*Ofra*: "is there some reasons to believe that the Collatz conjecture is undecidable?"
There is reason to believe a generalized version is undecidable.
This was explored by John Conway in "On Unsettleable Arithmetical Problems."^{1}
And this paper proves a version recursively undecidable:

Kurtz, Stuart A., and Janos Simon. "The undecidability of the generalized Collatz problem." *International Conference on Theory and Applications of Models of Computation*. Springer, Berlin, Heidelberg, 2007.
(Springer link.)

**Abstract**. The Collatz problem, widely known as the $3x + 1$ problem, asks
whether or not a certain simple iterative process halts on all inputs.
We build on earlier work by J. H. Conway, and show that a natural
generalization of the Collatz problem is $\Pi^0_2$ complete.

Here is their generalization:

^{1}Conway, John H. "On unsettleable arithmetical problems."

*American Mathematical Monthly* 120.3 (2013): 192-198.
(

Jstor link.)
Reprinted in the

*Best Writing on Mathematics 2014*.

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