Re: Is continuum completely filled up?

Andy Smith a écrit :
In message <eomd8t$9nu$1@xxxxxxxxxxxxxxxxxxxxxxxxxx>, Dave Seaman <dseaman@xxxxxxxxxxxx> writes
On Wed, 17 Jan 2007 22:13:40 GMT, 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?

I don't understand what it is that you think has not been answered. You
seemed to be struggling for some way to reduce the distance between
points to 0. I did three things: (1) I showed you how to reduce the
distance to 0 by considering sets of points instead of individual points,
(2) I pointed out that reducing the distance to zero is not sufficient to
preclude gaps, and (3) I showed you how the concept of completeness
addresses the problem of gaps without using distance.

Yes, I understood that.

I am not sure. Why, then, dont you see there is a difference between R and Q ?




If that isn't answering your question, then I evidently have not
understood your question. Please ask it again.


Thanks. Virgil pointed out that my use of the word "gap" is inadvisable. My simple train of thought (I can't see how you can ever cover a continuous line with a set of points, however many points you have) went e.g.:

- take two reals. There is an open interval between them
- insert 2 more reals between them. There are now 4 numbers, 3 open intervals.
- iterate N-2 times - you now have 2^N reals and 2^N - 1 open intervals.
- if you have a countably infinite number of bits, then after a countably infinite number of iterations, you still have as many open
intervals as distinct real numbers?


This is the usual mistake : taking "limits" where it is not allowed


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).

As I said earlier, I am sure that this line of reasoning is not original and is fallacious, but what is wrong with it?

First, what is wrong with it for rationals ? Second, you are only rediscovering Zeno's paradoxes.







.

Relevant Pages

  • 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)
  • 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
    ... 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: Existence of reals and observation of them
    ... Not zero? ... Then Venkat is obviously unable to deal with open intervals. ... If the distance is zero, it means there is no break or gap. ...
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