*Summary*

*The axiom of mathematics*

* *

*The set of natural numbers is infinite*

* *

*i.e.:*

* *

*Abstract: **The question, how unconditionally mathematic
can be developed, is equivalent to the question, what has to be presupposed as
something given to do so. Mathematics begins with the natural numbers. These
natural numbers are referred to as something simply existent, even where we
have to do with a justification of these numbers. Taken out of consideration is
thus the system that produces all of these numbers. Mathematics reduces itself
thereby to a formalism that does not come-up to the solution of certain
questions any longer. For example the continuum hypothesis is this way
surrendered to an indifferentism strange for mathematical matters. .*

*Mathematics will only be able to scope with the
whole of its possibilities if it begins as originally as possible, that is if
it does not begin with the natural numbers as something given but with what
creates and generates these numbers.*

1. The operation as central mathematical (limit-) operation presupposes the existence of the natural numbers in explicit and definitive material representation. This operation cannot be set simply formally.

2. There is infinite only by a serial law, in which infinite is already anticipated to its entirety from the beginning. Therefore, infinite that is in itself unfinished by definition can be regarded as something ready one either. Infinite cannot be established step by step. Infinite stands outside time. Any time-relatedness contradicts infinite.

3. The serial law of the natural numbers is the law of digit sequences like – for example and especially – the sequence 1, 2, 3, … These sequences follow all a certain system and a certain procedure. They consist of all finite settings of single digits, which are taken from a predetermined finite set of digits. The repeated setting of the same digit within a single sequence is possible. None of these sequences however is – read from the left to the right – allowed to begin with a certain one of these digits: 0.

4. The mathematical formalism is able to deal only with a model of the natural numbers, in which each such number is represented by a certain – finite – sum of 1-ones. Sums of this kind can however not become calculated within the framework of these models. Only the + signs can be neglected, leading to a model, in which each natural number is represented by just as many digits, say 1-ones, as is denoted by the cardinal number, which is associated naturally with each natural number. This cardinal number does not depend on the sequence in which the digits are set, of course, all digits being identically equal. We can see only on the number of digits set in a sequence, and therefore this system is accessible for a usual set-theoretical treatment. In such a model natural numbers are understood as equivalence classes of equally mighty finite sets. Even the Peano-axioms indebted the thought of counting exceed the range of such a model.

5. All these models are not communicative, that is one cannot speak in the language of these models about the numbers represented therein. Moreover, it cannot be counted in it either. These models are only good enough for a ‘mayor-minor-estimate’ regarding the number of indications set in the single "numbers". In regular classical representation, we can on the contrary immediately infer from every digit sequence its position within the infinite sequence of all finite sequences of the system.

6. Numerical values appear neither in the mathematical nor in the philosophical models of the natural numbers. Furthermore – and that is crucial for the whole thing – we cannot go to infinity with these numbers in these models, simply because we drop out of the natural numbers thereby. A performance of the limit process in these systems leads to an infinite sequence and such a sequence cannot be connected with a natural number any longer. This infinite sequence is in this system part of this system, that is it becomes really "adopted" within the limit-procedure as regular members of the series. The sequence appears not only as unreachable limit there, that is: this procedure can then no longer be regarded as performing the operation n goes to infinity:

7. The classical representation takes care of this danger. In this system a blockade is integrated to prevent, that the whole system could "degenerate" into infinite digit sequences. This system knows itself committed to the production of all possible finite digit sequences from a predetermined finite set of digits. Therefore the development of every single sequence shares the attention of this system in equal parts with every of the other infinite many finite sequences. Only by undivided attachment to a single sequence, the entire procedure could result in an infinite sequence. So, all these infinite sequences figure as limit set of the whole procedure. Moreover, if we read sequences that „finish“ on infinitely many zeros as finite sequences then this limit is formed by all infinite sequences as well as all finite ones.

