Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
The following is an English translation of a paper by Julius König (Footnote: König, Julius. “Über die Grundlagen der Mengenlehre und das Kontinuumproblem” Mathematische Annalen 61, no. 1 (1905): 156-160.) which appears to have be the first to raise the question of whether there can exist numbers that can have no finite representation. The original German can be viewed at Über die Grundlagen der Mengenlehre und das Kontinuumproblem.
English translation by James R Meyer, copyright 2018 www.jamesrmeyer.com
On the foundations of set theory and the continuum problem
by Julius König
It is only after overcoming serious reservations that I have decided to publish the following. But regardless of how the notion advanced herein might be received, I think that the questions raised concerning the further development of set theory cannot be ignored.
The fact that the word “set” is being used indiscriminately for completely different concepts and that this is the source of the apparent paradoxes in this recent branch of science, and that furthermore, set theory itself cannot dispense with axiomatic assumptions any more than any other precise science, and that such assumptions are subject to a certain degree of arbitrariness, as is the case in other disciplines, even if they lie far deeper here - I do not want to present any of this as a novel stance. But even in this sketchy provisional communication, I believe that some new points of view are being presented concerning these questions. In particular, the special theory of well-ordered sets might be considered as not being fully developed with a complete foundation so long as the questions dealt with in Section 4 below are not clarified.
If a1, a2, … ak, … is a countably infinite sequence of positive integers (of type ω), the entities
(a1, a2, … ak, …)
shall define the set called the “continuum”. Or if some other definition of the continuum is assumed, then the entities (a1, a2, … ak, …) are symbols that on the one hand unambiguously determine the elements of the continuum and on the other hand, are precisely distinguished from each other.
An element of the continuum will be said to be “finitely defined” if, by means of a language capable of giving a clear expression of our scientific thinking, we can specify in a finite timespan a method (law) that conceptually distinguishes that element of the continuum from any other one, or - to state it differently - an arbitrarily chosen k gives rise to the existence of one and only one associated number ak.
However, it must be clearly emphasized that the “finite” conceptual distinction required here should not be confused with the requirement for a well-defined method, or even a finite method for determining ak.
It is very easy to show that the finitely defined elements of the continuum determine a subset of the continuum with the cardinality ℵ0, which from now on will be referred to as E.
Such a finite definition must be completely given by a finite number of letters and punctuation marks, of which there is also only a definite finite number. One can further rearrange those various finite definitions so that any given definition corresponds to one and only a certain positive integer as (finite) ordinal number.
Each finitely defined element of the continuum is such that, for such an element there can exist, and in fact do exist several “finite” definitions, so that such elements determine a series of positive integers, among which the smallest is clearly determined; and conversely, this integer, by its associated combination of signs, uniquely determines the corresponding finitely defined element of the continuum.
E is therefore equivalent to a subset of the set of positive integers. But since
(a, a, … a, …)
where a can be any positive integer that is a finitely defined element of the continuum, it follows that:
e = ℵ0
where e stands for the cardinality of E.
However, since the continuum is by its definition, non-denumerable, there must exist elements of the continuum that cannot be finitely defined.
Although at the present moment, I am not able to give a precise and systematic account, it is an absolute necessity that the axiomatic assumptions contained in the previous train of thought should be precisely specified.
Above all, the “fact” has been assumed that there are processes taking place in our consciousness that satisfy the formal laws of logic and are called “scientific thinking” and that there are among others, some that are associated with other equally unique processes, the generation of those previously described strings of symbols.
The question of "how" this relationship comes about, or even how far these relationships can be extended, is not considered here at all. (Metalogical axiom).
Any “arbitrary” sequence of the type ω formed from positive integers and the “entirety of all these sequences” which we call the “continuum” are “possible notions”, i.e., they are such that they do not lead to a logical contradiction (Continuum axiom).
A further, thorough analysis of this claim is, I believe, contained in the developments which Hilbert presented at the International Congress of Mathematicians III (1904) in Heidelberg.
In particular, this definition of the continuum entails the assertion that its cardinality is ℵ0ℵ0.
And also that ℵ0ℵ0 > ℵ0 as is proved by the method in my article “Zum Kontinuumproblem“ (On the Continuum Problem), Math. Ann. Vol. 60, (1905) p. 177.
I am consciously placing myself in direct opposition to the assumption that it is not permissible to go beyond “finite laws”. In my view, that assumption denies the existence of the continuum and the continuum problem. On the contrary, the assumption I use here is that there are elements of the continuum which we cannot “think through to their end” but which are, in spite of this, without any contradiction; i.e., they are “ideal” elements, if one permits this expression to be used in quite a new sense.
If, according to previous assumptions, we can speak of an “arbitrary” element of the continuum, we can now employ the logical antithesis: “Any element of the continuum is either finitely defined or this is not the case”. According to a ) and b) this antithesis cannot be rejected, especially since without altering our conclusions, it can also be subjectively interpreted as: “For any element of the continuum, either it is certain that a finite definition exists or else this is not the case.”
