Imagine that we should prove some statement. But how are we going to do that. Let’s just use a bunch of rules set in a proper order. According to the first (weak) Gödel’s incompleteness theorem there will always be some sort of statement, which we are unable to prove whether it is true by using just selected axiomatic system. What shall we do if we are unable to prove something? Well, we have two options: just agree that a statement is baseless or adding it as a new axiom to a previous set of rules. However, the number of statements impossible to prove is never-ending. Despite all of our efforts to make our set of rules comprehensive and complete, we will never be able to claim that it is considering all the cases, providing we will base solo on itself. By showing that all rules are taken into account indeed, we will have to go out of the box and add some extra axioms. That is what the second (strong) Gödel’s incompleteness theorem tells us. It provides us with an example of an unrealistic statement to prove, it is impossible to prove that an axiomatic system is complete, regardless of how complicated and comprehensive it is.

British scientists John Lucas and Roger Penrose used Gödel’s theorem to demonstrate human intelligence superiority of computer. Artificial intelligence acts strictly according to the preassigned algorithms and always remains inside a formal (binary, consisting of 0 and 1) logic, while our brains are not limited to these. According to Gödel’s theorem there will always be a statement, which will be impossible to process by a computer. On the other hand, a human will be able to define truthfulness by using expertise and intuition. Because of that, John Lucas, in his article published in the 1960s, assumed that the human brain, belonging for example to a math scientist and working on a problem, must have been something totally different than a machine. This exact aspect might interest Artificial Intelligence specialists. In the 1960s and 70s scientists including Terry Winograd, an American AI researcher went to great lengths to create a computer, which would have operated exactly the same as human brain. Nevertheless, Roger Penrose in his books “The Emperor’s New Mind” and “Shadows of the Mind” suggested developing ideas by John Lucas. According to Penrose, even machines built of great number of logical schemes (consisting of 0s and 1s) are following exclusively formal logic. If we want to create an authentic human brain model, then our efforts to build like a computer would be useless. By presenting this idea, Roger Penrose started to research in his works, which exact element distinguishes the human brain from a silicon machine. In his opinion, human consciousness has a quantum nature. Penrose’s hypothesis has generated a great amount of discussion: some scientists, such as the American neurobiologist Stuart Hameroff or the Swedish American cosmologist and astrophysicist Mark Tegmark, have begun to experimentally test whether structures in the human brain can indeed exhibit quantum properties. Other specialists, such as Douglas Hofstadter, have an opinion that the idea of being based on the Gödel’s theorem itself might be wrong: human mind is not at all required to be a consistent formal system or even a formal system to which Gödel’s theorem could be applied. Hofstadter emphasises that algorithmic thinking of a computer is inferior to the one of a human in terms of proving theorems, as described by Lucas and Penrose, the computers, on the other hand, are able to produce something unreachable by humans.

In the early 20th century, many mathematicians supposed that soon all axioms required to answer all the mathematical questions would be listed in the future. Whitehead and Russel from the UK even created a three-volume maths textbook, where they tried their best to collect all axioms into a comprehensive logical system. Gödel’s theorem showed that their efforts were of no effect. Despite the fact that Gödel’s theorem has a vivid mathematical formulation and belongs to the formal logic section, its results turned up to be interesting for philosophers, biologists, computer scientists and artificial intelligence specialists. Photo: akspic.ru