How do humans know that a mathematical sentence is true? There is only one way: by proving it. Otherwise, it will be deemed a hypothesis and not a (true) theorem. But then again, we are still able to correctly detect some Gödel sentences, i.e. sentences that are true but not provable, but that requires a rather special situation, such as for example, in the case of the Goodstein's theorem.https://en.wikipedia.org/wiki/Penrose%E ... s_argument
Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing machine that works on Peano arithmetic because the latter cannot see the truth value of its Gödel sentence, while human minds can.
The language in which Goodstein's theorem is phrased, is Peano Arithmetic theory (PA). However, the language in which its proof is phrased, is Zermelo-Fränckel set theory (ZF). Its proof uses infinite ordinals, which are defined in ZF but not in PA. Hence, Goodstein's theorem belongs to PA but its proof does not belong to PA. Its proof belongs to ZF. That is why we know that Goodstein's theorem is true in ZF and therefore also in PA. Hence, from the standpoint of PA, Goodstein's theorem is indeed true but not provable, i.e. a Gödel sentence.
A Turing machine could also use ZF to prove an otherwise unprovable theorem in PA. Therefore, it is not something that only human minds can do. What if there is no alternative theory available to prove the Gödel sentence from? In that case, both humans and the machine will not be able to know that the Gödel sentence is true. They will both consider it to be just a hypothesis.
Conclusion. The ability to see the truth of Gödel sentences is not different between human minds and Turing machines. In the general case, they will both fail to do it. The human mind may still be superior to Turing machines but not for its ability to see the truth of Gödelian sentences.