Re: Is continuum completely filled up?


"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.
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.
( 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.
I'm afraid that I am perfectry wrong . Any comments are appreciated.

Regards OT


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