(x) <-> Ey[y <= x)
> (*)P(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)]
> Finite(x) <-> ~(*)P(x)
> Infinite(x) <-> ~Finite(x)
>
> Can you quit babbling now?
Yours is deaf dumb and silent as to whether 0000....9999 is a finite-
number
or an infinite-number. And, yours obviously fails wildly for the
Hensel P-adics. Basically
you have a "order induced definition" whereas the concept of finite
and infinite is grounded
in "metric or measure" not "order".
You need to go back to the drawing board and base a definition on
"metric and absolute value" not one grounded in "order".
My precision definition of 10^500 is based on Physics of Planck Units
that no physicist
ever needs to work with any number larger than that. That is a
grounding in Measure theory
such as "absolute value" of mathematics. Did you have a chance to
study measure theory
and absolute value in school? Or did you waste your time in scribbling
symbols of logic
that never get off the ground?
P.S. Hopefully, Nam, Peter Nyikos can come to your aid and rescue.
Archimedes Plutonium
Archimedes Plutonium wrote:
> Brian wrote:
( all snipped)
It is good that Brian posted those comments, because he set me to
thinking about
a precision definition of finite-line in Euclidean geometry. It has
never really been done.
We all call it a line-segment and we all assumed that it was finite
because it had two
ends and which were represented by Real numbers and which we could
take an
absolute value thereof. But we never really gave finite-line versus
infinite-line a precision
definition.
I think the precision definition of finite-line will guide the
precision definition of finite-number.
Now I looked up the Planck Unit for the Planck Length and found it to
be
1.616 x 10^-35 meters. Now if we take the inverse of that for the
Macroworld
of 1.6 x 10^35 meters I doubt that such is the maximum distance of the
entire
Universe. But I am certain that 10^500 meters is far and away the
total length of
the Universe and then some. I think the total lenght of the Universe
is reckoned
to be somewhere between 10^80 and 10^120 meters and so 10^500 meters
is
vastly longer than what physics reckons.
So here we have a simple means of defining Finite-line as a line in
Euclidean
geometry that is equal to or less than 10^500 meters.
Now when we draw any line-segment we automatically know it is finite-
line since
it is far less than 10^500 meters.
Now we cannot measure 10^500 meters because no physics experiment
could undertake
that project. So what is an infinite-line? It is most assuredly simply
one unit larger than
10^500 meters.
In other words, the concept of Finite line or finite number is marked
at a boundary by 10^500 and the concept of Infinity is simply a number
larger than 10^500.
Now with finite-line, unlike finite-number we really do not need a
Incognitum zone sandwiched
in between finite and infinity. We simply can never reach 10^500
meters anyway nor reach
10^-500 meters in the microworld. Since we can never reach that
extreme, it is easy to
recognize that Infinity is simply one more unit beyond 10^500.
Another comment is that there is no human nor any machine nor any
other physical entity that
can count to 10^500. By counting, let us say they write down each
number from 0 to 10^500. Since noone can do that, then 10^500 + 1
should be considered an infinite-number since it is
unobtainable. Just like the speed of light is unobtainable for any
object with mass.
So in this perspective we gradually begin to see that Infinity is not
necessarily mean
endlessness, although it can be endless, it rather means that Infinity
is just larger than
any finiteness.
This would reconcile the idea that the numbers 50000.....00000 and
9999....999 are both
infinite yet one is about 1/2 the other, just as both 5 and 10 are
finite yet one is twice the size of the other. So many people think
that infinity is a unique object that only one number
is infinite which the Peano Axioms promotes that false idea that this
set {0, 1, 2, 3, . . . .}
has only finite elements yet the entire set is alleged to be infinite
with only one
entity represented by sideways 8 as infinite.
In AP-adics we have all varieties of infinite-numbers such as
1212....3434 or 5555...8888
etc etc.
But when we define Finite Number and Finite-Line as that of 10^500,
then infinity no longer
is this mysterious entity but simply a reflection of merely going past
10^500 where Physics
can no longer take place.
So in a sentence, one can describe Infinity as to where Physics can no
longer use numbers
and that is the boundary at 10^500.