In this way, the whole procedure results in the material all real numbers are built up from. Let the set of all infinite sequences be denoted by, and then the real numbers consist of the Cartesian product, if additionally finite many zeros may appear after the decimal point in fractional expansion. That is simply a matter of pure construction then. The material the set of real numbers is “constructed” from is made available by the limit set of this – unique – procedure for the production of the set of natural numbers. We cannot think only formally in these things. We have rather move in a concrete model then. For this purpose, there is only this one and unique system. Therefore:

There result three immediate consequences from this formula:

1. The only possible existence as well as uniqueness proof of the set of real numbers is included in this formula. The real numbers exist in this form as well as by this way or it does not give them at all. Difficulties in this context arise from the irrational numbers, i.e. the non-periodic fractions. These fractions do not exist yet for the simple reason that we have a criterion that separates these fractions from the rational ones. In matters of infinite, one cannot conclude from property to existence. We need the serial law in every such case. Moreover, if we have to do with an infinite set, then it needs in turn again a serial law for all this serial laws. If this infinite set is furthermore denumerable like the set of irrational numbers, this serial law cannot be given by means of a sequence, i.e. by definition a mapping from the natural numbers to this set or by another construction in dependence of the natural numbers.

Such a serial law of the series remains necessarily outside the formal constructive possibilities of mathematics. This serial law can only be a law of the production of all non-periodic infinite digit sequences representing all these irrational numbers. This procedure provides us really with all these sequences. That this can be done only per limit-procedure does not meet this observation since infinite can be attained only per limit-procedure anyway.

2. In addition, this procedure provides us at the same time with the only possible positive proof of the non-countability of the real numbers. The "classical" proofs of this theorem proceed all indirectly, showing that the assumption of the countability of these numbers is self-contradictory. In matters of infinite this is the only possible procedure for taking evidence. Infinite can never "be gone through" element for element. The proof of the countability of an infinite set could never be set this way. Infinite can only be let counted by itself. A set is uncountable if and only if all the natural numbers are not enough to reach in order, i. e. by regular mapping all elements of this set.

In matters of infinite it may seem strange to proof a specific property of a set by assuming the set to have this property only to show that this proves the thesis – necessarily – wrong. Is that not kind of a vicious circle? This kind of proof may work in finite matters where we can demonstrate and manipulate a set element by element quite according to our dispositions. Infinite sets however lie completely beyond our mental facilities and abilities.

The situation is quite another if we consider the procedure for the production of finite sequences representing the infinite set of natural numbers. The exclusiveness of the relation: guarantees that the same result cannot be obtained by a regular and formal mapping.

3. Finally the continuum hypothesis can be derived from the exposed procedure as well. The continuum hypothesis contains an existence statement. Therefore, this hypothesis depends naturally on the existence of the real numbers. If these numbers do not exist, then this hypothesis immediately would become completely meaningless, simply because it then makes no sense to mind about the existence of a set that lies – corresponding to its mightiness – between the natural numbers and the real numbers. That would simply not add up then. Only the proof of the existence of these real numbers turns this continuum hypothesis into a justified question. This proof derives from our procedure, and just as this procedure performs – and as it can only perform – it confirms the continuum hypothesis.

We reach the real numbers only by means of our limit procedure. However, such a procedure does not catch up analytically, that is gradually with infinite. In other words: The step of the natural numbers to the real numbers consists only of one – final – step, even if this one step gets together from infinitely many single steps. We always are on either one side or the other of this one limit-step respectively limit-side.

We cannot place any cut, that could let us look within this procedure where we actually and exactly are in the transition of the natural numbers to the real numbers. We cannot do that. Therefore, there is no set that is more mighty than the set of natural numbers but less mighty than the set of real numbers. That is the situation anyway, just like it represents itself for the human mind, and this mind is and remains the upper canon of our thinking and our intelligence. About a more powerful – an infinite – mind, a mind in which all this possibly shows itself differently, we can – only (?) – speculate. Such speculations would in any case no longer be task of mathematics but philosophy. To the mathematical truths that keep their validity equally like also for the divine mind, this non-reconstructiveness of limit processes is not to be count necessarily, too.