>>13765812Weak logics (in the sense of not being able to describe Peano arithmetic), or really any logic not subject to the Incompleteness Theorem. There was some interest in it for a while in the mid-20th century. But then they realized that nobody, not even 99.99% of mathematicians, gave a shit about syntactic completeness anymore, and for some reason people got bored researching alternative methods of "effecting" a logic, i.e. beyond the traditional "finitary" sense.
The field is so dead that most logicians are unaware of research into other kinds of semantics, and they kind of just assume,
>oh, you mean true Second-order semantics?>isn't that computationally equivalent to standard first order semantics?>didn't they prove it's incomplete in both senses, and therefore useless?because SURELY nobody would be insane enough to research something beyond the usual senses of "completeness", and SURELY nobody would consider that perhaps e.g. second-order logic being able to prove the uniqueness of infinite models might betray a structure both absent in first order logic, and more useful than just for limiting models.
