Proof theory for second-order logic seems to be problematic, and I have a formalist stance towards mathematics in general, which leads me to suspect that the standard definitions of second-order logic are somehow smuggling in uncountable infinities, rather than justifying them.
But I admit second-order logic is not something I’ve studied in depth.
Yeah, second-order logic is basically set theory in disguise. I’m not sure why Eliezer likes it. Example from the Wikipedia page:
There is a finite second-order theory whose only model is the real numbers if the continuum hypothesis holds and which has no model if the continuum hypothesis does not hold. This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality. This example illustrates that the question of whether a sentence in second-order logic is consistent is extremely subtle.
Yes, that’s my understanding as well.
Proof theory for second-order logic seems to be problematic, and I have a formalist stance towards mathematics in general, which leads me to suspect that the standard definitions of second-order logic are somehow smuggling in uncountable infinities, rather than justifying them.
But I admit second-order logic is not something I’ve studied in depth.
Yeah, second-order logic is basically set theory in disguise. I’m not sure why Eliezer likes it. Example from the Wikipedia page: