In this theory, The Groundlessness of Infinity, I introduce by three separate proofs a totally innovative concept regarding the nature and relations among (pure) numbers. The applications of this wholly new knowledge conclude to proving and solving major up-to-now unresolved problems, e.g. the Riemann Hypothesis, Zeno\'s Paradoxes, the P Vs NP Problem, and other major open issues. The core knowledge, as by the three above mentioned proofs is that a number cannot be composite, it cannot bear any properties at all, and there cannot be any kind of relationship among numbers. This leads to the abolition of the numeric set of any kind, and subsequently to the abolition of the numeric continuum and infinity. Nevertheless, it is the same logic that abolishes infinity, which abolishes the notion of a supposed "ultimate end", by introducing a new way of regarding the "limit" in counting. The only logical basis I use about the proofs is the arithmetical identity. Therefore in this theory, the new Foundations of Arithmetic are established. Also, the Euclidean space has been proved here: the notion and the entity of the straight line, the angle and the square have been proved. In extension, the circle is proved to be a non-Euclidean object, and the proof of the genuine calculation of the circle is given. This leads to answering the, surprisingly, so far unanswered question of how a bicycle works.