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### Infinity as a Skolem Function

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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?

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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/
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Associated Topics:
High School Functions
High School Number Theory
High School Sets

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