Goedel's Incompleteness Theorem?

Question

can some1 please try to explain to me what this theorem is?
(no copy paste job please)

i had a brief look at his theorem and it had been brought up in several of my discussions as far as i have been told it is a "proven" theorem that shows that nothing can be "proven".

help me out here please.

Answer 1

From your summary, I can see the people are rather misrepresenting Gödel's Incompleteness theorem a bit.

Here is a fairly straightforward way to put it: For ANY logical or mathematical system, it is possible to formulate a statement in that system which is true, but which cannot be proved by the system. Thus no such system can be both consistent and complete.

In other words, you cannot develop a system which proves EVERYTHING. No matter what you do, some things will be true, but you won't be able to prove it.

This does NOT mean that NOTHING can be proven! All the proven theorem states is that there will be one (1) statement which is unprovable. As with mathematical systems, you may be able to prove even an infinite number of things, and this proof will be reliable. Just not EVERYTHING.

Curiously enough, it does mean that if someone comes up with a theory which does explain everything, then is MUST be internally inconsistent. It's not much of a limitation for most scientists and mathematicians aren't willing to sacrifice consistency for completeness and are pretty accustomed to having many, many problems that they don't know the answer to. Instead, it may be more inhibiting to other kinds of systems... those who DO think they have all the answers...

Answer 2

To really understand Gödel's Incompleteness Theorem, that is, to understand why it should be true, you need quite a bit of education in formal logic. I only took one class in logic, so I can't help you much. I can tell you basically what it is, but not why it is.

The gist of the idea, as simply as I can put it, is that no axiomatic system can be both sound and complete. Arithmetic is an example of an axiomatic system. Gödel proposes that for any such system, an example can be constructed that is true, but not provable within the system. This means, in essence, that arithmetic is incomplete. It can't express everything that had thought, prior to this theory, that it could.

The actual impact of this theory on mathematics is negligible. It absolutely does not show that "nothing can be proven." It has been widely misunderstood as undermining the fundamental nature of mathematics, which it does not do. Two plus two definitely equals four, in other words.

Answer 3

As I understand it, he proved that in every logically consistant system there is at least one true statement that cannot be logically proven.

Answer 4

beware philosophical paradoxes