Co-induction is the mathematical or logical suggestion from category theory that induction be extended to worlds of things that have no kernel value, like the natural numbers have .
This kind of thinking is usually criticized for being circular logic, which it is. (Sometimes circular logic is enough in an argument; self-consistency might be all you get from analyzing a situation. Then you know simply to step out of the context.) However, there are important ways of thinking about the world that need this kind of thought process to sort out, like information coming in from an endless stream. When you receive something from a stream, you wind up with that thing plus another stream. There's no end to dividing up the stream until you get to some basic thing; you simply get more streams.
One other nice thing about this is to be able to think of paths that are travelled that might lead you to places you've already seen as containers for future possibility, sometimes called possible worlds.