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. $