The circularity I am looking at is that my ability to have "actual questions you could ask a computer program that it can't answer" is reliant on my ability to generate them and verify the results (as in, that they answered the question correctly or not) using algorithms, which inherently limits the cardinality of the set of possible questions to the cardinality of the set of possible answers, but I feel only because we are starting from inside of the algorithmic box.
To me, it is very similar to claiming (which many do; I won't, btw, go so far as to say that it is a poor way of looking at the world: this may be more practical) that there is no such thing as a set that has infinite cardinality (whether we are talking countably, uncountably, or something even more infinite than that) because no one would be able to count how many elements are in the set, nor would the set have been able to be constructed in the first place.
Maybe "circular" isn't the right term, but it does seem to rely on an assumption of the result (hence my usage of "begging"). My goal, then, in switching from "problem" to the more specified "language" is that, if you believe in the concept of "infinity" in the first place, I feel justified in my ability to prove that the set of languages is uncountably infinite, and the fact that I can't enumerate (or possibly even choose) elements from that set is not relevant.
To me, it is very similar to claiming (which many do; I won't, btw, go so far as to say that it is a poor way of looking at the world: this may be more practical) that there is no such thing as a set that has infinite cardinality (whether we are talking countably, uncountably, or something even more infinite than that) because no one would be able to count how many elements are in the set, nor would the set have been able to be constructed in the first place.
Maybe "circular" isn't the right term, but it does seem to rely on an assumption of the result (hence my usage of "begging"). My goal, then, in switching from "problem" to the more specified "language" is that, if you believe in the concept of "infinity" in the first place, I feel justified in my ability to prove that the set of languages is uncountably infinite, and the fact that I can't enumerate (or possibly even choose) elements from that set is not relevant.