"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 don't know if this is true, but I once had a lecturer tell me that there used to be considerable debate over the question of whether a derivative of a differentiable function was necessarily continuous, which ultimately boiled down to the two sides having different definitions of continuity, but not realising it since neither had ever fully set down their axioms and definitions.

I've heard that before also but haven't seen a source for it. But keep in mind that that's a question about functions on the real numbers, not about what is being called "numbers" here which seems to be a substitute for the integers or the natural numbers.