Re: [Math Noise] (fwd)

> Infinity does not have a predecessor, so it makes no sense to
> count back from it a finite number of steps.

If infinity does not have predecessors (ie is immune to normal arithmetic
operations) then it is not possible for a sequence to approach it by adding
a finite amount to succesive terms in order to approach it. This means that
a sequence can not meaningfuly be asymptotic with infinity (meaning I have to
be able to draw a asymptote, at least in theory, in order to demonstrate the
limit).

> If one constructs the Ordinals, which are isomorphism classes of
> well-ordered sets, and the Cardinals, which are equivalence
> classes of equipotent sets, one will automatically end up with
> all sorts of transfinite numbers.

If infinity is not a number, how is it possible to have a definite number
(ie transfinite) which is larger than it?

My contention is that number theory as you present it is playing fast and
loose with the concept of infinity not being a number or visa versa.


                                                  Jim Choate
                                                  CyberTects
                                                  ravage@ssz.com

Jim Choate <ravage@EINSTEIN.ssz.com> writes:

 > If infinity does not have predecessors (ie is immune to
 > normal arithmetic operations) then it is not possible for a
 > sequence to approach it by adding a finite amount to
 > succesive terms in order to approach it. This means that a
 > sequence can not meaningfuly be asymptotic with infinity
 > (meaning I have to be able to draw a asymptote, at least in
 > theory, in order to demonstrate the limit).

 > If infinity is not a number, how is it possible to have a
 > definite number (ie transfinite) which is larger than it?

 > My contention is that number theory as you present it is
 > playing fast and loose with the concept of infinity not
 > being a number or visa versa.

The problem here is that the terms "infinity" and "number" are
used to refer to many different things in mathematics.  We use
"infinity" as a term for transfinite Cardinals and Ordinals, but
also use it when describing convergence on the reals, to indicate
that a sequence increases without bound.  We use it in calculus
to describe the limits of integration over all real numbers, a
set which does not include any infinite numbers at all.

Similar we use the term "number" to refer to Cardinals, Ordinals,
Reals, Integers, Complex, Quaternians, or whatever, hoping that
what we mean by it will be clear from the context.

With respect to the Real numbers, infinity is not a number, but
simply a way of saying something increases without bound or that
we wish to include all the positive or negative reals when
performing some mathematical operation.

With regard to Ordinals and Cardinals, not only is infinity a
number, but there are an uncountable number of different
infinities, which cannot be placed in 1-1 correspondence with
each other.

--
     Mike Duvos         $    PGP 2.6 Public Key available     $
     mpd@netcom.com     $    via Finger.                      $

Jim Choate allegedly said:
> 
> 
> > Infinity does not have a predecessor, so it makes no sense to
> > count back from it a finite number of steps.
> 
> If infinity does not have predecessors (ie is immune to normal arithmetic
> operations) then it is not possible for a sequence to approach it by adding
> a finite amount to succesive terms in order to approach it. This means that
> a sequence can not meaningfuly be asymptotic with infinity (meaning I have to
> be able to draw a asymptote, at least in theory, in order to demonstrate the
> limit).

Jim, the problem here, as elsewhere in your posts, is that you 
confuse the prosaic meaning of terms with the mathematical meaning.  
"Approaching" infinity is sort of inane, mathematically speaking, but 
if it did have a meaning, it wouldn't mean "getting closer to".

> > If one constructs the Ordinals, which are isomorphism classes of
> > well-ordered sets, and the Cardinals, which are equivalence
> > classes of equipotent sets, one will automatically end up with
> > all sorts of transfinite numbers.
> 
> If infinity is not a number, how is it possible to have a definite number
> (ie transfinite) which is larger than it?
> 
> My contention is that number theory as you present it is playing fast and
> loose with the concept of infinity not being a number or visa versa.

Your contention is precisely what you are doing.  The term "infinity" 
means several things mathematically speaking, but the various 
meanings are precise.  Your comments demonstrate that you don't 
really know what those precise meanings are, and thus it is difficult 
for people who are used to them to communicate with you.

So perhaps you could define *exactly* what you mean by "infinity"?

-- 
Kent Crispin				"No reason to get excited",
kent@songbird.com,kc@llnl.gov		the thief he kindly spoke...
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