It applies to systems that are complex enough to formulate the Godel sentence, i.e. “I am unprovable”. Gödel did this using basic arithmetic. So, any system containing basic arithmetic is either incomplete or inconsistent. I believe it is still an open question in what other systems you could express the Gödel sentence.
Doesn’t that only apply for sufficiently complicated systems? Very simple systems could be provably self-consistent.
It applies to systems that are complex enough to formulate the Godel sentence, i.e. “I am unprovable”. Gödel did this using basic arithmetic. So, any system containing basic arithmetic is either incomplete or inconsistent. I believe it is still an open question in what other systems you could express the Gödel sentence.
I think it’s true for any system. And I’d say mathematics or just logic are simple enough. Every system stems from unprovable core assumptions.
Propositional logic as a system is both complete and consistent.