The Continuum

The concept of the “continuum” has not only played an important role everywhere in the development of the sciences, but has also always caused the greatest differences of opinion and even angry disputes. Perhaps this is due to the fact that its conceptual basis appears quite different to the dissenters, and the reason for this is that the precise and complete definition of the concept has not been communicated to them. It is perhaps also the case, which seems most probable to me, that the concept of the continuum originated with the Greeks, but without the necessary clarity and completeness that would obviate the possibility that subsequent thinkers would perceive it in diverse ways. Thus we see that Leucippus, Democritus and Aristotle regard the continuum as a composite, which consists of endlessly divisible particles (ex partibus sine fine divisibilibus), whereas Epicurus and Lucretius consider that it is a composite of atoms that are finite things. From this a great dispute later arose among philosophers, some of whom followed Aristotle, while others followed Epicurus, while others, in an attempt to steer clear of the dispute, sided with Thomas Aquinas[Footnote17] in the claim that the continuum consists neither of an infinite number nor of a finite number of parts, but of no parts at all. This opinion seems to me more of a tacit admission that one has not got to the root of the matter, preferring to avoid it in a polite manner, rather than being any explanation of the facts. Here we observe the medieval-scholastic origin of a view that we still find to be held today, according to which the continuum is an indivisible concept or, as others put it, a notion that is purely a priori, and hardly amenable as a basis for any definition. Every attempt at an arithmetical determination of this mystery is regarded as an inadmissible interference and rejected with all due intensity, and those of a diffident disposition get the impression that the “continuum” is not a mathematical-logical notion but rather a religious dogma.

Far be it from me to conjure up these issues again, nor would I have the space to discuss them more precisely within this restricted framework. I only see myself obliged to develop here, as briefly as possible, the concept of the continuum as logically and soberly as I understand it and only as it is required for the mathematical theory of sets. This process has not been easy for me because, of all the mathematicians whose authority I would like to appeal to, not a single one of them has dealt with the matter more closely than I in the sense that is required here.

On the basis of one or more real or complex continuous quantities (or, I believe, more correctly expressed as sets of continuous quantities) one has the concept of a continuum which is dependent on a single or many such sets. The concept of a continuous function has developed in the best possible way in the most diverse of directions, and in this way the theory of so-called analytical functions has arisen, as well as of more general functions with very remarkable properties (such as non-differentiability and the like). But the continuum itself as an independent entity has only been assumed by mathematical authors in its simplest form and has not been subjected to any more detailed consideration.

But first, I have to explain that in my opinion, the use of the concept of time or the intuition of time is not applicable when discussing the much more fundamental and general concept of the continuum. In my opinion, time is a notion which requires for any clear explanation of it, a presupposition of the concept of continuity, a concept that is independent of it. Even with the aid of the concept of continuity, time cannot be conceived either objectively as a substance or subjectively as a necessary a priori form of perception. Time is nothing more than an auxiliary and relational concept through which, by way of our perceptions, we establish the relationship between different motions occurring in nature. A thing such as objective or absolute time does not occur anywhere in nature and therefore time cannot be used as a measure of movement. Such motion might be regarded as a measure of time, were it not for the fact that time, even in the modest role of a subjectively necessary a priori form of perception, has not been able to bring about any useful, incontrovertible benefit, even though since Kant there has been plenty of time for this to happen.

It is also my conviction that one cannot do anything with the so-called intuition of space in order to gain information about the continuum, since space and the structures imagined in it can only achieve such information along with the help of the concept of an already completed continuum - in which case they are not the subject of mere aesthetic considerations, philosophical insight, or imprecise comparisons, but more sober and exact mathematical investigations.

Hence I have no choice but to attempt, with the help of the real number concepts defined in § 9, to form a purely arithmetical concept of a point continuum that is as general as possible. The basis for this, as it cannot be otherwise, is the n-dimensional plane arithmetical space Gn, i.e: the set of all value systems:

(x1 | x2 | … | xn )

in which each x can receive all real numerical values from - ∞ to + ∞ independently of the others. Every specific value system of this kind is called an arithmetical point of Gn. The distance between two such points is given by the expression:

| √(x1 - x1)2 + (x2 - x2)2 + … + (x1 - x1)2 |

An arithmetical point-set P contained in Gn is to be understood as any rule-based definition of a point-set of the space Gn. The analysis therefore comes down to establishing a precise definition for when P is to be called a continuum, but which is at the same time as general as possible.

