Re: Is continuum completely filled up?

David R Tribble writes

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?

Yes, but isn't this a bit of a chicken and egg argument. every time I produce a gap, you produce (an infinity) of reals in it. But then I identify more gaps between those reals, and so on. If one takes a gap oriented view of your argument, for any two unlike reals there is a gap between them, so then the logical conclusion is that there are as many gaps as reals?

I don't actually trust any such arguments one way or another.
--
Andy Smith

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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 ...
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  • 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?
    ... So while there are an infinite number of reals ... What's true for any finite process is not necessarily ... of procedure for finding a gap. ... reals, you then find another real point to "fill" it. ...
    (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)