Rules concerning ordering

The concept of the well-ordered set is fundamental for the whole theory of sets.

It is always possible to bring any well-defined set into the form of a well-ordered set.

I shall come back to this law of thought, which seems to me to be fundamental and momentous, and which is particularly remarkable due to its universality. Here I limit myself to the demonstration of how the basic operations for the integers, whether they are finite or actual-infinite numbers, result in the simplest way from the concept of the well-ordered set, and how the rules for these derive immediately with absolute certainty from inner intuition.

If 2 well-ordered sets M and M1 are given, to which the ordinal-numbers correspond as numbers α and β, then M + M1 is also a well-ordered set, which arises when the first set M followed by the set M1 and is and united with it; a certain ordinal-number corresponds to this set M + M1 in respect of the resulting ordering of its elements.

If α and β are not both finite, then α and β is generally different from β and α.

The commutative rule ceases to be generally valid even for addition. It is now easy to form the concept of the sum of several summands given in a certain sequence, whereby this sequence itself can be actual-infinite, so that I do not need go into more detail at this point. I therefore only note that the associative rule generally proves to be valid, specifically: α + (β + γ) = (α + β) + γ

If one takes a succession, determined by a number β, of similar sets which are similarly ordered, in which the ordinal-number of elements is each equal to α, then one obtains a well-ordered set, the corresponding ordinal-number of which gives the definition for the product βα, where β is the multiplier, and α is the multiplicand.

Here, too, it is found that βα generally differs from αβ, that is, the commutative rule is generally invalid for the multiplication of numbers.

On the other hand, one finds that the associative rule also applies in multiplication, so that one has: α(βγ) = (αβ)γ.

Of the new numbers, certain of them are distinguished from the others by the fact that they have a prime number property, but this must be characterized here in a somewhat more specific way by understanding a number α where the decomposition α = βγ (where the multiplier is β) is not possible unless β = 1 or β = α.

On the other hand, in general, even with prime numbers where α is the multiplicand, there will be a certain margin of indeterminacy, and it is in the nature of things that this cannot be resolved.

Nonetheless, in a later paper it will be shown that the factoring of a number into its prime factors can always be done in an essentially unique manner, even with regard to the sequence of the factors (as long as they are not finite prime numbers that occur adjacent in the product).

Two types of actual-infinite prime numbers emerge, of which the first type is closer to the finite prime numbers, whereas the prime numbers of the second type have a completely different character.

Furthermore, with the help of the new knowledge, I will soon be able to provide a strict justification for the proposition regarding the so-called linear infinite sets that is cited at the end of my article Ein Beitrag zur Mannigfaltigkeitslehre.[Footnote6]

In the previous part of this work (Part 4),[Footnote7] I proved a proposition for point-sets P which are continuous in an n-dimensional region, that uses an application of the new, previously defined terminology, which is as follows:[Footnote8]

“If a point-set whose derivative P(α) vanishes identically, where α is any integer of the first or second number-class, then the first derivative P(1), and therefore also P itself, is a point-set with the cardinal-number of the first number-class.”

It seems to me most remarkable that this proposition can be reversed as follows:

“If P is a point-set whose first derivative P (1) is the cardinal-number of the first number-class, then there are integers α belonging to the first or second number-class for which P (α) vanishes identically, and of all such numbers α for which this phenomenon occurs, there exists a smallest such α.”

I shall publish the proof of this theorem in the near future, in response to a kind invitation from my esteemed friend, Herm Prof. Mittag-Leffler in Stockholm, in the first volume of the new mathematical journal (Acta Mathematica) of which he is the editor.

Following upon this, Mittag-Leffler will publish an article in which he will show how, on the basis of this theorem, his and Prof. Weierstrass’s investigations into the existence of unambiguous analytic functions with given singularities can be given considerable generalizations.