Re: Is continuum completely filled up?


toshiaki wrote:
"David Marcus" <DavidMarcus@xxxxxxxxxxxxxx> wrote in message
news:MPG.201ed0ebcdd5321d989bd5@xxxxxxxxxxxxxxx
Gaps that I mean, are spaces remained between distinguishable reals, or
line
itself.

Defining one word by using other undefined words doesn't accomplish
much. Please define "spaces", "remained", "distinguishable reals", "line
itself".

"Spaces" is coordinates. "remained" is spaces other than points.
"distinguishable reals" are these points.

You still manage to avoid giving a useful definition of what you
are trying to express. I realize that there is a language issue here
(I admire your effort) so let us try to minimize use of English and
maximize use of symbols.

By "gap" do you mean
"There exists reals x and y such that x - y > 0 and there are
no reals z such that x < z < y"?

Does "distinguishable" mean "|x - y| > 0"?

I take up a position that, line and number system embeded on it are
different things.

They are. The line is just one possible representation of R.
The set of decimal expansions is another.

Various way to constuct reals is designed to fill line,
but there are no gurantee for actually being so, even though that is
complete ordered field.

Then maybe "gap" means something like this: When the
reals are represented as points on a line, there are points
which do not correspond to any x in R.

To even make this sentence meaningful, we need some
independent definition of line not using real numbers.

Though points are measure 0, they can picture line in any precision we need.
And remained spaces become meaningless accoding to that. Remained digits of
decimals which are used to discribe it, are still infinite, but they lose
its meaning gradually. When we correspond numbers to reality, we must take
in account its meaning.

Here I do not follow what you are saying. Both the line and
the real numbers do not "lose meaning" at any finite
precision level.

If there might be microscopic subworld that is simillar to ours, within our
world, and there is anoter far more microscopic world, within that
world......, we can depict all of them only with infinite decimal. Infinity
is such a thing for me.

- Randy

.

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