Julius König’s paper:
Über die Grundlagen der Mengenlehre und das Kontinuumproblem
• English Translation •
On the foundations of set theory and the continuum problem (Part 2)

On the foundations of set theory and the continuum problem

by Julius König

 

This article closely follows the developments that I published under the same title a few months ago (Math. Annalen, Vol. 61, pp. 156-160). However, the method of inference (Footnote: This conclusion must and can be reshaped in the sense of the new conceptual developments given here.) given there in Section 3 will not be used here at all. In its place is a new method which confirms my viewpoint which has remained unchanged, and which in essence is based on a clarification and generalization of the term “finitely defined”.

 

First I would like to preface the following arguments with a point of principle. The fundamentals of set theory deal with the formalization and logical establishment of matters that are taken from our inner conscious perception, so that our “scientific thinking” is itself the object of scientific thinking. This connection between set theory, logic and epistemology is indissoluble and is already evident in the elements of arithmetic.

 

Although the algebraization of logic, which has up to now been carried out according to mathematical analogies, is useful in this direction, by itself it cannot assist us in overcoming the existing difficulties. The “facts” and “rules” that underlie our scientific thinking must be examined more closely than has been the case up to now, and we must - above all - create a discipline which, by an analogy with “mathematical physics”, I would consider to be a theory of logical evidence.

 

I hope to further pursue this course in due time in a more detailed publication.

 

1.

The finitely defined elements of the continuum form an enumerable set in the ω type that can be written as follows:

The a_ik here are any positive integers, since we are taking the continuum to be the set of things defined as:

(a₁, a₂, a₃, …, a_k, …)

where a_k can be any positive integer. For the series (I) we can now apply the rule based on Cantor’s diagonal method:

from which a new continuum element (a₁, a₂, … ) = a^(d ) can be defined if d > 0 is a fixed positive integer. However, the definition of a^(d ) is only consistent if one assumes that a^(d ) does not occur in the series (I), i.e. it is not finitely defined. If a^(d ) (e.g. at the n^th position) occurred in (I), then (II) could not be satisfied for k = n, since a_n = a_n n, but at the same time a_n is defined by a_n = a_n n + d. It thus seems that the definition of a^(d ), which we expressed using a finite number of signs, is in fact self-contradictory, i.e. impossible. On the other hand, it is equally impossible that we should reject as incorrect the “fact” which arises directly from the concept that a new continuum element can indeed be formed by the application of the diagonal method. But this extremely strange apparent paradox leads directly to a very precise and fundamental deepening of the logical methods that are to be used in set theory. The sense of the diagonal procedure is clearly unchallengeable, the contradiction only arises from the requirement of the sense in the form of a finite definition. It is impossible to fulfill this requirement. But if we could - without changing the logic - alter the form of our definition in such a way that it is not a finite definition, then we would be dealing with a real and consistent definition of the continuum element a^(d ). Accordingly, we have to perfect our “language” as in fact is often the case for any significant clarification of our scientific thinking, and indeed this can be done as follows.

 

For the finite definitions we use certain “pseudo-finite” definitions, which consist of an (enumerable) infinite number of characters (words, letters) that are to be in such a way that from a certain n^th position (n a finite ordinal number) only a certain character (N.V.) (Footnote: N.V. indicates not variable.) occurs. In order to give these definitions a meaning (content), we state: this pseudo-finite definition is said to be equivalent in sense to the finite definition which arises from the infinite if all (N.V.) are omitted, i.e. which is formed from the first n characters. The infinite string is a logical definition if and only if the corresponding finite string, which we shall also call the Main Part (Hauptteil) of the pseudo-finite definition, defines in sense (but not necessarily also in form) a continuum element without arbitrariness or contradiction.

 

In this way, various pseudo-finite definitions can arise from a finite definition by adding the (N.V.) ω times, ω + 1 times, ω + 2 times, …, in general α times, where α is any number of the second number class Z(ℵ₀).

 

 

2.

According to the above, there are elements of the continuum that are completely determined by the finite Main Part of their definition, but whose definition is only in a consistent form if this finite Main Part (H) has the sign (N.V.) supplemented ω times. The definition of such an element is fully characterized by the “concept” H(N.V.)^ω; however, this concept must be strictly distinguished from the definition itself, since the latter, in order to be formally consistent, must contain an infinite number of characters. The totality of those elements of the continuum whose definition is given by such a notion H(N.V.)^ω must - even after considering all finitely defined elements - form an enumerable set, since two different elements must certainly also correspond to different finite Main Parts and these all form an enumerable set.

 

However, that enumerable set of elements of the continuum is given - both by all of the finitely defined elements - and also by the finitely defined Main Parts that are of the type ω. On this enumerable quantity of the ω type, which in full is:

the diagonal method can also be used again. The rule giving the form

b_k = b_k k + d

is again not a consistent definition for (b₁, b₂, … ) = b^(d ). But the contradiction does not disappear even if (N.V.) is added ω times, since the definition must also appear in (III) in this form, e.g. at the n^th position, so that b_k = b_k k + d cannot be satisfied. But if (N.V.) is added ω + 1 times or “more times”, we get a consistent definition. If one proceeds in this way, one obtains finite character sequences which can only give a consistent definition by adding (N.V.) ω + 2 times, ω + 3 times, … .

 

Now let α be the smallest number of the second class of numbers such that the definite finite character string H (the Main Part) becomes a consistent definition if (N.V.) is added α times; we then say that H has the Rank of α. We want to replace the resulting pseudo-finite definition with

H(N.V.)^α.

Of course, this is not to be understood as the definition itself, because it must contain an infinite number of characters. It is only a conceptual characterization of that definition. If such an H belongs to an α, then we say that the α is of the first kind. Now, if the concept

H(N.V.)^α.

were that for a consistent pseudo-finite definition, then for every α′ < α, then

H(N.V.)^α′

would still generate a formal contradiction.

 

 

3.

The discussion up to now finally leads to the result that the second class of numbers, if it is to be understood as a set, i.e. as a totality of conceptually separate elements, would also have to be enumerable. We prove that under this assumption, on the one hand, the numbers of the first kind are enumerable and, on the other hand, that every number of the second class of numbers is of the first kind.

 

In fact: the set of finite character strings H is enumerable, and so the same is true for the set of numbers of the first kind, since by definition different α are associated with different H.

 

On the other hand, if every number of the second class of numbers were not of the first kind, there would in any case be a smallest (first) α₀ which is not of the first kind, while ω, ω + 1 etc. are of the first kind.

 

But this leads to a contradiction. The finite character strings that have a number of the second class as rank are enumerable, since this applies to the set of all finite character strings. So they can be written as follows as the ω type: H1, H2, … . Let α₁, α₂, … be the corresponding rank. The pseudo-finite definitions

should now define the following continuum elements:

By the Diagonal method we have:

a_k = a_k k + d

which gives (IV'), and we denote this by H₀. Now, we have that there is no contradiction in

H(N.V.)^α₀

since according to our assumption, in the series (IV) there is no definition of this form H(N.V.)^α₀. Any α_i less than α₀ appears in the series (IV); and the definition H(N.V.)^α_i thus yields a contradiction. i.e. H₀ is of the range α₀; in other words the transfinite ordinal number α₀ is of the first kind. The assumption that the second class of numbers also contains numbers of the second kind results in a contradiction.

 

Budapest, January 2, 1906.

Footnotes: