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A Brief History of Infinity – A Paradox

A Brief History of Infinity - by Brian Clegg

A good little read.  This book taught me a lot of things about set theory and Georg Cantor.  It also taught me about a very interesting paradox within set theory, that was identified by Bertrand Russell.

It goes something like this.

Set theory is something so simple that we take it for granted.  It merely points out that certain things, although not identical to one another, are similar enough to be grouped into a “set.”

In fact, set theory is where the working concept of numbers in general finds canon.  For instance, the number one looks like this in set theory:

{{Ø}1}

The symbol we as a people have for the concept of the number one (1) is a “set,” which is contained by brackets.

The “Ø” is called the empty set.

Brackets contain the emptiness of the empty set, and it is denoted by {}.  The empty set being “Ø,” and Ø being equal to zero for most intents and purposes.

{} = Ø

The number two in set theory could be written like this:

{{{Ø}1}2}

The number two contains the number one, which contains the empty set.  The pattern here is that the number three would contain the number two, which contains the number two, which contains the number one, and so on.  This can be repeated to infinity, in nested brackets form.  Cantor considered numbers this way.  And came up with the whole idea of transfinite numbers, which are numbers beyond infinity.  The book above goes over this concept quite nicely.

Now that we have some contextual knowledge of set theory, the foundation of all mathematics.  Here’s what’s wrong with it!

Consider the set of all <somethings>.  <somethings> could be anything you want.

Let’s say “llamas,” the set of all llamas.

I denote this set, the set of all llamas, in red for easier reference.  Trust me, this gets kinda twisted.

So please consider the set of all llamas.  Now the fact that there is a set of all llamas implies that there is an inverse set, a set of all NOT llamas which shall henceforth be denoted in green.  Now lots of things fall into the set of all NOT llamas category: you, me, a planet, electrons, horses, etc.  Anything that isn’t a llama falls into this category.  You’re not a llama.  I’m not a llama.  Planets aren’t llamas, and vice versa, even though a llama might exist on a planet.  And the same goes for electrons.   There  are obviously a lot of electrons that might be considered “in” or “part of” a llama, but a llama is definitely NOT an electron and vice versa.  The same goes for the set itself.

The what?

The set itself.  The container.  The thing that we named “set of all llamas.”

A set is a concept, it’s a thought.  Isn’t that a thing?

But wait, thoughts and feelings are ephemeral.  They’re not real things.

Are then none of your thoughts real things that can be considered part of a set?  And how do these non-things keep making and changing real things?  I mean, if you want to get technical about it, thoughts  are a bio-chemical reaction and that’s definitely about as physical and real as anything.

So anyway,

Please consider the fact that the set of all NOT llamas is a member of itself?

A member?

Yes, an element, a constituent, a part of the whole.  The set itself is not a llama, and so that set is a member of itself.

So let’s generalize, and consolidate, what knowledge we’ve gleaned.

There are at least two types of sets.  There are sets that ARE members of themselves, and sets that are NOT members of themselves.  For instance, the set of all llamas is not a member of itself, but the set of all NOT llamas is a member of itself.  And this general rule applies to all sets.  That’s why we chose “llamas,” arbitrarily.

So let’s just consider the sets that are NOT members of themselves, for a second.  We’ll call this super set the set of all sets that are NOT members of themselves.  

Now let’s ask ourselves the same question we asked about the other sets to categorize them.  Is this set a member of itself, or not?

Let’s explore both possibilities.

The set of all sets that are NOT members of themselves IS a member of itself.

– Well then it’s actually not a member of itself by definition.  If it’s a member of  the set of all sets that are NOT members of themselves, then it can’t be a member of itself…if it is a member of itself.

The set of all sets that are NOT members of themselves IS NOT a member of itself.

– Paradoxically, it IS a member of itself since it’s NOT a member of itself.

Bertrand Russell pointed out this paradox within set theory, and proved it mathematically.

He doesn’t propose a new theory of “the way things work,” but merely points out that set theory albeit useful might not be the way things actually work.

15. September 2011 by Kiv
Categories: Math | Tags: bertrand russell, infinity, math, paradox | 18 comments