This is a key concept and will crop up several times again. Thus, clear thinking for the simple problems will make thinking about the complicated ones much more productive. The moral to this story is to take each solution and ask the kinds of questions that would reveal why it is impossible. Remember that evolution actively functions to eliminate the bad, not to select for the good. In the beginning there were antigens, then came antibodies, and now there are immunogens, epitopes, paratopes, antigenic determinants, immunoglobulins, and the list grows.
Many names are sales gimmicks, though some are ultimately useful.
I tend to think of pathogens as real antigens, and all else as an attempt to mimic a pathogen. Experimentalists like to avoid pathogens because they are hard to handle, particularly if you are a little sloppy. Then came the real chemical era when proteins could be synthesized chemically and all manner of bizarre structures were made. It was clear from the earliest days that there were vast numbers of antigens, even as pathogens, and certainly vastly more than there could possibly be genes to encode one antibody for every antigen. So, if evolution had to get by with less than one, we might ask how few.