Re: Is continuum completely filled up?

Andy Smith wrote:
I

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.

Thank you, understood that, and I can see that things are inevitably
more tricky than they might seem. But I don't see that that answers my
simplistic argument about gaps between the numbers?

Between *what* numbers, is the question. You seem to have
this idea that between real number x and the "next" number y,
there has to be some sort of gap because x =/= y. Leaving aside
the question of why you should think that's necessary, the fact
is, you can't define your gap that way because there is no such
y as that. Your gap can't be something that requires two points
in its definition -- each gap must be associated with *one* point
only, if it exists at all. Perhaps you could think of it as a little
zero-thickness helmet that each point wears, to keep it from
coalescing into any other point. But, I don't see what that buys
you. Points are just mathematical objects whose properties are
up to us. Why can't we just say that points per se have the
property that they don't coalesce? No helmets (no gaps) needed.

OTOH, maybe you just want to define your gap as the ability
to put in a knife blade and cut the line at a point, so that the
point and everything below it is to one side, and everything
above it is on the other. This is fine -- you don't have to
specify anything like your nonexistent y above; you just say
all of the points above are on that side of the cut. And clearly,
there is one such gap (as in ability to make a cut) for each
real number. Well guess what -- this is nothing more than
a Dedekind cut, and mathematicians *do* use it. But I think
the term "gap" for this would not be popular, as a gap connotes
a void that needs to be (or at least could be) filled with something
else, and Dedekind cuts aren't like that.

.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... Distinct points on the line cannot have a separation of 0. ... requiring a distance to be zero is not sufficient to ... guarantee the absence of a gap. ... This is why the notion of completeness does not rely on distance. ...
    (sci.math)
  • Re: Non-zero gaps between real numbers
    ... numbers with zero gap between them. ... Let's locate all the reals on the line as points. ... But, by choosing the suitable reals/points, the distance can be ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... requiring a distance to be zero is not sufficient to ... guarantee the absence of a gap. ... completeness uses the least upper bound property. ... reals that is bounded above. ...
    (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 ...
    (sci.math)
  • Re: Non-zero gaps between real numbers
    ... numbers with zero gap between them. ... Let's locate all the reals on the line as points. ... But, by choosing the suitable reals/points, the distance can be ...
    (sci.math)
Loading