Why is residues modulo 2 a model for natural numbers I am reading the book "Introduction to Metamathematics" by Kleene. I am now in Chapter 8 which is about "Systems of objects". I must warn you that this chapter is meant to be an introduction and, therefore, it is written in an intuitive language. Still, I would like to understand the details.
There, he defines: 


*

*By a system of objects, we mean a set or class or domain $D$ of objects among which are established certain relationships. For example, the natural number sequence (given in his previous chapters by something very similar to Peano axioms) constitutes a system of the type $(D,0,')$, where $D$ is a set, $0$ is the member of the set, and $'$ is a unary operation. 

*Any specification of what the objects are gives a model of the system, i.e. a system of objects which satisfy the relationships and have some further status as well. 

*Two models of the same abstract system are isomorphic, i.e. can be put into 1-1 correspondence preserving the relationships. 
Then, he gives an example of a model for the abstract system $(D, 0, ')$ to be the one with two distinct objects $0$ and $1$, where $0' = 1$ and $1' = 0$, and calls it residues modulo 2.
I do not see how this fits with the definitions. 
I guess that usual natural numbers, i.e. sequence $0, 1, 2, ...$ is a model for the natural numbers. 
Question 1: Are natural numbers a model for natural numbers $(D, 0, ')$ ? Intuitively, I feel that they are, because I give some information about the objects of the system, in a sense that I give names for each of the objects. 
From definition 3, two models of the same abstract system can be put into 1-1 correspondence. Now, if residues modulo 2 have only two objects then I do not see how that could be given a 1-1 correspondence to a sequence $0, 1, 2, ... $, which I assume is a model for the abstract system of natural numbers. 
What intuitively I would think could solve the problem is, that, for example, we have ordered pairs $(0, 0), (1, 0), (0, 1), ...$, where the first coordinate would be the way we wanted it to be, but the second coordinate shows which time $0$ or $1$ is produced. 
Question 2: How can the model of residues modulo 2 be given a 1-1 correspondence with other models, such as natural number sequence?
I would appreciate your help and any comments.
 A: I've had a look at this now and agree that Kleene's terminology is a little confusing. (Aside: your reference isn't quite right: it's section 8 and is in chapter II.)
Kleene goes on to say that "two different systems $(D_1, 0_1, '_1)$ and $(D_2, 0_1, '_2)$ of type $(D, 0, ')$ are (simply) isomorphic if there exists a $1$-$1$ correspondence between $D_1$ and $D_2$ that [preserves all the structure]".
What Kleene is calling an "abstract system" is an isomorphism class of models in more modern terminology. When he says "model of the abstract system" he means "representative of the isomorphism class". His notation is a bit confusing: when he says "of type $(D, 0, ')$", he just means what we would nowadays call (a model) with signature $(0, ')$. In modern terminology, explicit representations of the natural numbers and of the residues modulo 2 give two different models for the signature $(0, ')$. In Kleene's terminology the two representations belong to two distinct abstract systems of the same type.
