How should I address a professor's inconsistencies?
This is an uncommon situation. It happens that a professor has a blackout in front of class or screws a proof - but this sounds much worse.

It is hard to tell from the distance, but to me it seems like the professor is having serious problems (maybe health issues, personal issues, ...) distracting him.

I would suggest talking to another trusted faculty member (if you don't know whom to adress: There are usually dedicated ombudspersons at every department / university) and tell them your observations.