Infinity as a Skolem Function

Date: 10/28/2000 at 00:58:23
From: Todd Zaba
Subject: Is infinity an absolute concept, relative, or both?

Could one reasonably say that infinity is relative? The following 
discussion came up today:

There is an infinite set of odd numbers  (1, 3, 5, 7, 9, ...)
There is an infinite set of even numbers (2, 4, 6, 8, 10,...)

So, each set is infinite, yet 1/2 of another infinite set... 

Does this make infinity relative? Or does this get back to the 
"infinity is a concept, not a number" that I've run across on your 
site so much?

Thank you for your time.

Date: 11/03/2000 at 18:41:57
From: Doctor Ian
Subject: Re: Is infinity an absolute concept, relative, or both?

Hi Todd, 

I'm not sure that what you mean by 'relative' is any clearer than what 
mathematicians mean by 'infinity'. Certainly some infinities (the size 
of the set of real numbers) are larger than others (the size of the 
set of integers). But there are exactly as many integers as there are 
even integers.

Personally, I think 'infinity' is a skolem function.

What do I mean by that?

In logic, you can set up relations between entities, such as:

     is( alice, mother-of( bob ) )

And you can use these relations to discuss entities whose identities 
you don't actually know:

     is( ?, mother-of( bob ) )

That is, we know that Bob has a mother, even if we don't know who it 
is. The function mother-of is called a skolem function. 

Skolem functions are really useful, for the reason just described - 
sometimes we want to talk about something, but we aren't yet able to 
identify it: the winner of the next lottery drawing, or the candidate 
who was ahead in the polls yesterday, or...

But sometimes they can cause problems, as when they are used to refer 
to things that don't actually exist, and may never exist: the cure for 
cancer, the cause of war, the solution to the education crisis, the 
sexiest man alive, the creator of the universe...

In other words, skolem functions let people talk nonsense without 
realizing it.

So, what does this have to do with infinity? Well, when we talk about 
sets, we refer to attributes of the sets: the first element, the last 
element, the size, and so on. So size-of is a skolem function:  

     equals( 435, size-of( House-of-Representatives ) )

     equals( 88, size-of( piano-keyboard ) )

     equals( ?, size-of( employees-of( IBM ) ) )

It just so happens that when we talk about sets that don't have finite 
numbers of elements, we can glibly use the skolem function size-of to 
talk about something that doesn't really exist, e.g., the 'size of' 
the set of integers:

     equals( 'infinity', size-of( integers ) )  

Actually, this only became a problem when Cantor showed that

     equals( 'infinity[1]', size-of( integers ) )

     equals( 'infinity[2]', size-of( real-numbers ) )

     not-equals( 'infinity[1]', 'infinity[2]' );

Infinity is a process, an algorithm, a mapping, a limit, a 
singularity, a destination... infinity is a lot of things, but it's 
not a number, even though you can assign it a symbol that makes it 
look like a number. The confusion starts when you try to treat it like 
one.

I'm not sure that I answered your question. Write back if you'd like 
to discuss this further, or if you have any other questions. 

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