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.

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The present theory is a purely personal work of mine and it is not based on any kind of existing knowledge – it is a parthenogenesis. This means that the reader is not expected to be acquainted with any scientific field in order to comprehend the present manuscript. Moreover, the present manuscript constitutes the first case in History where we have record of purely scientific and orthologically proving knowledge, which is not based on axioms at all. This for the whole logic and the proofs of the present manuscript are exclusively based on the arithmetical identity, which, as it is known, does not exist and is not apprehended in any axiomatic way. An axiom is a logical/scientific demand, that is an arbitrary acceptance, regardless its being actually correct or mistaken. As to this, the present theory does not run the risk of even the least logical arbitrariness, so the knowledge offered here is literally absolute and perfect.

As by the title of the theory “The Groundlessness of Infinity” one is likely to consider that, if the title is actually valid, everything (time, space, mass, energy, etc) has an ultimate end/barrier. Nevertheless, the proofs that abolish infinity are the same proofs that introduce the wholly new consideration of the notion of counting and of what is called limit. The notion of the “ultimate end” has no place in the present manuscript, and specifically, in this manuscript, we refer to the difference of The Groundlessness of Infinity with “Finitism”, which does not prove anything.

This theory –The Groundlessness of Infinity– proves the numeric substance of infinity, thus the infinite in general terms, not to exist. That is nothing can be considered as being of an infinitely large or infinitely small quantity. This is rendered clear and definite by revealing the single quality and nature of the number; the quality which has been unknown since the first time Man has considered numbers and arithmetic. The nature of the number is its being one-part; indivisible. As to this, numbers cannot share anything at all with each other, since, being partially similar (or different), requires their being parted, which is proved impossible by this theory. Therefore, e.g. 3 and 4 cannot be included in one definition, for a common definition of them would require a common element between them. Thus the most common and fundamental definition of infinity “for each number n there is n+1” cannot be valid, for, in order to accomplish it, we need one definition for each number of it. There is no possibility that someone makes infinitely many definitions. And furthermore, the definition of infinity “for each number n there is n+1” constitutes some kind of numeric set. But a set of numbers is not possible. This for, e.g., 3 and 4 as being one-part each, they cannot constitute a set.

The problems like the Goldbach conjecture or the Riemann hypothesis are given the complete solution through the abolition of the numeric set, in a negative way. That is they cannot be proved because they are based on the concept of the numeric set. The P Vs NP problem and is also solved here by applying the numeric nature in the logic of the problem.