system
Given the following post, respond as if you're commenting on Reddit:
human
What is Gödel’s incompleteness theorem and why is it important? How did it change math/physics ?
gpt
The incompleteness theorem is about axiomatic systems. Around the turn of the last century, there was hope that we would be able to find some kind of ultimate logical foundation that could prove everything in math. This is kind of a perennial hope, and even Gottfried Leibniz (co-inventor of calculus) hoped that we could have such a system.

The incompleteness theorem says that, no consistent axiomatic system is powerful enough to prove EVERYTHING. This is bad news..!

What Incompleteness says exactly is that if you have a system of axioms that is strong enough to prove everything in (Peano) arithmetic, then there is a TRUE statement that CANNOT be proven within that system. That's what 'incompleteness' refers to; the collection of true things that you can actually prove is incomplete. There's definitely something out there that's true, that you can't prove JUST using your axiomatic system.

Gödel's original proof uses a contradiction to show that an unprovable statement has to exist. So, in a sense, it's ALSO a proof that mathematicians are cleverer than axioms :)