The Infinite

“Grundlagen einer allgemeinen Mannigfaltigkeitslehre”

My theory of sets depends on extending the notion of actual integers beyond the previous limits.

This extension lies in a direction that no one has previously sought to tread.

Without this expansion of the concept of number, I could not have taken the smallest step forward in set theory.

Such an expansion is a justification, or perhaps an excuse, for the fact that I introduce seemingly alien ideas into my work, because it involves an extension, a continuation of the sequence of numbers beyond the infinite.

I assert that this extension will over time be regarded as completely simple, correct and natural.

I turn the mathematical infinite into a variable quantity, which either:

• grows beyond all limits or
• is diminished to any desired diminutiveness, but always remains finite.

I call this infinite the non-actual-infinite.

However, in more recent times, in geometry and in the theory of functions, another equally justifiable concept of the infinite has been developed.

For example, analytic functions of a complex variable use a single point that lies at infinity (i.e: an infinitely distant but definitive point) in the plane of the complex variables.

The behavior of the function in the vicinity of that point is examined, just as in the vicinity of any other point.

It turns out that the function shows exactly the same behavior in the vicinity of the infinitely distant point as at any other finitely distant point, so that in this case we are fully justified in thinking of the infinite as placed at a completely specific point.

When the infinite appears in such a definitive form I call it the actual-infinite.

The emergence of these two manifestations of the mathematical infinite has led to great advances in geometry, in analysis, and in mathematical physics.

• The first form, the non-actual-infinite, occurs as a variable finite.
• The other form, the actual-infinite, is a completely determinate infinite.

The notion of actual-infinite integers (which I will define later) was with me for many years before I realized that they are concrete numbers of real significance.[Footnote3]

They have absolutely nothing in common with the non-actual-infinite; rather, they have the same property of determinativity that we find in infinitely distant points in the theory of analytic functions, i.e: they belong to the realm of the actual-infinite.

But while the point at infinity of the plane of complex numbers is isolated from all points that lie in the finite region, here we have not simply a single infinite integer but an infinite sequence of them, and which are clearly differentiated from each other, and are in accordance with rule-based relations of number theory both among themselves and to finite integers. But these relationships are not such that they allow themselves to be reduced essentially to the relations of finite numbers with each other.

These relations in fact appear frequently, but only in the different intensities and forms of the non-actual-infinite, for example, in functions of a variable x that grows infinitely large or infinitely small, where they become certain numbers of finite order as they become infinite.

Such relationships in fact can be considered as concealed relations of the finite, or in any case immediately reducible to the finite. On the contrary, the rules for the actual-infinite integers, which will be defined later, are initially different from those that apply in the finite, but this does not exclude the possibility that the finite real numbers themselves can obtain certain new properties with assistance from the actual-infinite numbers.

The two principles of generation by which, as will be shown, the new actual-infinite numbers are defined, are of such a kind that, through their combined application, every limitation in the conception of the formation of actual integers can be overcome.

Fortunately, as we shall see, a third principle applies in opposition, which I call the principle of constraint or limitation, by which certain limits are successively imposed on the absolutely endless process of generation, so that we have a natural segmentation of the absolutely infinite sequence of integers, into what I call number-classes.

The first number-class (I) is the set of finite integers 1, 2, 3, …, v, …, which is followed by the second number-class (II), consisting of specific infinite integers following upon one another in a determinate sequence. Once the definition of the second number-class is given, the third follows, and then the fourth, and so on.

The introduction of these new integers seems to me of the greatest significance for the development and advancement of my concept of “cardinal-number” as introduced in the previous parts of this work and elsewhere.[Footnote4] According to this concept, every well-defined set has a determinate cardinal-number; two sets have the same cardinal-number if they can be correlated with one another element for element.

In the case of finite sets, the cardinal-number coincides with the number of elements, since such sets are known to have the same cardinal-number of elements regardless of any ordering.

In the case of infinite sets, on the other hand, no-one has made mention of a precisely defined cardinal-number of their elements, either in my work or anywhere else, but a certain cardinal-number, completely independent of any ordering of their elements, can also be ascribed to them.

The smallest cardinal-number of infinite sets must be ascribed, as is easy to justify, to those sets which can clearly correspond to the first number-class and which therefore have the same cardinal-number as it. On the other hand, an equally simple, natural definition of the higher cardinal-numbers has so far been lacking.

Our above-mentioned number-classes of the actual-infinite integers are now identified as the natural, uniformly presented representatives of the cardinal-numbers of well-defined sets, and which increase according to definite rules.

I will show most definitely that the cardinal-number of the second number-class (II) is not only different from the cardinal-number of the first number-class, but that it is actually the next highest cardinal-number, and consequently we can call it the second cardinal-number or the cardinal-number of the second number-class.

Likewise, the third number-class gives the definition of the third cardinal-number, the cardinal-number of the third number-class and so on and so on.