It is easy to exhibit an uncomputable function, the busy beaver function is the standard, easy example. It is immediately obvious that a solution to the halting

It is easy to exhibit an uncomputable function, the busy beaver function is the standard, easy example. It is immediately obvious that a solution to the halting problem make the busy beaver problem computable: just stick to running the programs that halt and pick the largest output. The unsolvability of the halting problem is way easier than Godel.