No memory in the system (between old and new), no problem. No
  paradox. We're the missing link. Our memory. Oh and the naming
  thing. "What's in a name?" Context answers it easily enough. One
  context, same ship. Another context, different ship. Metaphor
  for continual self-renewal of self as we're the same people we
  were once, yet entirely different, both physically and
  psychologically. But if there was no history, no memory, no
  records... well.. there's be no old and no new. It would simply
  "be". Paradox resolved via amnesia. Not that you asked 'cause
  you didn't but I always answer my own questions, even when I'm
  answering someone else's. If it happens to resemble an answer
  they wanted to see, well, huzzah