Re: Is continuum completely filled up?

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?


Russell wrote:
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.

Indeed, it looks like every time you find a "gap" between two
reals, you then find another real point to "fill" it. So if you
continue this quest for gaps indefinitely, won't you end up
filling them all up and thereby produce a completely "filled"
line?

Looking at it another way, for any two unlike reals you can
always find a third real (actually an infinite number of reals)
that lies between them. So the logical conclusion is that there
is no "gap" between any two reals because there is always a
real there to fill that gap.

So we ask, where are all these gaps between reals you are
talking about? Does a "gap" mean a place in the real line
where there is no real point?

.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... Andy Smith wrote: ... If any 2 real numbers are different, there is a gap between them, ... So while there are an infinite number of reals ...
    (sci.math)
  • 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 ...
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  • 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)