Re: Is continuum completely filled up?

Russell <russell@xxxxxxxx> writes


Sticking my head over the parapet again ...I can't see how you can make
up something continuous from something point like, however many points
you have.

If any 2 real numbers are different, there is a gap between them,
in which of course there is another real. But there are also now
2 gaps. So while there are an infinite number of reals
there must also be an infinite number of gaps?

Here, I think, is the fallacy: you go in one fell swoop from 2
to infinity. What's true for any finite process is not necessarily
true for an infinite one.

Plus, I think you have to be crystal clear what you mean by
a gap. How would you know if the gaps were still there after
you "go to infinity" with the process? You need some kind
of procedure for finding a gap. I think if you try to define one,
at least if it's one that agrees with our notion of a line with
order topology, you'll discover that everywhere you look for
a gap, there is instead a real number there.

Or is this fallacious
because implicit in this type of construction of the reals is a
countably infinite process?

If I understand you right, I don't think that's it. You can
construct the reals as the set of paths in an infinite binary
tree. The depth of that tree is countably infinite.



I don't see any difficulty in specifying the "gaps" in such a binary tree construction - it is just the open interval between two reals on the previous layer on the binary tree? The construction is infinite, so there is never a final layer on the tree, but that layer has as many gaps as there are reals. So at no stage can you say that there are no gaps - if there is no gap between 2 points they are the same number.

In any event I am 100% sure a) that this elementary line of argument is not original and b) that it was sat on 100 years ago. I just wondered what the refutation was, hoping that it was elementary and didn't involve reference to some detailed theorem in infinite set theory.
--
Andy Smith

.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... Andy Smith wrote: ... So while there are an infinite number of reals ... there must also be an infinite number of gaps? ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... So while there are an infinite number of reals ... there must also be an infinite number of gaps? ... But the same is true of the rationals. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... Sorry to be dim/simplistic/ignorant, but if you are going to cover e.g.with points (reals U rationals) but no gaps, then that must imply that each successive point touches its neighbour, with no gap between them? ... the gaps between the reals ... dimension 2.0 and a line has dimension 1.0. ... So if you quantise the real line into N points, you have N numbers and N-1 gaps, halve the scale, you have 2N numbers, 2N-1 gaps, and so on, so the fractal dimension is 1 for both numbers and gaps. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... is wheather a line is build from points or not. ... So while there are an infinite number of reals ... there must also be an infinite number of gaps? ...
    (sci.math)
  • Re: Attempts to Refute Cantors Uncountability Proof?
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