Encoding n as the set of all m < n is called "von Neumann ordinals". Encoding (positive) n as just the singleton set containing n's immediate predecessor is called "Zermelo ordinals". The main advantage of using the former representation rather than the latter is that it allows uniformly encoding not just finite ordinals, but also transfinite ordinals, many of which do not have an immediate predecessor. E.g., in the von Neumann ordinal system, the infinite set of all finite ordinals may itself be interpreted as an ordinal value larger than every finite one. (And then the set of finite ordinals ∪ {the set of finite ordinals} becomes yet a larger transfinite ordinal still, and so on...)