 # AI Cancelled?

Turns out that the development of computer learning algorithms has stumbled upon an obstacle. This obstacle is the set theory that cannot be solved due to fundamental reasons.

Amir Iegudajov and his colleagues from the University of Tel Aviv found out that machine learning algorithm has stumbled upon a fundamental mathematical paradox discovered by Georg Cantor and Kurt Goedel, the great mathematicians of the 19-20 centuries. The issue was addressed in the article published in Nature Machine Intelligence on January, 7.

Background: Famous paradoxes of the 20th century

The best way to illustrate the paradox discovered by the mathematician Bertrand Russell is with the problem on two catalogs. The problem situation is the following. All the books in the library must be added to one of the two catalogs. The first catalog is meant for the books that include a self-reference. The second catalog is meant for the books with no self-reference. Since the catalogs are books, they must be cataloged too. As for the first catalog, you can either add a self-reference to it or leave it as it is. Either way, the condition of the problem is met. As for the second catalog, it belongs in neither catalog and you can’t leave it uncatalogued. Either way, the condition is violated.

Pondering over Russell’s paradox, Kurt Goedel invented his famous “incompleteness theorem.” Let’s take a system of mathematical axioms and list all possible mathematical statements that follow from the axioms. (It’s something like a library catalog). There will always be a true mathematical statement that doesn’t belong on the list, just like the second catalog in the example above. Therefore, any system, even an infinite one, is always incomplete. There always will be a true statement that doesn’t follow from the axiom system. Mathematicians call such statements “undecidable.”  Even if you call categorize such an undecidable statement as an axiom and add it to the list, the new system will still be incomplete, because there will be an indefensible and irrefutable statement for its tool. One example of Goedel’s undecidable statement is the continuum hypothesis by Georg Cantor. Comparing infinite sets of numbers, the German mathematician discovered that they are different in “power.”  For example, sets of natural, rational and real numbers are infinite. However, while you can compare natural and rational numbers (because these sets are of equal power), you can’t do the same with real numbers because they’re more “densely” packed.

Cantor formed the following question: Are there sets that are more powerful than a set of natural numbers but less powerful than a set of real numbers? Unfortunately, Cantor failed to find the answer to his question. However, in 1940, Goedel proved that this is actually an example of an undecidable statement in the set theory. One can say that there is no such thing as sets of intermediate power, and this statement will become a part of a consistent mathematical system. At the same time, one can say just the opposite, and that will also be a consistent system of statements, although different from the first one.

The English mathematician Alan Turing elaborated Goedel’s idea by applying it to computing algorithms. He proved that the list of “all possible algorithms that solve the problem” will lack an algorithm that would determine whether the problem can be solved by a random algorithm. Based on that discovery, the British mathematician Roger Penrose made a well-founded hypothesis saying that human reasoning cannot be imitated through algorithms. Therefore, the artificial intelligence, in the strict sense of the word, is impossible to create because some of the problems solved by the human brain might be Turing’s undecidable algorithms.