# Is there a semi-decidable statement equivalent to the Collatz-conjecture?

We cannot rule out that the Collatz-conjecture cannot be proven. But we also cannot rule out that it is false and we cannot prove this in the case the sequence diverges for some start-number.

Is there a provable semi-decidable statement equivalent to the Collatz-conjecture ?

There is at present no $$\Sigma^0_1$$ (= provably true if true) or $$\Pi^0_1$$ (= provably false if false) sentence which is known to be equivalent$$^1$$ to the Collatz conjecture.
Note that checking whether a sentence is $$\Sigma^0_1$$, or $$\Pi^0_1$$, is decidable: it's a purely syntactic property. The issue is the "equivalent" bit.
$$^1$$Equivalence of sentences isn't a meaningful notion without specifying a "base theory:" two sentences $$p,q$$ are equivalent over $$T$$ if $$T$$ proves $$p\leftrightarrow q$$. The naive "theory-free" definition of equivalence - "$$p$$ is equivalent to $$q$$ if $$p\leftrightarrow q$$ is true" - leads to all true statements being equivalent and all false statements being equivalent. When we replace "true" with "provable," we get something more interesting, but then we have to specify a theory: provable in what system?
I haven't specified a base theory above. However, there is no reasonable $$T$$ over which the Collatz conjecture is known to be equivalent to a $$\Sigma^0_1$$ or $$\Pi^0_1$$ sentence. If you like, take $$T$$ to be ZFC.