You should be careful with addition and multiplication—to use them, you would have to define them first, and this is not trivial if you have the natural numbers plus A->B->C->A, infinite chains and so on.
In addition, “group” has a specific mathematical meaning, if you use it for arbitrary sets this is quite confusing.
You don’t have to define addition and multiplication—you can just make them be a part of your language. In fact, in first order theories of arithmetic, you have to do so because you cannot define addition and multiplication from successor in first order logic.
In other words, the difficulty is with the language not with whether you happen to be using a standard or a non-standard model. This is a general rule in model theory (and for that matter everywhere else): what you can express has to do with the language not the subject.
You should be careful with addition and multiplication—to use them, you would have to define them first, and this is not trivial if you have the natural numbers plus A->B->C->A, infinite chains and so on.
In addition, “group” has a specific mathematical meaning, if you use it for arbitrary sets this is quite confusing.
You don’t have to define addition and multiplication—you can just make them be a part of your language. In fact, in first order theories of arithmetic, you have to do so because you cannot define addition and multiplication from successor in first order logic.
In other words, the difficulty is with the language not with whether you happen to be using a standard or a non-standard model. This is a general rule in model theory (and for that matter everywhere else): what you can express has to do with the language not the subject.