# Why is $EQ_{cfg}$ not recognizable but is co-turing-recognizable

I've seen the proof that $$EQ_{CFG}$$ is not recognizable but its complement is, my problem is that in the proof that it's complement is recognizable, it says that we test every string in $$\sum^*$$ and checks if it's generated by one of the $$CFG$$s but not the other. My question is that why don't we use this method to solve $$EQ_{CFG}$$ problem, where we iterate on all possible strings in $$\sum^*$$ and confirm that it either gets generated or not by both $$CFGs$$

An algorithm that returns "true", must always do so in finite time, hence, it will never be able to go through all strings in $$\Sigma^*$$ and confirm that it is either generated not by both CFGs.
However, for the complement problem, the requirement changes drastically: It is enough to show one string $$w$$ such that $$w\in L(G_1)$$ and $$w\notin L(G_2)$$ (or vice versa) to know that $$L(G_1)\neq L(G_2)$$. This allows us to find this string $$w$$ and halt when we see it, without the need to continue and check more strings.