Math Forum

From: haoyuep@aol.com (Dan Hoey)
Date: Mar 14, 2000 9:56 PM
Subject: Re: 1.9...is it 2?

On a discussion drawing a distinction between the real number 1 and
the integer 1, David W. Cantrell wrote:

> Alan, I agree with you (and would guess that Derek would agree
> as well) that the point being made is indeed quite insignificant.
> Nonetheless, there is no reasonable way to construe "0.999..." as
> representing an integer.  Of course, it may be construed as the
> real number (an "integer" real number, so to speak) which could
> be more simply represented as "1".

On the contrary, it is quite reasonable to consider the integers a
subset of the reals.  Many mathematicians, myself included, consider
them to be so.

I am aware that some other mathematicians consider the integers to be
a non-identical isomorphic copy of a subset of the reals.  The issue
is generally considered moot.  That is to say, that mathematics is
carried on in such a way that it remains valid whether or not we
consider the integers to be a subset of the reals or merely isomorphic
to such a subset.

In case you are wondering, the foundations of the system I prefer are
perfectly sound.  Whereas some people (for instance) construct the
reals as a partition of the rational Cauchy sequences, I consider such
a construction to be only the penultimate step of the construction of
the reals.  The final step is to take the reals as the union of the
rationals and the irrational equivalence classes.  To prefer one of
these constructions to another is no more material than to prefer a
Cauchy sequence construction to Dedekind cuts or Conway options.

Of course, there are other systems that have "integers", and perhaps
some of those sets of integers can be isomorphic to Z without being
usefully identified with Z.  There are plenty of other structures with
isomorphic copies of well-known structures.  But when I speak of _the_
naturals N, _the_ integers Z, _the_ rationals Q, _the_ algebraic
reals, _the_ reals R, and _the_ complex numbers C, I am speaking of a
tower of subsets.  You should understand that they act just like the
ones you speak of, except for some equalities that you would treat as
isomorphic images.

> >1 is a positive integer, a rational, a real, whichever you need
> >it to be.

> Yes, assuming that you mean that "1" can represent either a positive
> integer, or a rational, or a real number, or...  No, if you are
> thinking that the multiplicative identity elements of Z+, Q, and R
> are the same.  They are not; they are very different types of entities.

It is reasonable for you to consider them different, just as it is
reasonable for me to consider them identical.  I cannot force you to
consider my point of view to be reasonable, but I will consider you
unreasonable if you do not.

Dan Hoey posted and (hopefully) e-mailed
haoyuep@aol.com