Re: Non-terminating numbers

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 03/14/05 

Date: Sun, 13 Mar 2005 22:50:58 -0700

In article <1B8Zd.3305$N8.115371@news20.bellglobal.com>,
 "Robert Finch" <robfinch@sympatico.ca> wrote:

> > No. We can DEFINE a number as the limit of a series --- this does not
> > mean that we know how to evaluate the series; compare above. You do not
> > 'evaluate a set of rules out to infinity', you just apply a given
> > theorem which tells you that the series you have defined has a limit.
> > How to actually estimate this limit is a different question altogether.
> >
>
> > Anyway, I am digressing. The point is that it is possible to show the
>
> Digressing some more.....
>
> Am I right in thinking that the number line isn't continuous and is composed
> of a series of discontinuities ?

I am not sure you mean what I would mean, but in the usual sense the
answer is no. The reals have the least upper bound property, which is
usually interpreted as guaranteeing "continuity". A linearly ordered set
satisfies the least upper bound property when for every non=empth subset
that is bounded above there is a least upper bound ( member of the
larger set which is as large as any element in the subset,but is smaller
than any other such upper bound)
>
> In the limit of infinite? precision a - b <> 0 if a and b are different
> numbers, otherwise they would have to be the same number.

Trivially true for the reals.

By the density of the reals, if two numbers are not equal, there are
infinitely many other real numbers beween them

>
> Doesn't that mean that a number like pi would eventually have to fall
> between two discontinuities ?

Not unless ther arctually are such discontinuities, but there are no
such things in the set of real numbers, at least not by anything like
that name.
>
> Can you take the limit of a series out to an infinity when dealing with
> numbers ? Don't you have to stop at infinity - 1 ? Otherwise there would be
> no distinction between numbers ?

There is no such thing as infinity -1.

There are lots of series which will converging to any particular limit
value. An "infinite" series that stopped after less than infinitely
many terms would not be infinite, would it?

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