Turing Machine, human mind and Turing Test

Logic (first order predicate calculus) and the notion of set are enough to create all known mathematics. It was also shown that we could start with the notion of function and define set. Does this mean Turin Machine can construct all of mathematics?

After Turing proved the Halting Problem we knew Turing Machine could not compute at least one function. Of course now we know the set of computable functions is smaller than the set of all functions (set cardinality comparison). Thus, Turing Machine represents only the set of computable functions and for that reason we cannot consider it as a representation (definition) of notion of function. Therefore, Turing Machine is incapable of representing (defining, being equivalent to) the notion of set.

The notion of set is a creation of human mind. If Turing Machine were equivalent to human mind, it should be able to start with the notion of set and create all of mathematics. The problem is that, abstractions are not computable, while Turing Machine is limited to mechanizing computable functions. Simply put, not everything human mind can do is possible to mechanize.

One characteristic of human mind is that we can learn to become a mathematician. There is so much written about machine learning, and even the ability of machines to think. As far as abstract notions, learning and thinking require the ability to communicate. Stated differently, the ability to communicate is fundamental to the ability to create and learn abstract notions. Thus, Turing Machine can never communicate for if it could it would also be able to learn the abstract notion of set (which we showed is not computable).

The ability to communicate is necessary for learning abstractions, but not sufficient. Given that Turing Machine can never learn to communicate, the only audience a Turing Test can fool would be those who cannot learn abstractions. Otherwise put, Turing Test is unachievable.