1. I don't think human reasoning is consistent in the technical sense, which makes the incompleteness theorem inapplicable regardless of what you think about us and Turing machines.
2. The human brain is full of causal cycles at all scales. Even if you think human reasoning is axiomatisable, it's not at all obvious to me that the set of axioms would be finite or even computable. Again this rules out any application of Gödel's theorem.
3. Penrose's argument revolves around the fact that the sentence encoding "true but not provable" in Gödel's argument is actually provably true in the outer logical system being used to prove Gödel's theorem, just not the inner logical system being studied. But as all logicians know, truth is a slippery concept and is itself internally indefinable (Tarski's theorem), so there's no guarantee that this notion of "truth" used in the outer system is the same as the "real" truth predicate of the inner system (at best it's something like an arbitrary choice, dependent on your encoding). Penrose is referring to "truth" at multiple logical levels and conflating them.
In other words: you can't selectively chose to apply Gödel's theorem to the situation but not any of the other results of mathematical logic.
1. I don't think human reasoning is consistent in the technical sense, which makes the incompleteness theorem inapplicable regardless of what you think about us and Turing machines.
2. The human brain is full of causal cycles at all scales. Even if you think human reasoning is axiomatisable, it's not at all obvious to me that the set of axioms would be finite or even computable. Again this rules out any application of Gödel's theorem.
3. Penrose's argument revolves around the fact that the sentence encoding "true but not provable" in Gödel's argument is actually provably true in the outer logical system being used to prove Gödel's theorem, just not the inner logical system being studied. But as all logicians know, truth is a slippery concept and is itself internally indefinable (Tarski's theorem), so there's no guarantee that this notion of "truth" used in the outer system is the same as the "real" truth predicate of the inner system (at best it's something like an arbitrary choice, dependent on your encoding). Penrose is referring to "truth" at multiple logical levels and conflating them.
In other words: you can't selectively chose to apply Gödel's theorem to the situation but not any of the other results of mathematical logic.