First of all, each point of this project, has been checked as OK, unofficially, for at least, two different persons that said they were mathematicians. The problem is that they are not the same group of persons, and each one believes the mistake is in a different point, while others consider that concrete point, as totally correct. Each comment, each guessing you can made, probably have been taken into account. The unique problem of this project is its crazy goal and the incapacity to put in the same room all the people I have talked sometime.We are going to present an alternative technic to compare infinite cardinalities. Different from bijections or injections, and not totally equivalent in potential, able to work when the previous ones can't. Applied to an obvious case we are going to create. After that, show the equivalents points between the obvious example and the real example we really want to study: P(N) vs N. To show how P(N) has not a cardinality bigger than N.I would like to remember now the first paragraph.We can apply the technic to different sets, with different natures and transfinite cardinalities. Always compared with N.The injectivity concept says to us that is impossible to create a relation r: A -> B, where |B| < |A|, and r being injective. In every possible relation, we must have a minimum quantity of pairs of elements from AXA, with the same image. At least one pair. The core of the idea of the TPI technic (Unlimited Transference of Pairs) is to create a mathematically correct process that shows that minimum is not bigger than 0. Considering the case of 0 pairs with same image, the case of a perfect injection. Remember: It will not be the same as an injective relation.The technic must be considered as valid. And after that, other documents will be recovered, each one per each equivalence, between the technic I will show in this document, and the real case we want to study really. Each equivalence must be considered as valid too. Finally, we will apply the same conclussion we obtained in the obvious example, to the real case.