## General - if a formal system is consistent, then it's incomplete - (there are some statements that can't be proven) - we assume a consistent formal system, then Godel gets this statement with Godel numbering: - "This statement can't be proven (my note: proven as true)" - if the statement is false, it means the statement *can* be proven as true, but this this would mean the statement is both false and true - note we assumed a consistent formal system in the beginning, so this is impossible, meaning the statement can't be false - if the statement is true (the only possibility), this means that in any consistent formal system, there are true statements that are not provable $QED$ ## Intuitive Corrections - this first theorem shows that in any [[Formal System]], there are truths that cannot be proved within the system itself - there is a common misunderstanding to say that there are truths in general that can't be proved, this is simply not what the result says - **for any statement $S$ unprovable in a specific formal system $F$, there are trivially other formal systems where $S$ is provable (you could take $S$ as axiom in that system)**