Re: Why is an integer finite entity?

From: Daniel Grubb (grubb_at_lola.math.niu.edu)
Date: 02/11/05 

Date: 11 Feb 2005 15:43:53 GMT

>Why is the definition of Z the way it is, and not some (perhaps absurd)
>definition which would include "integers with infinitely many digits".
>I saw no reply that said nothing more than "because that is the way it
>is defined". My friend said something like "Well, I am not satisfied
>with that definition. It seems to delimit things arbitrarily to ensure
>reaching the desired conclusions." I threw up hands "Okay, you win".
>You don't throw up hands, but you say "Look, the rule of the game is
>that you accept all given definitions without questioning them.".

>Some reasonable approaches could be "because that does not seem
>natural", or "because that's hard to define formally", etc. My approach
>is the first: I explain that the finite-ness of an integer has a lot to
>do with the integers in the real-world, as an idealized model of all
>fractionless quantities.

Fair enough. First of all, the 'finite' integers have a lot of intertesting
properties. For example, you can do add or multiply two finite integers
and get another finite integer, they have unique factorization into primes,
and you can prove things about them using mathematical induction, and basic
algebra works. All good properties to have. There are many, many more.

Now lets look at those with infinite expansions. For example, let
x=1000.... with an infinite number of 0's (working base 10, but other
bases have similar difficulties). What is 10x? Is it the same as x?
If not, how do you distinguish the two? If so, then 10x=x, which algebraically
leads to x=0. Hmmm, not good. So even simple multiplication of such
infinite integers and finite integers leads to problems. Next, what
would x times x be? Again, it could either not be x, in which case how do you
distinguish the two, or it could be x, which leads to x^2=x, so x=0, or
x=1. So multiplication of two infinite integers has problems also.
It is easy to see that addition has similar difficulties.

Again, if you let y=3333....3, what is 10y+3? Is it y again? Is so, then
10y+3=y, so y=-1/3. Are you sure this is what you want? Otherwise,
what is the expansion of 10y+3 and how does it differ from that of y?

Next, it is very difficult to figure out whether or not an infinite
integer should be a prime (since multiplication has problems). So
there is no clear way of even talking about factorization, let alone
unique factorization into primes. Which primes divide x above? How many
times? What about 11x? What about x+1? x+2?

Finally, mathematical induction fails. In fact, it is almost the
defining property of the finite positive integers that induction
works. No good property of infinite integers is around to take its
place.

I hope this helps your discussion.

--Dan Grubb