However, in this case there are, of course, numbers larger than 0. But you seem to think that "there is no number less than 0" means only that "there are numbers that are more than 0" and it does not.and that wont exactly define 0...
The problem with that is that you would first have to define "less than every y". You can do that, of course, but are you still in First Order logic?And since the structure only contains natural numbers, then I dont think we needed
to worry about numbers less than 0.
But I was thinking about something like this though in First Order: "There is one and only x less than every y"
Then x has to be 0 because its less than every number. Could this be it? Or can someone think of something better?