Re: Is continuum completely filled up?

Dave Seaman <dseaman@xxxxxxxxxxxx> writes:

On Tue, 23 Jan 2007 03:09:42 +0900, toshiaki wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:eovoec$8j5$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Reals don't have size. The "size" (Lebesgue measure) of a single point
on the line is 0.
Sorry what I meant is size of number. number 2 has size 2.

Then I don't understand how you conclude that the reals must have gaps
between them.

Thank you for your detailed reply and many information. I shall reply little
by little.
Gaps that I mean, are spaces remained between distinguishable reals, or line
itself. Reals which don't have any figuers that caracterize them, and can be
manipulated only by AC, and among their sea, specifiable reals are scattered
whose amount has measure 0, is very similar to my difinition of gap.

Between any two computable reals there are uncountably many noncomputable
reals. Is this what you mean by a gap? If that is the case, then surely
what you have described is a gap in the computable reals, not a gap in
the reals. There is no gap in the reals.

Of course, even such a "gap" in the computable reals is not an *atomic*
gap: between two computable reals lies (for example) their computable
average. Does a gap with stuff in it really *count*?

Lee Rudolph
.

Relevant Pages

  • Re: Non-zero gaps between real numbers
    ... assert that there is no non-zero extent gap devoid of real numbers in ... use RAA to prove that there is no non-zero extent gap devoid ... This is a real number, because + - are closed over the reals, and / is ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... requiring a distance to be zero is not sufficient to ... guarantee the absence of a gap. ... reals that is bounded above. ... there are not enough bits to address the reals (and you would still always have as many open intervals as numbers). ...
    (sci.math)
  • Re: Non-terminating numbers
    ... When I think of a gap I think of a break ... You can always find real numbers between other reals. ... that implies a gap of some sort. ... The reals filling the gaps are at the next level of precision. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... requiring a distance to be zero is not sufficient to ... guarantee the absence of a gap. ... reals that is bounded above. ... Of course if that was true, then you could still continue the process ad infinitum, but that would mean that there are not enough bits to address the reals (and you would still always have as many open intervals as numbers). ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... it looks like every time you find a "gap" between two ... for any two unlike reals you can ... requiring a distance to be zero is not sufficient to ... completeness uses the least upper bound property. ...
    (sci.math)