Sure, and that's the age-old argument for why we should not take second-order logic at face value.
I'm not familiar with the literature of this argument. (It was probably clear from the tentativeness of my comment that I was thinking my own murky way through this issue.)
You seem to take it as the default that we should take second-order logic at face value. (Now that I know what you mean by "face value", I see that you did acknowledge this issue in your earlier comment.) But I should think that the default would be to be skeptical about this. Why expect that we have a canonical model when we talk about sets or predicates if we're entertaining skepticism that we have a canonical model for integer-talk?
Why expect that we have a canonical model when we talk about sets or predicates if we're entertaining skepticism that we have a canonical model for integer-talk?
We don't. Skepticism of sets, predicates, and canonical integers are all the same position in the debate.
A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).
ETA: It would seem that rationality quotes are no longer desired. After several days this thread stands voted into the negatives. Wolud whoever chose to to downvote this below 0 would care to express their disapproval of the regular quotes tradition more explicitly? Or perhaps they may like to browse around for some alternative posts that they could downvote instead of this one? Or, since we're in the business of quotation, they could "come on if they think they're hard enough!"