What is undecidability What does it mean that some problem is undecidable?
For instance the halting problem.
Does it mean that humans can never invent a new technique that always decides whether a turing machine will halt?
If not, what techniques are allowed such that halting problem is still undecidable?
For instance induction is a good technique, why cant one discover some new technique?
I have trouble understanding how some new invention cannot solve the halting problem.
Given some computer and a program, is there really insufficient information stored in it to determine if it will halt?
It seems like a purely mechanical problem

In principle, there might be a notion of computability that goes beyond the notion of Turing computability.  That there is not is usually called the Church-Turing Thesis.  There is an enormous amount of evidence for the Church-Turing Thesis. But unless we declare that by definition computable means Turing computable, the Thesis cannot be proved, since it says that a certain informal notion (computability) coincides with a formal notion.
