Confused about Logic Machine in The Lady or the Tiger by Smullyan In The Lady or the Tiger by Raymond Smullyan, on p.172, it says: "I am working in a language that contains names of various sets of numbers--specifically, positive integers. There are infinitely many sets of numbers nameable in this language. For example, we have a name for the set of even numbers, one for the set of odd numbers, one for the set of prime numbers, one for the set of all numbers divisible by 3--just about every set that number-theorists are interested in has a name in the language. Now, although there are infinitely many nameable sets, there are no more nameable sets than there are positive integers. And to each positive integer n is associated a certain nameable set A_n. We
can thus think of all the nameable sets arranged in an infinite sequence A_1, A_2,..., A_n..."
I'm confused about the bolded part. Wouldn't the number of nameable sets in this situation be the power set of the positive integers? If so, wouldn't it be uncountable and therefore have a larger cardinality than the positive integers? And wouldn't that mean that these nameable sets couldn't be arranged in a "1, 2, ..., n..." sequence?
I haven't read the rest of the book, and despite being a lover of logic puzzles, I don't really understand Godel's Incompleteness Theorems or have formal education in advanced mathematics. I was wondering if I'm missing something or interpreting something wrong. Anyone care to provide enlightenment on the situation?
Edit: I just realized something. Does it have to do with the number of expressible permutations in a phonetic language? Or whatever other restrictions a language would probably have?

The comment in your edit is right. You need to think about what a "name in a language" might be.
If the language has countably many characters or words and each name is a finite string of characters or words then the number of names is countable, which is the point Smullyan is making.
If you have a broad enough definition of "language" then each subset of the integers could name itself in your language and your argument about the power set would be correct.
