Bob is coming from the perspective of Brouwerian Intuitionism, and so he is not really that interested in a closed "formal system". So, "proofs" in the setting that Bob cares about are not derivations in a formal system, so Gödel's result doesn't really apply.
In bob's setting, the computation system is not determined or fixed. It is open-ended...
In intuitionistic mathematics, we of course accept that there is no Turing computable halting oracle, but we do not rule out the possibility that there is some other effective oracle that can decide halting. (This is in contrast to recursive mathematics / Russian constructivism)
