Missing Things

21 February 2010

Kurt Gödel

I've been really bad about updating this - last week I was sick, but yesterday I was just lazy. Shame on me. But - I'm sworn not to become one of those people who starts a blog and never updates it, so.

Time to start talking about the other half of the title.

The Library of Babel, if you'll remember, is a metaphor for the inherent vagueness of truth and falsehood - you cannot know that a statement is true simply by looking at it. You have to compare it to itself, and to other statements that you know to be true... which you don't necessarily really know to be true, either.

So doubt everything. Test every book you read. Can it describe itself? Or does it rely on another book that can, or another book that relies on another book that relies on an entire series that can? If it can survive your earnest flame - and you must be in earnest to avoid deceiving yourself - you may trust it with your life. If not, it was not worth keeping to begin with, as useful to you as the belief that you can live without breathing.

And if you think that's no way to live... I have a bridge to sell you.

There's this construct called mathematics, which is basically the above taken to extremes. It's not just for numbers - that would be the subject known as arithmetic - but the rigorous calculation of fact. Set theory is probably one of the most general sub-categories, dealing with absolutely anything that can be grouped; but also predicate logic, with begins with tautologies like "if A is true, then A is true" and "either A is true or A is not true", and builds from there. Because of this, mathematical proofs are absolutely reliable... and this leads to problems.

Let's talk set theory. Any group that can be described constitutes a set - the set of all rational numbers, for instance, or its subset that contains only the numbers 4, 18, and 6... or the set of all the books in my library, or the set of everyone who has ever had the name "Julius Caesar". You can even define a set whose elements are {white, 14, Literacy, [you]}, as long as you don't include yourself (or literacy, or white, or 14) more than once.

There are also sets whose elements are other sets - the set of {white, 14, Literacy, the set of all the books in my library}, for instance. This is where a few important distinctions come in:

(1) 14 is not {14}. A set containing a single element is not the same as that element; saying {14} + {7} = {21} is like saying that {apple} + {orange} = {some bizarre sum equal to apple+orange}.

(2) {14} is a subset of {14, white} because all the elements of {14} are also elements of {14, white}. 14 is not a subset of {14, white} because it is not a set. More weirdly, 14 is an element of {14, {14, white}} and {14} is a subset, but {14} is an element of {{14}, {14, white}} but not a subset. The brackets are important!

(3) {14, white} and {white, 14} contain exactly the same elements (i.e. they're subsets of each other), which means they must be the same set. Order doesn't matter.

Confused yet? If you are, ask me and I'll try to explain better.

1 comment:

Go ahead, say something! I'm the shy one here.