Extending the current notions of the infinite

The extended integer sequence can, if required, be easily completed into a continuous set of numbers by adding to every integer α all real numbers x that are greater than zero and less than one.

Since in this way a certain expansion of the domain of the real number into the infinitely large is attained, perhaps a connected question will be whether one could not also with the same success define certain infinitely small numbers, or what might amount to the same thing, finite numbers which do not coincide with the rational and irrational numbers (which appear as limit values of sequences of rational numbers), but could be included into what one assumes are intermediate points in the midst of the real numbers, just like the irrational numbers in the realm of the rational, or like the transcendental numbers into the structure of algebraic numbers?

The question of the production of such interpolations, on which a lot of effort has been applied by some authors, can, in my opinion and as I will show, only be answered clearly and unambiguously with the help of our new numbers and, in particular, on the basis of the general concept of the number of well-ordered sets.

Previous attempts are based partly on:

• an erroneous confusion of the non-actual-infinite with the actual-infinite, and
• a completely uncertain and fluctuating basis.

The non-actual-infinite has often been called the “bad” infinite by recent philosophers, wrongly in my opinion, since it has proved itself to be a very good, extremely useful instrument in mathematics and in natural knowledge.

The infinitely small quantities have so far only been developed for use in the form of the non-actual-infinite, and as such are capable of all those differences, modifications and relationships which are used and expressed in infinitesimal analysis and in the theory of functions, in order to establish the abundance of analytical truths there.

On the other hand, all attempts to force this infinitely small into some actual-infinitely small would have to be given up as being pointless. If, otherwise, actual-infinitely small sizes exist, i.e: they are definable, they are certainly not directly related to the usual, infinitely decreasingly small quantities.

In contrast to observations on the infinitely small and the confusion of the two manifestations of the infinite, there is an opinion regarding the essence and meaning of numerical quantities, which is that no numerical quantities are believed to actually exist, apart from the integers of our first number-class (I).

At most, the rational numbers that follow directly from them are granted a certain degree of reality. But as far as the irrationals are concerned, they are assumed to have a purely formal meaning in mathematics, in that they serve to a certain extent only as arithmetical symbols that assign properties of groups of integers and to describe them in a simple, uniform way. According to this viewpoint, the true substance of analysis is formed exclusively from finite integers, and all truths found in arithmetic and analysis or still awaiting discovery are to be understood as relationships of integers to one another. Infinitesimal analysis, and with it the theory of functions, is considered legitimate only to the extent that its terms can be demonstrably interpreted as rules governing integers. With this viewpoint concerning pure mathematics, although I cannot agree with it, there are undoubtedly certain advantages attached to it, which I would like to emphasize here, and its proponents also argue for its importance.

If only the finite integers are real, but all the rest are nothing other than types of relationships, then it can be required that the proofs of the analytical theorems be checked for their “number-theoretic validity” and that every gap that occurs in them indicates how they must be filled according to the principles of arithmetic.

The true touchstone for the legitimacy and perfect certainty of the evidence is seen in the efficiency of such completion. It cannot be denied that in this way this rationalization can be perfected and that it can bring about other improvements in methods in various parts of analysis. By observing the principles emanating from that view one can also see that it is a safeguard against any kind of inconsistencies or mistakes.

In this way, a definite, if rather sober and obvious principle is established, and which is recommended to all as a guideline; it should serve to show the trajectory of the mathematical desire to speculate and conceptualize within the true boundaries, and where it avoids the danger of falling into the abyss of the “supernatural”, where, as is said with fear and a healthy concern, “Anything might be possible”.

It is left as an open question whether or not it was just an attitude of expediency that led to the authors of this attitude seeing themselves as acting as an effective safeguard regulating the rising forces, which can be so easily led into danger by arrogance and extremism, and as an effective protection against all errors, even if no useful principle can be found in it.

That they began with these principles even when they were discovering new truths is an assumption that, for me, is out of the question, because regardless of how many good things I might get from these maxims, I must, strictly speaking, consider them to be erroneous. We do not owe any real progress to them, and if it had actually been the case that they had been followed exactly, then our knowledge would have been held back or at least confined within the narrowest of limits.

Fortunately, things are in fact not so bad, and though both the praise and the observance of those rules are useful under certain circumstances, their assumptions have never been taken literally. It is also striking to observe that up to now, there has been a lack of anyone to undertake a formulation more complete and better than has been attempted here by myself.

Similar views were often represented and found as early as in Aristotle’s works. It is well known that in the Middle Ages, the “infinitum actu non datur” (the actual-infinite does not exist) was always represented by all scholastics as an irrefutable sentence taken from Aristotle.

If, however, one considers the reasons which Aristotle[Footnote9] gives against the real existence of the infinite (see for example his Metaphysics, Book XI, Chapter 10), then they essentially lead back to one presupposition, involving one petitio principii (begging the question), based on the assumption that there are only finite numbers, which he concluded from the fact that he was only aware of things in finite quantities.

I believe, however, to have proved above, and this will be shown even more clearly in the remainder of this work, that definite countings can be made on infinite sets as well as on finite sets, provided that the sets follow a certain rule which is that they can be well-ordered sets.

Without such a rule regarding the ordering of the elements of a set no counting can be done on the set - this is in the nature of the concept of counting. Even in the case of finite quantities, a counting can only be carried out for a certain sequence of the counted elements, but here it shows as a special nature of finite quantities that the result of the counting - the ordinal-number - is independent of the respective arrangement; while in the case of infinite quantities, as we have seen, such independence in general does not apply, but the number of an infinite set is an infinite integer which is also determined by the rule of counting.

This is where the essential difference between the finite and the infinite lies, which is based upon nature itself and can therefore never be eradicated, but the existence of the infinite will never henceforth be denied on account of this difference, while the existence of the finite must be admitted - if one is permitted to fall, then we have to do away with the other also - if so, where would this road take us?

Another argument used by Aristotle against the reality of the infinite consists in the assertion that the finite, if it existed, would be abolished and destroyed by the infinite, because a finite number is supposedly annihilated by an infinite number.

As one will see clearly in the following, in truth the fact is that for an infinite number, if it is considered to be perfectly determined, a finite number can very easily be added and united with it, without eliminating the finite number; the infinite number is modified by the addition of a finite number to it. It is only the reverse process, when the finite number is set first in the addition, that causes the elimination of the finite number without causing any modification of the infinite number.

This correct state of the matter with regard to the finite and the infinite, which Aristotle completely misunderstood, could lead to new ideas not only in analysis but also in other sciences, especially in the natural sciences.

While I am interested in the concepts of the infinitely large in the form of increasing without limit and also in the closely related form of the convergent infinite sequences (first introduced in the seventeenth century), but I am also impelled, almost against my will, to perfect the definitive form of the mathematics of infinite numbers.

In contrast to traditions that are valuable to me, through the course of many years of scientific endeavors and attempts I have been logically compelled by many years of scientific endeavors and attempts to do so, and because of this, I also believe that no argument can be asserted against them that I would not know how to counter.