for all problems, there exists an axiomatic system to prove or disprove it
given an axiom (we assume to be true), we extend it with theorems
incompleteness theorem:
- for all axiomatic systems, there exists a statement that can’t be proved or disproved.
- one of them is if the system is consistent or not
consistent system:
- we can both prove / disprove statement?
- suppose we prove an axiomatic system is consistent, then we need to prove the larger system is also consistent. this is an infinitely extensible argument, so it fails at some point
”mathematics is a very honest science. it’s the only field that proved its incompleteness”