Goldbach's conjecture states that
$$\text{Every even integer greater than 2 is the sum of two primes}$$
Is it true to say that Goldbach's conjecture is formally claiming that:
$$\forall 2(k+1),\exists\; p_1,p_2 \in \mathbb P:k \in \mathbb N^+ \implies 2(k+1)=p_1+p_2$$
Or $$\forall n \in \mathbb E,,\exists\; p_1,p_2 \in \mathbb P: n=p_1+p_2$$ Where $$\mathbb E:=\left\{n:n=2(k+1), k\in \mathbb N^+\right\}$$
And $\mathbb P$ is the set of prime numbers.
$\forall 2(k+1)$ is not a correct logical formulation. You could write instead $$ \forall\ k \in {\Bbb N}-\{0,1\}\ \exists\ p_1, p_2 \in {\Bbb P}\quad 2k = p_1 + p_2 $$ But of course, the original formulation without any logical symbol is highly preferable.