Shadows of the Mind: A Search for the Missing Science of Consciousness

mere algorithm, formal system, computer, or robot. A few minutes of reading and pondering is a small price to pay for this distinction. Penrose presents a version of the theorem in which the true but unprovable sentence is an assertion that a certain computer program runs on forever. The technical formulation is in terms of an ideal computer known as a Turing machine. A Turing machine is a computer equipped with unlimited memory and with a program for computing some function. Assign a numerical value to an input variable n and start the machine. The machine may eventually give an answer; we say in this case that it halts. It also may just go on computing forever; computer programmers will recognize this as a real possibility. There is a special kind of Turing machine that is universal; this is the equivalent of a general purpose computer that can run any program. The universal Turing machine has two input variables q and n. Given any Turing machine with one input variable n, there is a value for q so that the universal machine simulates the given machine. Consider the universal Turing machine in which the input variables have the same values n and n. Let c(n) be the assertion that this machine does not halt. This is a definite mathematical assertion about each natural number n. However, it may be difficult to verify such an assertion—one can run the machine, but what if it runs for a long time without halting? Can one conclude that it will never halt? The difficulty of this problem is the key to this version