Re: Non-terminating numbers

From: Robert Finch (robfinch_at_sympatico.ca)
Date: 03/15/05 

Date: Tue, 15 Mar 2005 04:16:31 -0500

>> A gap to me implies a space. When I think of a gap I think of a break
>> between two points where there is open space. Like a bridge with the
>> centre
>> span missing. When I think of a discontinuity I think of a sudden change
>> in
>> level, like a stairstep with no (horizontal) space between the levels.
>
> There are no sudden changes in the reals.

I guess I'm a bit dense. What is the definition of a number ? When I think
of numbers I think of them as being different than other numbers. You
wouldn't be able to have different numbers if there were no differences
between them. Why can't the idea of calculus be applied to this ? As one
continues to zoom in on the number line, more and more distinct numbers
appear. You can always find real numbers between other reals. The faster and
further one goes, the more numbers there are to be found. But just because
more and more numbers can be found, doesn't mean that it's a continuum.
There is always a difference between the numbers, otherwise at some point it
wouldn't be possible to find them. As long as there is a difference between
numbers, that implies a gap of some sort. Maybe the difference isn't a real
number ? It doesn't matter even if it is. It'd just be the next level of
precision. The reals filling the gaps are at the next level of precision.
Don't the comparision have to be kept to the same level ?
But I think the fact remains that one can always tell the difference between
numbers because that's the definition of a number. As soon as one can tell
the difference, that's a gap.

I guess I need more proof about the continuity of the real number line. Any
good suggestions would be welcome.

> This analogy doesn't apply, because there is no such thing as a pair of
> reals being "next to" each other. There are always infinitely many other
> reals in between.

Does that prove continuity ? The infinitely many are still different from
each other, they don't merge together. a-b still <> 0.

> There are no heights that are unaccounted for. It's like walking up a
> smooth ramp, not a staircase.

It's more like jumping from the top of one pole to another. Between the two
poles 500 feet below are infinitely many more poles and so on.

> The reals are not a sequence. In a sequence there is always a next term,
> but that doesn't hold for the reals.

> We don't need to add more precision, because the reals are already
> infinitely precise to begin with.

Okay.

> Your description of "continuity" of the reals depends on the erroneous
> assumption that it is possible for two real numbers to be "next to" each
> other, with no other numbers in between.

At any given level of precision that's true I think. There are no other
numbers between 3 and 4 at that level of precision. It's only when one goes
deeper that more numbers appear. You can go as precise as you like and it's
the same thing. At the same level of precision the numbers can be placed
next to each other.

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