>>11177675First of all, it should be said that this has been a popular subject of nerd discussion on the internet for many years, which involves lots of subtleties. Lots of anons have therefore misunderstood or misrepresented the theorems themselves. I myself don't fully understand the content of the theorems, despite having read about them. Therefore, take my sketch with a grain of salt and do your own reading if you want to learn more. I recommend Goldstein, Incompleteness and esp. "Gödel's Proof".
Basically any time you have some mathematical language-system which is "enough like" (this has very precise definitions) what we call arithmetic, algebra, etc, there crop up statements that can neither be proven nor disproven in that system. The system is "incomplete" in the sense that you can construct statements in it which can't be decided one way or other. And you can PROVE that certain such statements can't themselves be proven, by giving examples of such statements. You can prove that there are things you can't prove, in a given math/language format.
This is significant in math, because one of the central activities is to prove things. Proofs aren't the end-all be-all, but they are central to math as we currently understand it.
The second theorem says something like: you can never PROVE that such-and-such a math/symbol system can never give rise to inconsistent (self-defeating) statements. Obviously, here I am being more vague.
The Gödel's Proof book concludes with a useful philosophical takeaway, following proof details which I (obviously) didn't fully understand on a first read-through: the significance of the theorems is not that they "ruin math", and it's also not that they give oxygen to any vague hippie bullshit. Rather, you can still discover/construct useful and interesting systems, but just prepare: you may not be able to achieve a perfect save-state and nab all accomplishments in your video-game of choice.
Did I screw anything up?
