illuminating science

Entropy Core (who in turn picked it up from a post on Lambda the Ultimate) has posted a link to an online book on Godel’s Incomleteness Theorem. Actually, it’s only the first 12 chapters so far, but that’s probably enough for most!

Godel’s first Incompleteness Theorem says (quoting the Wikipedia) “In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system.” In layman’s terms, what this says is that if you try and make a model that has a set of basic rules and includes the idea of counting numbers (-1,0,1,2,3,etc) then you can always think of statements that are neither true nor false, at least within the model you’ve made. A (sort of) example is the sentence “This statement is false.” If the statement is true, then the statement must be false. Oh. Well, then the statement is false. But then the statement says it must be true, which means it must be false, which means…And so on. It is neither true nor false, neither provable nor disprovable. What’s really interesting is that this theory is completely rigorous and mathematical - it’s not just a philosophical argument.

The book is quite readable for someone (like myself) who’s had experience with mathematical proofs and seen set theory before. If you’re not really a maths person, you might find it tough going. Other sources include the Wikipedia and Google turns up a number of pages. Unfortunately, a lot of it is either very “airy fairy” or hard to understand without a background in mathematics. If I find a good summary, I’ll post it.