Re: Exact-Point paradox
- From: SmarterThanTheAverageTroll@xxxxxxxxxxxxxxxxx
- Date: Mon, 06 Aug 2007 19:03:54 -0000
Said Alyami wrote, or had his pet monkeys type at random,
probably by hurling their feces at a keyboard:
How things can be finite and infinite at the same time
The phrase "be finite" has no meaning.
The phrase "be infinite" has no meaning.
Your alleged paradox is mere gibberish without meaning.
1- can the infinite really makes the finite?The phrase "the infinite" has no meaning.
The phrase "the finite" has no meaning.
Your apparent question is mere gibberish without meaning.
2- What is the Exact finite output for infinite addition ?The phrase "infinite addition" has no meaning.
Your apparent question is mere gibberish without meaning.
I'm guessing you're referring to infinite sequences,
and limits of such if they happen to converge,
especially infinite series which are really convergent infinite
sequences whose terms are equal to partial sums,
each of which is a *finite* sum,
an example of *finite* addition.
No infinite addition is involved.
Real numbers set, or real numbers line is a continuum =E2=80=9C
linearly ordered set =E2=80=9C that is "densely ordered", i.e.,
between any two members there is another, and lacks gaps
You've invented some of your own personal jargon, but it's
logically constructed from common English, so it makes sense. And I
agree with the two points you've made, that it's a totally-ordered
set, and that there are no gaps in it, at least no gaps of positive
measure. There is no set S of positive measure such that S
partitions the real numbers into two non-empty sets, the part
uniformly less than S and the part uniformly greater than S.
it is impossible to have neighboring real numbers
Correct.
This makes real numbers appear on the real numbers line ordered
with the relations < and > as: X<Y<Z<M<S<T <W<=E2=80=A6 to
infinity [but none of them is neighboring any other one]
This is very sloppy notation. Do you mean to say that there exists
a mapping from the natural numbers into the reals which is strictly
increasing? I.e. 1->X 2->Y 3->Z 4->M 5->S 6->T 7->W with the rest
of the image points not given explicit names here?
But what does "to infinity" refer to?
The fact that there are more than any finite quantity of natural numbers?
Or the the additional assertation that the image points have no upper bound?
Your language is completely ambiguous/meaningless on this technical point.
The existance of more-than-finite-quantity can be proved by
mathematical induction from what you said before, but the existance
of such a sequence that has no upper bound is something which
doesn't follow from what you said before, and would need to be stated
as another "fact" you're bringing into your calculation as a "given".
There is one exact point for each real number representing it on
the real numbers line.
This is nothing more than a definition, part of the definition of
the model of the classical Euclidean geometry in one dimension,
as the reals.
More commonly, we model two-dimensional geometry as ordered pairs,
wherein a line is the locus of a linear equation.
Or we model three-dimensional geometry as ordered triples,
wherein a line is the intersection of two non-redundant planes.
But in any case, this is just the definition of "point" in such a model.
(For those who have taken a Geometry class but haven't yet taken
an Analytic Geometry class: This is what Analytic Geometry is,
this particular kind of algebraic model for classical geometry,
which makes lots of geometric derivations reduce to algebraic trivialities.)
we specify a point on the real numbers line to represent a real
number , for example we specify the point X1 to represent the real
number 1.
OK.
There cannot be a very next point to X1 on the real numbers line,
so the point X2 does not exist, and because X2 does not exist then X3
does not exist too, hence X4 does not exist, hence X5, X6, X7, X8,=E2=80=
You haven't defined any of the notations
X2 or X3 or X4 or X5 or X6 or X7 or X8
so how can you say at this time whether any of "them" exist or not??
Please post a definition of X2 before trying to say anything about it.
the non-existence of the =E2=80=9Cvery next point=E2=80=9D goes
to infinity, like what happens in Domino tiles after the fall of
the very first one, then the effect of falling continue through all
the next other tiles.
Well, I agree that there's no next point after X1, and there's no
next-next point after X1, and no next-next-next point after X1, etc.
That kind of statement doesn't seem to lead anywhere. For sure
there is no infinite ascending sequence of points where each is
next to the previous when considered as real numbers. So what??
we can see that there will be no points at all after
the exact point X1 which represents the real number 1.
That does not follow. There are many many points after X1. Just
there are no **next** points after X1. None of those many points
after X1 is **next** to X1, and none of those many points after X1
is next to any of the other points after X1. No two points after X1
are next to each other.
You've committed a horrible logical mistake.
You have shown that there are no polkadot elephants
You have falsely concluded from that, that there are no elephants whatsoever.
(Substitute /polkadot/nextto/ /elephants/numbersgreaterthan1/)
You correctly concluded there are elephants,
and falsely concluded there are no elephants,
consequently claiming a contradiction.
There is no contradiction, except between a true result and a false result.
Your alleged contradiction (paradox) is nothing more than a mistake you made.
real numbers express our real world
That has never been established.
At best, real numbers provide a very accurate model for several
aspects of our real world, namely the passage of time at ordinary
scales (femtoseconds to billions of years), and the span of space
at ordinary scales (large-molecule dimensions to billions of light
years) when far enough away from strong gravitational sources such
as black holes and when more recent than the epoch of Guth inflation,
and various combinations of the two used in Einstein's relativity.
.
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