Archimedes Plutonium
Archimedes Plutonium wrote:
> Nam Nguyen wrote:
>
> >
> > You've challenged few of us to come up with the definition of "finite-
> > number" or else you'd go on with your ignorant babbling. So here they are,
> > the definition of properties Finite(x) and Infinite(x):
> >
> > P(x) <-> Ey[y <= x)
> > (*)P(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)]
> > Finite(x) <-> ~(*)P(x)
> > Infinite(x) <-> ~Finite(x)
Alright, let me use Nam's symbols to define both finite number and
finite line:
Finite-Number(y) <-> Ay[y <= 10^500)
Finite-Line(y) <-> Ay[y <= 10^500)
Infinite-Number(x) <-> Ax[x > 10^500)
Infinite-Line(x) <-> Ax[x > 10^500)
jmfbahciv
tidbits. Thanks.
/BAH
Nam Nguyen
Apparently you didn't know birds are dinosaurs. What a shame.
Nam Nguyen
>
> Archimedes Plutonium wrote:
>> Nam Nguyen wrote:
>>
>>> You've challenged few of us to come up with the definition of "finite-
>>> number" or else you'd go on with your ignorant babbling. So here they are,
>>> the definition of properties Finite(x) and Infinite(x):
>>>
>>> P(x) <-> Ey[y <= x)
>>> (*)P(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)]
>>> Finite(x) <-> ~(*)P(x)
>>> Infinite(x) <-> ~Finite(x)
>
> Alright, let me use Nam's symbols to define both finite number and
> finite line:
>
> Finite-Number(y) <-> Ay[y <= 10^500)
> Finite-Line(y) <-> Ay[y <= 10^500)
> Infinite-Number(x) <-> Ax[x > 10^500)
> Infinite-Line(x) <-> Ax[x > 10^500)
You don't know that Mathematics is abstract, do you?
Marshall
I think you missed the fact that he was honestly thanking you.
The "snort" was for the idea that AP would ever stop babbling,
which you have to admit is at least modestly far-fetched.
Marshall
Brian
<plutonium.archime...@gmail.com> wrote:
> Brian wrote:
> > You have the number of ways q quanta can be arranged. Next you have
> > the number of ways of arranging these arrangements. Continue long
> > enough and you'll eventually surpass the "largest number" in finitely
> > many steps.
Try again, this time actually addressing this point.
>
> Finite-number, not finite set.
A number is finite if it is the cardinality (cardinal number) of a
finite set. (If we're talking natural numbers.)
David R Tribble
>> You have never given any definition of finite-number versus infinite
>> number? You said you would, but apparently you fail at this also.
>
Nam Nguyen wrote:
>> You've challenged few of us to come up with the definition of "finite-
>> number" or else you'd go on with your ignorant babbling. So here they
>> are,
>> the definition of properties Finite(x) and Infinite(x):
>> P(x) <-> Ey[y <= x)
>> (*)P(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)]
>> Finite(x) <-> ~(*)P(x)
>> Infinite(x) <-> ~Finite(x)
>> Can you quit babbling now?
>
jmfbahciv wrote:
>> <snort> After pigs and dinosaurs fly. I enjoyed the useful
>> tidbits. Thanks.
>
Nam Nguyen wrote:
>> Apparently you didn't know birds are dinosaurs. What a shame.
>
Marshall wrote:
> I think you missed the fact that he was honestly thanking you.
> The "snort" was for the idea that AP would ever stop babbling,
> which you have to admit is at least modestly far-fetched.
Nam can be forgiven, though, since he obviously expected the
usual knee-jerk, post-before-thinking response from AP.
Which he did get, it's just in a different post.
Nam's problem is that he's using long-accepted mathematical
notation, instead of taking the path of genius and inventing his
own notation, terms, rules, and logic, like AP. He can't expect
anyone to know how smart he is if he keeps using the notation
of the old accepted math.
A
AP has already claimed (in this thread) that the notion of number is
not logically dependent on the notion of sets--evidently he doesn't
think numbers are what we use to count things.
Jesse F. Hughes
I'm afraid I don't quite get it.
A number is infinite iff (*)P(x) holds, right?
That is, iff Ey[y <= x] & AyEz[(y <= x) -> (z < y)].
Now, I assume that we know that (Ax)( 0 <= x ), right?
As well, (Az)~(z < 0), right?
Thus, for all x, ~Ez(( 0 <= x ) -> (z < 0)) and hence
~AyEz(( y <= x ) -> ( z < y )).
Hence, for all x, ~Infinite(x) and thus Finite(x) <-> P(x). (Note
that P(x) is trivially true, as well, since x <= x.)
Am I missing something here? You seem to have defined Finite(x) as
the always true predicate.
--
"Looking at their behavior I see them endangering not only their own
futures, but that of their families, and now, considering my latest
result, the future of people all over the world." -- James S. Harris,
on the shortsightedness of his mathematical critics