"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.

Various theorems, lemmas, and other principles equivalent to the Axiom of Choice (e.g. Zorn's lemma) were argued over until it was established (by Kurt Gödel and Paul Cohen) that the AoC is entirely independent of the ZF axioms, i.e. ZFC and ZF!C are both consistent systems. I think this is the canonical example.

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona