Re: counter example in analysis

Eckard Blumschein wrote:

My arguments are quite simple and hopefully compelling:
Let's consider like an example the interval of real
"numbers" between 0 and 1.
As long it contains just arbitrarily many numbers,

This is not true. There are c many numbers between
0 and 1, which means we can't have "arbitrarily many".

these numbers can be different from each other and may
be called rational numbers.

At this point you're going to run into trouble when
communicating to other people. The term "rational
numbers" already means something specific. You should
choose a different term. Also, you never specified what
your "rational numbers" are, unless you're intending
this phrase to mean all the numbers between 0 and 1,
which seems like a pointless thing to do, pardon the pun.

There is no largest amount of them.

Do you mean there is no largest among them? What you
wrote doesn't make sense. If you're talking about a
set, it has at most one cardinal number (also at least
one if we're in ZFC), but since you used the plural
"them", I'm not sure what you mean.

In this case, I may and I have to distinguish between
open and closed intervals.

You can, but they might be the same, depending on what
set of numbers you're talking about. If you're talking
about the usual rational numbers, and b is an irrational
number between 0 and 1, then the open interval (b^2, b)
of rationals is equal to the closed interval [b^2, b]
of rationals.

Between two munbers, there is always a potentially
infinite sauce-like continuum of locations.

I guess, except the use of the word "potentially" seems
misleading to me, besides being unnecessary.

Nonetheless, according to Dedekind's wording, any
countable amount of rational numbers is overall dense.

No, and he never said this either. Perhaps you could
give a precise quote and reference?

Any finite set is countable and not dense. It is easy
to get countably infinite sets that are not dense,
such as the range of a convergent sequence. In fact,
the Cantor set is uncountable and not only fails to
be dense, it fails to be dense in every interval.

I just envison two possibilities to cover the whole
line with points. The first one is motion. Remember
the fluxus of indivisibles for instance tought by
Pythagoreans, Cavalieri, Descartes, Torricelli and Newton.

I also remember Zeus, Aphrodite, Apollo, etc., but I wouldn't
use them to support a modern biblical argument for something.

Okay, I need to get back to work.

[snip rest]

Dave L. Renfro

.

Relevant Pages

  • Re: Dense vs. Continuous
    ... mentioned that the rationals were "dense". ... function on the reals is a continuous function. ... If I were to graph the same function and said the domain ...
    (sci.math)
  • Re: counter example in analysis
    ... Since I referred to rationals, ... different not just from the irrationals but from the real ones, ... On the other hand, uncountable implies dense. ... If I consider the reals one by one, then they are not dense but countable. ...
    (sci.math)
  • Re: counter example in analysis
    ... of rationals is equal to the closed interval ... infinite sauce-like continuum of locations. ... Any finite set is countable and not dense. ... it fails to be dense in every interval. ...
    (sci.math)
  • Exotic functions (elementary)
    ... function whose graph is dense in the plane. ... A FUNCTION WHOSE GRAPH IS DENSE IN THE PLANE ... We define f from the reals to the reals as follows. ... "countability of the rationals", etc. ...
    (sci.math)
  • Re: counter example in analysis
    ... countable amount of rational numbers is overall dense. ... potentially infinite amount of rational numbers it is thought ... to be overall dense, and not even this corresponds to the original. ... of rationals, nor any other densely ordered set. ...
    (sci.math)
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