I'm asking some follow up question on the following;
Existence of a prime model for a theory
Because I'm doing a practice exercise for an exam that is extremely similar to this. Like what the OP asked; I wish to know how to find a prime model for the Theory $T$ in question, and the comeplete 1-type of it.
In the comments, the OP said " it is not hard to see that every model of $T$ contains (up to isomorphism) $\{…,−2,−1,0,1,2,…\}×ω."$ (I would assume $\omega$ refers to set of natural number). So my first difficulty is seeing this fact; I must have missed something obvious.
Next the OP said "Here we consider $\{k\}×ω$ as the elements $x$ such that $Pf^kx$ is true but $Pf^{k−1}x$ is false". I dont really understand what he meant by that.
Lastly; is there a way to show the existence abstractly instead of constructing one? And how does one describe the 1-types of the theory ?
As to the question of showing that a prime model exists, there are various relevant theorems. For example, Vaught showed that for a countable theory the existence of a prime model is equivalent to the set of principal types being dense in the type space. This paper of Knight and its bibliography might be of interest.
Now as to the specific problem, we begin with a general analysis of $T$. So fix $M\models T$ and let's see what we can say about it.
The key is to shift to a "geometric" description of $M$: specifically, axioms $(1)$ and $(2)$ say that we can think of $M$ as a bunch of copies of $\mathbb{Z}$ - basically, $f$ represents the successor operation on each copy. In more detail, for $x\in M$ let $$[x]=\{f^z(x): z\in\mathbb{Z}\}$$ be the set of elements of $M$ "connected" to $x$ by $f$; then for each $x$, the set $[x]$ looks like $\mathbb{Z}$.
Now all that's left to describe in $M$ is:
How many "$\mathbb{Z}$-copies" there are.
How $P$ behaves on each of the $\mathbb{Z}$-copies.
The axiom $P(x)\rightarrow P(f(x))$ tells us that there are three possibilities for how $P$ can behave on a given $\mathbb{Z}$-chain $[x]$: maybe $P$ holds everywhere, or $P$ holds nowhere, or $P$ holds "half the time:" there is some $z\in\mathbb{Z}$ such that $P(f^w(x))$ for all $w\ge z$ but $\neg P(f^w(x))$ for all $w<z$. Call these "type $1$," "type $2$," and "type $3$" $\mathbb{Z}$-chains respectively.
Now we see that $M$ is determined up to isomorphism by three numbers, $c_1, c_2, c_3$, where $c_i$ is the number of $\mathbb{Z}$-chains of type $i$ in $M$.
Our next step is to determine what possibilities we can have. Axiom $4$ says that we $c_3$ is always infinite; right away this gives us an idea for the prime model of $T$, namely countably infinitely many chains of type $3$ and no other chains. With a little work we can prove, based on this, the following:
Any collection of $\mathbb{Z}$-chains, infinitely many of which are of type $3$, yields a model of $T$.
$\aleph_0$-many chains of type $3$ and no other chains gives the prime model of $T$.
The above is a bit informal but it's not hard to make precise. In particular, we have an intuitive description of the prime model; the discussion at the linked question gives a detailed construction.
As to the $1$-types, the point is that in each chain of type $3$ every element looks different from every other. Specifically, each element is determined by its "distance from the switching point" (and which side its on); e.g. the last element on which $P$ fails, the second element on which $P$ holds, the seventeenth element on which $P$ holds, etc. This gives us a bunch of complete $1$-types, which we can think of as corresponding to integers if we want.