Thought provoking: "The expressive power of real numbers may be used to generate a simple recipe to obtain the values of a Turing-uncomputable function. Consid

Thought provoking: "The expressive power of real numbers may be used to generate a simple recipe to obtain the values of a Turing-uncomputable function. Consider that the digital expansion of a real number contains countably many digits. Hence, for any characteristic function (i.e., a function whose values are ‘0’ or ‘1’) defined over a countable domain, including all Turing-uncomputable such functions, there is a real number whose binary expansion encodes its values. This is because for all values of a characteristic function, the nth value of the function may be defined to be the nth digit of the binary expansion of a real number. Suppose you wish to know the value of a specific Turing-uncomputable characteristic function, such as the halting function for Turing machines, for its nth argument. Take the real number r whose digital expansion encodes the values of the halting function. All you need to do is obtain the value of the nth digit in the binary expansion of r and you have the result you desire. If you can do this, you may obtain any value of any characteristic function defined over strings, including all Turing-uncomputable such functions."