"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.

Is there really that much difference between Second Order Logic, and First Order Logic quantifying over sets, or over classes?

Put it another way, Eliezer is using SOL to pinpoint the concept of a (natural) "number". But to do that thoroughly he also needs to pinpoint the concept of a "property" or a "set" of numbers with that property, so he needs some axioms of set theory. And those set theory axioms had better be in FOL, because if they are also in SOL he will further need to pinpoint properties of sets (or proper classes of sets), so he needs some axioms of class theory. Somewhere it all bottoms out in FOL, doesn't it?

Why isn't it necessary to pin point FOL?