[qqqwww]

> A Turing machine has to be given the axiom of infinity to make this kind of inference, it cannot derive it in any way.

Why not? Are you aware of a proof of this? I think you are limiting the capabilities of Turing machines without evidence.

> Unlike the game AIs that repeatedly try to walk through walls.

Game AIs capabilities are a small subset of what a Turing machine can do. Most game AIs can't do speech recognition or solve math equations either.

> We programmers write halting programs with great regularity.

So do other programs. Writing a halting program is not an uncomputable problem, and doesn't require solving the halting problem.