I know that such a $2$-predicate formula exists since Enderton's have already constructed such a formula in his text on mathematical logic but it was not easy to remember so I wonder if there is other known ways to construct such a formula.
My questions is, Are there more than one way to express such a statement in FOL? If yes, give some examples to do that. I'd love to see if any of those ways (if exist) is simple enough to remember (or give an intuition that enables oneself to re-construct it)
The question can be reforumlated as: Is there a simple trick that enables us to express "$P$ is the $n$-th prime" in first order logic?
Here is the way Enderton's used to construct such a formula:
The function whose value at $a$ is $p_a$, the $(a + l)$st prime, is representable. (Thus $p_0 = 2, p_1 = 3, р_2 = 5, p_3 = 7, p_4 = 11$, and so forth.)
Proof. $p_a = b$ iff $b$ is prime and there exists some $с < b^{a^2}$, such that (i)-(iii) hold:
(i) $2$ does not divide $c$.
(ii) For any $q < b$ and any $r \leq b$, if $(q, r)$ is a pair of adjacent primes, then for all $j < c$,
$$q_j | с \iff r^{j+1} | c$$.
(iii) $b^a$ divides $с$ and $b^{a+l}$ does not. This equivalence is not obvious, but at least the relation defined by the right-hand side is representable.
One alternative way is to first prove that the number of primes $π(n)$ in the range $\{1\,..\,n-1\}$ is representable, and then "$x$ is the $n$-th prime" is represented by the formula:
Since $π(0) = 0$ and $π(n+1) = π(n) + \mathbf{1}_{prime(n)}$ for any natural $n$, and since $prime$ can be represented by a $Δ_0$-formula, we can represent this sequence $π$ by a $Σ_1$-formula using Godel's β function.