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This book is about artificial intelligence (AI), which we conceive as the application of mathematics to the modelling (primarily) of the functions of the human brain. We focus specifically on the question of whether modelling of this sort has limits, or whether – as proposed by the advocates of what is called the ‘Singularity’ – AI modelling might one day lead to an irreversible and uncontrollable explosion of evermore intelligent machines. As concerns the current state of the art, AI researchers are, for understandable reasons, immensely proud of their amazing technical discoveries. It therefore seems obvious to all that there is an almost limitless potential for further, equally significant AI discoveries in the future. Enormous amounts of funding are accordingly being invested in advancing the frontiers of AI in medical research, national defence, and many other areas. If our arguments hold water, then a significant fraction of this funding may be money down the drain. For this reason alone, it is probably no bad thing for the assumption of limitless potential for AI progress to be subjected to the sort of critical examination that we have here attempted. The result, we must confess, is not always easy reading. To do our job properly, we found it necessary to move to a rather drastic degree beyond the usual disciplinary borders, drawing not merely on philosophy, mathematics, and computer science but also on linguistics, psychology, anthropology, sociology, physics, and biology. In the ‘Approach’ section of the Introduction, we provide the rationale for this methodology and, where this is needed, for our choice of literature. In the ‘Glossary’ (pp. 352ff.), we provide what we hope are reader-friendly definitions of the technical terms used in the main text. We raise what we believe are powerful arguments against the possibility of engineering machines that would possess an intelligence that would equal or surpass that of humans. These arguments have immediate implications for claims, such as those of Elon Musk, according to which AI could become ‘an immortal dictator from which we would never escape’. Relax. Machines will not rule the world. At the same time, our arguments throw light on the question of which varieties of AI are achievable. In this respect we are fervent optimists, and one of us is indeed contributing to the creation of new AI products being applied in industry as this book is being written. In the final chapter of the book, we outline some of the positive consequences of our arguments for practical applications of AI in the future.