No matter how many articles I read on this subject I cannot comprehend how it proves what it proves. I do well with words and rhetorics, philosophy and science – but as soon as you add numbers my mind goes blank. Not very helpful when those fields often rely on equations and models for explanations and proof. I can somewhat understand equations if explained in a simple or cohesive way – but if at all possible analogies or just word-centric explanations would be very helpful.
In: 25
For thousands of years mathematicians have relied on proofs as the basis of mathematical systems. Statements can either be true or false, and finding ways to prove whether a given statement is true or false allows you to build larger and more complex systems.
But can all statements be proven true or false? Some mathematicians believed that should be true, and they set about to find or create a system that would allow them to do that.
However not all people believed that to be the case and Godel, with his incompleteness theorems proved (somewhat ironically) that such a system could not exist. That there would ALWAYS be statements which, while they could be true, could also not be proven.
How did he do it? Well on a very abstract level Godel created a mathematical version of the statement “This statement can not be proven”. If it COULD be proven it would set up a paradox, because it would be simultaneously false (because it had been proven) and true, because there was a proof that it was true. Instead the statement had to be true, but also unprovable (I. The mathematical sense). He further proved that this would be the case for any mathematical system, that no matter how you tried to adjust it to account for true but unprovable statements like his own, you’d simply introduce more.
The YouTube channel Veritasium has a great, though lengthy explanation in the following video:
Latest Answers