Re: Exact-Point paradox

Dear William Hughes and Laureano Luna:

for the number 0.999... , you cannot deny there are
INFINITE REAL
NUMBERS contained in this 0.999... every one of them
is bigger than
zero , which leads to k* 00 = 00
Before anyone can agree or disagree, you will have to say what you mean by one number "contained" in another.
And are you talking about NUMBERS or a particular decimal representation of a number? If you are talking about numbers themselves why not just say "there are INFINITE REAL NUMBERS contained in 1"? 0.999... is just a (slightly unusual) representation of 1.


( 00 means INFINITY )

better explination is this:

1) For 0.999... there are infinite 9s , every 9 of
them is a
positive real number by itself "no matter how small
it is "

2) Every 9 of them is NOT JUST A POINT, it is a
positive real number
bigger than zero, which means, if any one of those 9s
is represented
on real numbers line, then it has a start point and
an end point and
between those points there are infinite points like
any other real
number.

The "9"s in .999...are neither points nor numbers- they are numerals, symbols representing particular things.

If, on the other hand, you are talking about the numbers represented by .9, .99, .999, .9999, etc. then, yes we can think of them as points on the number line and there are an infinite number of points between each pair of them.

3) because there is an infinite number of positive
real numbers bigger
than zero " the 9s in 0.999.." , then in 0.999.. we
have the
addition of infinite positve real numbers which must
be INFINITE .
Where did you get that idea? Most students in pre-calculus learn how to sum infinite series- the sum of an infinte set of positive real numbers is NOT necessarily infinite. In a good secondary school algebra class you should learn that, if -1< r< 1 then the "geometric series", summing ar^n, although the sum of infinite numbers is the finite number a/(1- r). In particular, 0.999... is, BY DEFINITION, the sum 0.9+ 0.09+ 0.009+ ... which is a geometric series with a= .9 and r= .1 It's sum is .9/(1- .1)= .9/.9= 1.

4) from (3) there is a contradiction because
0.999... is a finite
real number which is 1

The "contradiction" seems to be based on your mistaken belief that an infinite sum cannot be finite.
A good algebra course should help.


HOWEVER, TO DECLARE MY STAND IN THIS ISSUE, I WILL
SAY THE FOLLOWING:

I think we can agree on one point: There is a big
mystery in this
issue,

No, the only mystery is your lack of knowledge in this subject.

" there is an exact point, there come points after
it,, but none of
them is
the immediate successor",when I made my post, my
y concern was not to
falsify any strong premise
in this issue, my real concern was to indicate there
are "lost
premises" in the big whole picutre,
and if we find those premises, then the whole issue
will get back to
be understood and make sense again.

problem with many mathematician is they are in love
with the statement
"things do not need to be understood to be true!"
?? I would not put it that way- the fact that YOU do not understand something does not mean it is false!
If you are saying there are "many mathematicians" who believe things to be true even if THEY do not understand them, please back that up by giving evidence: name some. I have never met any.

well this is just an escape button was made to give
e execuses for
those lazy who do not want to dig deeper
to find what make the whole issue clear and
understood,,,the target
for science is to make things understood.
and by the way ,in explaining those things that are
"un-understood"
there hide huge potentials. Even in explaining
little mysteries come very big revolutions in
sciences.

All the mathematicians I have met, if they do not understand something are not satisfied until they DO understand it.

Best example for this come from the great work of
Albert Einstein,
when "Lorentz transformation" could not explain
the mystery of light speed in Electromagnetic waves
equations," speed
of light looked constant regardless of
the observer speed" ,,, Einstein could just relax and
say " things do
not need to be understood to be true " ...
Okay, now you are naming a paraticular person (a physicist, not a mathematician). You may be misunderstanding what he said (I'd like to see a reference to that particular quote). It is certainly true of physics that the result of an experiment is "true" even if you do not understand why.

putting in mind no body could claim that
"Electromagnetic waves
equations" or " Lorentz transformation " no body
could
claim any one of them to be wrong. But Eistien
instead of using the
escape button , he was brave and persistence enough
to find the lost premise which made this mystery
disappear and make
things to be understood again ,,, the lost premise
was " Time is Relative not Absolute" and he did not
need to falsify
any solid premise or law.

Oh, wait, you are giving Einstein as an example of a person who did NOT believe something to be true even though he did not understand it? So, again, give some evidence for your statement about "many mathematicians".

I think the statement "things do not need to be
understood to be true"
is a TRUE statement but NOT COMPLETE,
the complete statement should be " things do not need
to be understood
to be true, but there most be premises to make them
understood in the whole big picture"

Okay, I would agree with that. I would add that one should then try very hard to find those "premises".

The main point is, for our "Exact-point" issue and
the mystery in it,
may be mathematicians have the privilege to say"
things do not need to
be understood to be true"
Once again, can you give any evidence that is true?

but people in physical
world unfortunately
do not have that privilege, because all physical
object consist of
mathematical points at their fundamental level " like
No. Nothing physical "consists" of any mathematical concept. Physical things can be "represented" by mathematical concepts, perhaps in many different ways.

the real numbers
line" , in fact real numbers line is a real physical
object in one
Absolutely NOT. The "real number line" is a mathematical concept, not a physical object.

dimension. So the problem of the existence of an
exact point and there
is no immediate successor, while at the same time
there are points
which come after that exact point,,, that mystery is
a real mystery in
the real physical world , and you cannot claim that
things are
abstract and we should not give much concern ,, you
cannot claim that
for real things in the real world,,,,
No mathematician claims anything about the real world.

from this,there came Zeno Paradoxes which in fact
t made good
claimsthat movement never happens ,,, and those
paradoxes were very
annoying for a long time. However, since the emerge
of Calculus ,,,
those kind of paradoxes set backward because we now
have the tool to
tell the exact output and say such movement happens,,
but the
philosophical part still never solved... it is still
I'm not sure what "philophical part" you are talking about. The "philosophical" point that movement exists does not require calculus- just get up and walk across the room!

a big mystery
HOW movement happen from one point to another point
through streams of
points and without passing any point as the first
point or any finite
order. You can read more about this subjectcalled
"the super task" and
how tricky it is, in Stanford University site:

http://plato.stanford.edu/entries/spacetime-supertasks
/

Interesting reference. Are you sure you understood it? It pretty much destroys your claims!

The exact-point issue , is just a reflection for the
basic dilemma of
seeing one object as finite .. and at the same time
that object is
consisted of infinite objects " the finite infinite
problem" ,,a
property and its negation at the same time for the
same object,
however, solving this exact-point mystery should
solve that problem
and other problems in mathematics and physics.

What I am sure of about this issue is ... there is at
least one missed
premise ,, which does not contradicts the other solid
premises ,and at
the same time makes things to be perceived in a clear
and understood
manner ..like how the premise of " Time relatvitiy"
solved mysterymade
by "electromagnetic waves equations" and "lorentz
transformation" ,
and like how the premise of "energy quanta" solved
the mystery
appearedfrom applying classic physics laws on "Black
Body Radiation
phenomenon" and made the big revolution called
"Quantum Mechanics" in
physics.

I would like to hear your stand from what I have
written here.

Regards
Said Alyami


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