I tried to Google Collatz and Gödel to see if anyone's done any work like this, and all I found was a now-private Youtube video with the description "Reframing the Collatz Conjecture using Gödel numbering..."

https://www.youtube.com/watch?v=VPQx1q9cfkU

...so maybe there's something to this line of research. I don't think this is a bad idea at all.