>>12567377Let's introduce two definitions (about logical systems):
Incomplete - An incomplete system has some statement T and its negation ~T, of which *neither* can be proved in the system.
Inconsistent - An inconsistent system has some statement T and its negation ~T, of which *both* can be proved in the system.
Gödel's first incompleteness theorem proves that any consistent system is incomplete - that is to say, the only way for a system to be complete is for it to be inconsistent (and inconsistent systems are trivial and useless, because every single statement that you can express in the system (and all their negations) are true, due to the explosion principle -
https://en.wikipedia.org/wiki/Principle_of_explosion). Essentially, every useful logical system is inconsistent.
Gödel's second incompleteness theorem proves that no consistent system can prove that it is consistent within itself. You could use a different system to prove that that system is consistent, however then you would need to prove that the other system you used is also consistent. This results in either an infinite regress of needing to prove consistency of another system (if you use a new system each time to prove consistency of the old one) or circular logic (if you use system A to prove system B's consistency and system B to prove system C's consistency and then system C to prove system A's consistency). So you essentially can't prove that any system is consistent.
Proof is left as an exercise to the reader.
