No, it doesn't. Is this a language difficulty? For example, I could say about the set {0} that "there is no number in the set less than 0". That does NOT mean that there are numbers in the set that are more than 0.

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" meansand that wont exactly define 0...onlythat "there are numbers that are more than 0" and it does not.

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?