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One of the most disastrous consequences of the actual infinity is the conception of space and time as a continuum infinitely divisible. Basic concepts of physics as motion or velocity were, and continue to be, founded on that assumption. Matter and energy were also initially conceived as a continuum infinitely divisible. But the obstinacy of facts finally revealed that rather than continue, they are discontinue (elementary particle and quanta, respectively). It is now being suggested from different areas of science that space and time could also be of a discrete and finite nature. If that were the case we would have to redefine the foundations of a considerable part of contemporary science. For these reasons it is of capital importance to definitively prove if the actual infinity is or is not an inconsistent notion. This book opens a new way of criticism of the actual infinity based on the notion of w-order, an example of which is the natural order of precedence of natural numbers: 1, 2, 3, dots In spite of its apparent formalism, the reader with a basic level of mathematics and logic will be able to understand all its content.