I am trying to construct a non-standard model < A,0,S,+,*,E,< > that has as its domain the natural numbers plus the letter a such that the model 1. Makes all of the axioms of number theory true (Say, Mendelson's S) 2. Makes a < a. So in this model the domain has already been specified. We take < to be just like < in the standard model except that we add the couple <a,a> to it. We take 0 to be the same as in the standard model. We take S to be just like S in the standard model with the addition that that S(a)=a. We take + to be just like the standard model except that a + a = a 0 + a = a for n > 0, a + n = n We take E to be the same as in the standard model with the addition that aE0 = S(0). Does that do the trick? I can't see where I've made any of Mendelson's axioms false.