Mathematical models for artificial intelligence (AI) are considered. It is assumed that AI is realized on the base of computers. Computers are controlled by algorithms in the form of software. Consequently, it is natural to assume that AI has to be realized by algorithms and it is reasonable to model AI by mathematical models of algorithms. Investigating these models, it is possible to predict abilities, limitations, and shortcomings of AI. In addition to this, such general models, which we call upper models of AI, can comprise different types and kinds of intelligence of human beings. There are diverse classes of mathematical models of algorithms: Turing machines, finite automata, recursive functions, RAM, etc. Consequently, AI may be realized by algorithms that belong to one of these classes. As different classes often have dissimilar computational power, capabilities of realizations might be different. There are three main types of mathematical models of algorithms: recursive (such as Turing machines or partial recursive functions), subrecursive (such as finite automata or real time Turing machines), and super-recursive algorithms (such as inductive Turing machines). Here we compare the capacity of AI based on recursive (Turing machines) and super-recursive algorithms (inductive Turing machines). It is proved that that AI based on inductive Turing machines is much more powerful than AI built by means of conventional algorithms. In particular, it is demonstrated that the famous Gödel incompleteness theorem for formal systems is true only when conventional algorithms are used for proofs. This explains how people solve problems related to incomplete systems with undecidable properties. Utilization of inductive Turing machines for proving theorems makes many such systems complete and decidable. As theorem proving and logic are used commonly in AI, these results have important implications for AI
