Re: Is continuum completely filled up?

Saurav wrote:
toshiaki wrote:
"Bob Kolker" <nowhere@xxxxxxxxxxx> wrote in message
news:4vvevuF1dt4kfU1@xxxxxxxxxxxxxxxxxxxxx
ooo wrote:
I am biginer in English and mathrmatics.
If real line is filled with points and each point is
distinguished,then each point has difference from every other points.
Therfore real line has void.

Thanks for advance.
There are no isolate points on the real line. And the real line is
dense. In addition every cauchy sequence of points on the real line
converges to a point on the real line. The real line is locally compact.

So to answer your question: no holes.

Bob Kolker

Thanks.
I can't imagine the condition that there are no isolated points , and they
have no contacts each other.
My imagination is that there exist condition whether
1. all points are isolated ,
2. or each points are undistinguishable each other.

Explain, first, what you mean by "there exist condition whether..". In
some kind of ordering on any set, the order topology assumes condition
1, it's true. For the second condition, I would tell you that, even if
there is a density in the set, the points are, as you said -
idealistically - distinguishable. This seems to contradict in some way
to our intuition. But remember, we do not always trust our intuition in
mathematics; and indeed, our natural intuition sometimes turns out to
be false, especially in the transfinite cardinality related areas.

This is only visualised idealistic explanation.
These are pictures that come from my idea that we can only deal with finite
objects.
Infinity is shown by following way.
countable infinity every number have its next. and assumption that there
exist set including all of them.
Von Neumann's folly!

Not merely! According to an assumption, which you have every reason to
disagree with, every set, however large, can be well ordered; and in
such an order, almost all elements in the set has a next element.

Well, that's true...

I actually think the H-riffics may be a well ordering of the reals after all...


( This assumption cause someone to imagine something completed total ).
uncountable infinity
reals > naturals assumption that new diagonal number is different from
all listed number because we can choose different numbers at every digits .
P(S) > S assumption that there exist one to one correspondence among sets
of infinte obects .
These assumptions couldn't be refuted logically . I restrict my argument on
reals.
At present objects that may be useful for our mathematical operation are
computable numbers. But ather more proper may be found.
And the rest are undistinguishable , but exist as closure.They cannot be
picked up in explicit form.
I hopefully think that this picture can avoid paradoxes which are based on
the assumption that each points are separable .
And we can use only countable axiom of choice.

First of all, you tell me whether there is at all an infinite set. And
if such one exists, are we elligible to talk about its properties?

No "pure" infinite set. There is always order where infinitude's implied. :)

Besides, you're diselligibulized :)

Remember, in our thought we seem to be rather finite beings than
infinite ones. Is it not a paradox? If we can talk about those, I
believe we can also talk about separating points.
I'm afraid that I am perfectry wrong . Any comments are appreciated.

Regards OT


Mmmm...for the infinite? Use x. Functions on x....

Tony
.

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