"I started to post a comment, but it got long enough that I’ve turned my comment into a blog post."

So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the forms of sentences but their meanings. In other words, you have to understand meanings before you can carry out the operation of inference. But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.

... it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.

I should have said this more carefully.

Ifone allows enough rules of inference so that all the logical consequences of the axioms can be proved, then there are no automated proof checkers. So you can have proof checkers, but only at the cost of restricting the system so that not all logical consequences (i.e. implications that are true in every model) can be proved.I think there's a definitional issue here. Eugine is using "proof checker" to mean an algorithm that given a sequence of statements in an axiomatic system verifies that the proof is formally valid. You seem to mean by proof checker something like a process that goes through listing all valid statements in the system along with a proof of the statement.