=James Propp There are only countably many sets of natural numbers that form (the nonnegative part of) a congruence classes. Here's a natural way to list them as bit strings: [...] We can use diagonalization to construct a set that isn't a congruence class: 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 ... Does this pattern continue?
Not sure what you're asking. Your strings for congruence classes all have the form of isolated 1's separated by blocks on N 0's. Since the constructed string doesn't have that form it isn't a congruence class, regardless of how you generated it. It appears to have the alternating form 1 0, 2 1's, 3 0's, 4 1's, 5 0's and so on. Are you asking if the diagonal selection process from the array has this apparent structure in fact?