What is Gödel’s incompleteness theorem, and why is it so infamous in Mathematics?
In: Mathematics
In math we have basic statements which are accepted as true called axioms. These are basically the fundamental ground rules of math. These, combined with rules of logic, allow us to deduce other, more complicated statements.
For a while in mathematics there was a big push by a handful of mathematicians/logicians to try and root *everything* that you could possibly prove into as few axioms as possible. Basically they wanted to take nothing for granted if it could be avoided and, from those few axioms, prove everything that could be proved true.
Stated more formally, they wanted a system that was both *complete* (everything that is actually true can be proved true) and *consistent* (nothing that is actually false can be proved true)
Gödel proved that this was false. That any but the most rudimentary systems will either be incomplete (there is something that is true that the system can’t prove true) or inconsistent (that there is something the system says is true but it actually isn’t).
It was infamous because it not only dashed the hopes of the top mathematicians at the time but it is somewhat counterintuitive to think that you can’t reduce all of math to a handful of simple axioms and rules.
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