Can Peano axioms be used to construct a model of Natural numbers? I understand that there exist first-order and second-order Peano axioms, but I am mainly concerned with latter. Specifically, the axioms as listed here.
As discussed here and here, it seems that the induction axiom is needed to create a set that matches our intuitive understanding of Natural numbers. 
Can we say that the induction axiom constructs set of Natural numbers using successor function and $\boldsymbol{0}$ , and hence is providing us with a model of Natural numbers?
PS. I know very little about logic and model theory.
UPDATE: Comments correctly pointed out that I should explain my degree of knowledge about logic and model theory (See @Asaf Karagila and @Skyking comments). Here is some more details my understanding of a model is that it provides meaning to the undefined terms of axiomatic systems. The undefined terms are defined in a way that  our informal intuitions are mathematically formalized (see this). 
In addition, I understand that if an axiomatic system has a model then it is consistent.
What I am trying to understand is that:


*

*What are the undefined terms in the Peano axioms? to me they are $\boldsymbol{0}$ and the successor function.

*Since the successor function and $\boldsymbol{0}$  are the concepts that can be understood by our intuitions (they are concrete concepts in our time. I understand they haven't always been like this), does this mean the Peano axioms are providing us a model for the Natural numbers. 

*I think $\boldsymbol{0}$ and the successor function are not enough to provide the model because as mentioned in the comments there exist sets that can be constructed using $\boldsymbol{0}$ and the successor function that arenot Natural numbers and hence the induction axiom is needed if we want our model to correctly math the Natural numbers.


There were also comments about the word construct, what I mean by construct is the use of already known objects to provide model for other structures. For example, Dedekind cuts of rational numbers to construct the reals. The use of he word construct for the Natural numbers occurred to me because of the wording here, in particular, the word generate, which I quote 
" However, considering the notion of natural numbers as can be derived from the axioms, axioms 1, 6, 7, 8 do not imply that the successor function [generates] all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number"
 A: No, the induction axiom does not construct the natural numbers, but in order for the induction axiom to hold (in addition to the other Peano Axioms), your model does need to be isomorph to the natural numbers.  Indeed, it is almost the other way around: the induction axiom does not generate or construct or force there to be infinitely many ordered elements in any model, as this is what the other axioms already do, but rather the induction axiom makes sure that in any model there aren't any further elements. That is, the induction axiom restricts the model to be 'nothing more' than the natural numbers.
To give at least some intuitive idea about this: Any model of the Peano axioms without the induction axiom is already required to contain an infinite number of elements that exist somewhere in an infinite chain of successors starting with $0$, i.e. something like the natural numbers. To see that, note that we need a '$0$' object (axiom 1) and that this $0$ object needs a successor (Axiom 6), and that this successor of $0$ cannot be $0$ itself (Axiom 8). So, we need a second object. But this second object needs a successor (Axiom 6), which cannot be $0$ (Axiom 8), but also cannot be itself, for we cannot have two different objects with the same successor (Axiom 7). So, we need a third object ... and  this process keeps repeating itself: every time we add the new object that we are forced to add as the successor of the last object we added before that, that new object needs a successor itself, and that successor cannot be $0$ (Axiom 8), nor can it be any of the other already existing objects, for then we would get two different objects with the same successor, which goes against Axiom 7. So, you could say that the natural numbers (or at least an infinite sequence of successive objects) is already 'constructed' by these 4 axioms.
OK, so where does the axiom of induction come in? Well, while the other axioms require for any model to contain an infinite sequence of successive element, these other axioms do not rule out any additional elements in the model: elements that do not occur  in this infinite sequence of successors. Indeed, without the axiom of induction you could have as a model $\mathbb{N} \cup \{ apples, bananas \}$, ... as long as you define how the interpretation of the successor, addition and multiplication function symbols apply to $apples$ and $bananas$ ... but that can all be done without problem.
However, the induction axiom says that 'if $0$ has some property $P$, and if $s(n)$ has property $P$ whenever $n$ has property $P$, then all objects have property $P$. Well, if you have that $0$ has some property $P$, and that $s(n)$ has property $P$ whenever $n$ has property $P$, then obviously all the objects that exist somewhere in the infinite chain of successors have property $P$, but how can you say that all the additional objects also have to have property $P$? You can't of course. So, the fact that the induction axiom says that we can conclude that all objects have property $P$ effectively rules out any of those additional objects in the domain, and hence your model needs to be isomorph to the natural numbers.
A: You will find a more commonly used presentation of Peano's Axioms here. The induction axiom doesn't "construct" anything. It is simply one of the essential properties of the set of nature numbers and the successor function defined on it. Without it, we would not be able to prove, for example, that no number is its own successor.