The assumptions developed thus far lead in a remarkably simple way to the conclusion that that the continuum cannot be well-ordered. If we think of the elements of the continuum as a well-ordered set, then those elements which cannot be finitely defined form a subset of that well-ordered set, which certainly contains elements of the continuum. This subset is also well ordered, and contains one and only one first element. Note, furthermore, that according to the assumptions now applying, the continuum, like any well-ordered set, defines a complete sequence of definite ordinal numbers, in such a way that each element of the continuum corresponds to one and only one such ordinal numbers also vice versa. Accordingly, “the ordinal number corresponding to a finitely defined element of the continuum” as well as “the element of the continuum corresponding to a finitely defined ordinal number” are both finitely defined. Hence there must be a first ordinal number in that sequence that is not finitely definable. But this is impossible.
For there is a definite (well-ordered) set of finite ordinal numbers, which follow each other from the first one onwards. But “the ordinal given by the series of all these first ordinal numbers” would be defined by what has just been stated, while - by the assumption above - it cannot be finitely defined.
The assumption that the continuum can be well-ordered has therefore led to a contradiction.
The direct objection against the correctness of the previous statements is that they can be applied, word for word, to every well-ordered and non-countable set, so that such sets cannot exist at all. But since Cantor's second numerical class Z(ℵ0) - “the totality of all order types of well-ordered sets of the cardinal number ℵ0” - defines such a “set” without contradiction, an error must have been made in the previous argument. It will be explained in more detail why it does not necessarily have to follow that this paradoxical result is due to an error .
The word “set” is used in the two cases for two completely different notions.
In the formation of the concept of the continuum, the “arbitrary” sequence (a1, a2, … ak, …) is the primary, original one. From the requirement that a1, a2, … are replaced by certain positive integers, this becomes a “definite” sequence, an element of the continuum, which we conceive of, if at all, as being separate to every other element. The additional requirement that we also conceive of the entirety of these "well-differentiated" objects then leads to the continuum.
The case for the number class Z(ℵ0) is quite different. Its “elements” are determined by the “property” of being order types of well-ordered sets of cardinality ℵ0 . While we know elements such as: ω, ω + 1, … but that property is only an abstraction, at best a means of distinguishing between class and other entities; but it is certainly not a method by which every element of Z(ℵ0) can be constructed. Here is the primary, primordial, collective concept, which, following Cantor’s nomenclature, I would not want to call it a “set” but instead a “class”; it is only then that elements belonging to the class are constructed.
That the second number class Z(ℵ0) can be defined as a “completed” set of well-differentiated elements, i.e., conceptually quite distinct elements cannot be described as being probable according to the state of our theoretical knowledge up to now. Insofar as the previously developed inferences are correct, this discussion would also include the proof that the second number class cannot be thought of as a completed set, i.e., as an entirety of well-differentiated elements that are conceptually distinct.
In concluding these sketchy arguments, I am pleased to note that, in spite of their partially dissenting character, they may, inasmuch as they are correct, only shed new light on the immense value of the ingenious Cantorian creation. The disagreement is only directed against certain assumptions by Cantor; the content of the theorems that he has proved remains completely intact. Finally, I note that the distinction between “set” and “class” as given here completely resolves the paradoxes mentioned (“set of all sets”, etc.).
The substance of the above was presented by me at the Hungarian Academy of Sciences on 20 June 1905.
In this fact lies, I believe, the origin of those paradoxes in the theory of ordinal numbers to which Mr Burali-Forti has drawn attention.
I should like to further add here a few short remarks which may assist the understanding of the content of Section 4.
Even the entirety of positive integers is originally given only as a “class”. This is how Mr. Hilbert (see as aforementioned) defines the “smallest infinity”. But it seems that the requirement to consider this class as a completed set is possible, i.e., it is consistent in itself.
On the other hand, the continuum would be conceivable only as a “completed set”, and the second set of numbers only as a class or (perhaps this term is permissible) an “emerging set”.
I would like to refer to a very elementary concept of collectiveness which certainly does not allow the conception of a “completed set”.
We begin with the entirety of all finite decimal fractions, but we write them as infinite decimal fractions, putting the digit 0 in all vacant positions.
Every such position is available in these entities, i.e., we can substitute for any digit any other digit without affecting the entirety of the defined entities.
Nevertheless, it would be utterly inadmissible to speak of an entirety of positions as available, obviously this would eliminate the "inhibition principle" that is included within the definition of availability. This inhibition principle can be summarized as follows: “The kth place can be occupied arbitrarily provided there exists a positive integer l > k, so that from the lth only the zero digit is used.”
The question: “how many freely available places are there?” cannot be answered by any cardinal number (in Cantor’s sense) and requires the creation of a new “concept of counting”.
Copyright © James R Meyer 2012 - 2018