Re: Is continuum completely filled up?

On Wed, 17 Jan 2007 21:26:38 GMT, Andy Smith wrote:
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?

Distinct points on the line cannot have a separation of 0. If a and b
are distinct real numbers, then | a - b | > 0.

However, there is a way to represent the concept of things being "next to
each other" on the line. That is to consider sets of points, rather than
individual points. For example, let

A = { x in Q : x < 0 } (the negative rationals),
B = { x in Q : x > 0 } (the positive rationals).

The sets A and B are distinct, but they approach each other more closely
than any positive real number. In fact, we can say that distance(A,B) =
0, where the distance between sets is defined by

distance(X,Y) = inf{ |x-y| : x in X and y in Y }.

Here, "inf" stands for "infimum", which means the greatest lower bound.

But notice, even though distance(A,B) = 0, there is nevertheless a gap
between the sets. The number 0 lies strictly between A and B and does
not belong to either.

The point is, requiring a distance to be zero is not sufficient to
guarantee the absence of a gap. After all, points have zero width and
therefore they can fit into a gap without causing any actual separation.

This is why the notion of completeness does not rely on distance.
Instead, completeness uses the least upper bound property. The set A
does not have a maximum element, but it does have a least upper bound,
and the LUB is a real number. The same is true of any nonempty set of
reals that is bounded above.

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


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

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