system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5:How to Kurt Godel manage to "break" mathematics?
gpt
Godel did not "break" mathematics, but rather he proved that there existed limitations to the degree we could root complex mathematics in basic, formal logic.

Philosophers and mathematicians wanted to devise a system where you could take a few basic axioms (unproved statements accepted as generally true), some rules of inference, and then construct every possible mathematical statement that was true.

There are two qualities we want in a system:

1. Consistency - The system never produces a result that is false (for example: 2+2=5)

2. Completeness - If something is true, then our system is capable of producing it as a result (for example: 2+2=4 is true, so we want our system to be able to prove that).

Philosophers and mathematicians spent a great deal of effort trying to develop such systems, until Godel proved that you cannot have a system that is both consistent or complete. A fully consistent system won't be complete (it won't be able to prove everything that is true) and a fully complete system won't be consistent (the system will "prove" things that are false).

Needless to say, it basically shelved such efforts to ground math in pure logic, and any logical system of mathematics has to settle for being either incomplete or inconsistent.