The Last Finite Number?

Date: 02/01/2004 at 10:59:46
From: Toxic
Subject: The last finite number?

I just read an article in a magazine that described a pair of numbers
that I found intriguing.  

One was called "psi", and it was supposed to be the "last" finite
number, i.e., the number just before infinity.

The second was called the "end number", which is supposed to be the
highest in the kingdom of numbers.  Nothing is larger than the end 
number because by definition it is the last number. 

Is there anything to this? 

Date: 02/02/2004 at 13:03:36
From: Doctor Vogler
Subject: Re: The last finite number?

Hi Toxic,

Mathematics is axiomatical, which means that it does not inherently 
reflect what "is true" or "is not true" but rather it describes the 
consequences of certain simple assumptions.  For example, we assume 
that there is some abstract quantity called a "one" and that you can 
add it to numbers to get other numbers, which, to give them names, we 
might as well call "two" and "three" and so on.  Then we define a
thing that we call multiplication, which has the curious effect of 
making certain numbers different from other numbers.  Some numbers 
can't be made by multiplying together smaller numbers, and we call 
these primes.  And so on.

The point of all that is that you can create different kinds of
numbers which behave rather differently.  For example, after the 
integers we define rational numbers, then real numbers, and finally 
complex numbers.  Then we can talk about vectors, which are sort of a 
different kind of number.  When you get into things like infinity, 
then you should look into what we call "cardinality."  I think you 
could learn a lot of information pertinent to your question by 
searching for "cardinality" in the Ask Dr. Math archives or on the 
internet.

But then when you define a set of numbers, you next need to describe 
what operations you have on these numbers, and what properties they 
satisfy.  If you can't define any operations that have any useful 
properties, then that is what we mathematicians would call an 
"uninteresting" set of numbers, because you can't do anything with 
them.  You define them and then that's the end.  So we like to stick 
to numbers that have useful properties.  In the field of abstract 
algebra, we list a few of the most useful properties and talk about 
sets of numbers which have them, and we call these things groups and 
rings and fields (depending on how many operations and properties they 
have).  For example, two useful properties that a set of numbers has 
is that they include a "unit" that we can multiply things by without 
changing them (that is, the number one), and that you can always add 
or multiply two things and get something else.

So what happens if you add one to psi?  If it is a finite number, then 
adding one shouldn't make it infinite.  But it would have to be bigger 
than psi.  Do you see why this would cause problems in defining an 
addition operation if you have a "last finite number"?  And what 
happens if you add one to the "end number"?  Okay, this is an infinite 
number, so you could argue that adding a finite amount to it shouldn't 
change its size.  But if you learn about cardinality, you'll find that 
the power set of a set of this size is necessarily bigger.  So what 
would that number be?

In other words, the article may have speculated about these numbers, 
and you could certainly define a set of numbers that contains them, 
but you could not define operations on these numbers that have any 
meaningful properties about them.  So they wouldn't really serve any 
useful purpose.

If you have any questions or need more help, please write back, and I 
will try to offer further advice.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/