I have proved (Crelle’s Journal, Vol. 84, p. 242) that all spaces Gn, no matter how large the so-called dimensional number n, have the same cardinal-number and consequently have the same cardinal-number as the linear continuum, such as the set of all real numbers in the Interval (0,1). The analysis and determination of the cardinal-number of Gn is therefore reduced to the same question, specialized to the interval (0,1), and I hope to be able to answer this soon by a rigorous proof that the cardinal-number sought is none other than that of our second number-class (II).[Footnote18] From this it follows that all infinite point-sets P have either the cardinal-number of the first number-class (I) or the cardinal-number of the second number-class (II). Another conclusion can be drawn from the fact that the set of all functions of one or more variables which can be represented by a given infinite series (regardless of the actual series), also only has the cardinal-number of the second number-class (II) and therefore is countable by numbers of the third number-class (III).[Footnote19] This proposition will refer, for example, to the set of all “analytical” functions, i.e: functions of one or more variables that result from the continuation of convergent power series, or to the set of all functions of one or more real variables that can be represented by trigonometric series.

In order to come closer to the general concept of a continuum located within Gn, I refer to the concept of the derivative P (1) of an arbitrarily given point-set P, as I first used it in the work: Mathematische Annalen Vol. V, then developed in Vol. 15, 17, 20 and 21 and expanded to the concept of a derivative P (γ), where γ is any integer from one of the number-classes (I), (II), (III) etc.[Footnote20]

The point-sets P can now also be divided into two classes according to the cardinal-number of their first derivative P (1). If P (1) has the cardinal-number of the first number-class (I), it becomes apparent, as already noted in § 3 of this article, that there is an integer α of the first or second number-class (II), for which P (α) disappears. But if P (1) has the cardinal-number of the second number-class (II), then P (1) can always, in only one unique way, be split into two sets R and S, so that:[Footnote21]

P (1) = R + S

where R and S have very different properties. R is such that it is capable of continuous reduction through repeated application of taking the derivative, it eventually disappears, so that there is always a first integer γ of the number-classes (I) or (II), for which:

R(γ) = 0

I call such point-sets R reducible. S, on the other hand, is such that with this point-set the repeated application of the derivative process produces no change whatsoever, in that:

S = S (1)

and consequently also:

S = S (γ)

I call such sets S perfect point-sets. We can therefore say: if P (1) has the cardinal-number of the second number-class (II), then P (1) reduces to a definitive reducible perfect point-set.

Although these two predicates “reducible” and “perfect” are incompatible in the same point-set, on the other hand as one can easily see with a little attention, irreducible is not the same as perfect, and similarly imperfect is not exactly the same as reducible.

The interior of the perfect point-set S is definitely not always what I have called “everywhere-dense” in my work mentioned above;[Footnote22] therefore, they are not suitable on their own for the complete definition of a point continuum, even if one must immediately concede that a point continuum must always be a perfect set.

Instead, one more concept is required in order to define the continuum along with the preceding one, namely the concept of a connected point-set T .

We call T a connected point-set if for every two points t and t′ of it, given an arbitrarily small number ε, there is always in multiple ways a finite number of points t1, t2, … tv of T where the distances tt1, t1t2, t2t3, … tvt′ are all smaller than ε.

It is now easy to see that all geometrical point continuums known to us also fall under this concept of a connected point-set; but I now believe that I have determined that the necessary and sufficient features of a point continuum are given by these two predicates “perfect” and “connected” and therefore I define a point continuum within Gn as a perfectly-connected set.[Footnote23] Here “perfect” and “connected” are not mere words, but quite general predicates of the continuum which are conceptually characterized in the most precise possible way by the preceding definitions.

Bolzano’s definition of the continuum (Paradoxien § 38)[Footnote24] is certainly incorrect; it only expresses one single property of the continuum, which is also satisfied in sets which result from Gn by “isolating” some point-set at a distance from Gn (cf. Mathematische Annalen Vol. 21, p. 51); it is likewise satisfied by sets which consist of several separate continuums. It is evident that there is no continuum in such cases, although according to Bolzano this would be the case. So here we see a violation of the sentence: “ad essentiam alicujus rei pertinet id, quo dato res necessario ponitur et quo sublato res necessario tollitur; vel id, sine quo res, et vice versa quod sine re nec esse nec concipi potest.” (It is the essence of any entity that, if something is judged necessary to it, then if that thing is taken away, the entity is lacking that thing; or for any other similar thing, without which the entity can neither be nor be conceived, but with them it can be.)

Similarly, it seems to me that Dedekind rather one-sidedly emphasizes only another property of the continuum in his article (Stetigkeit und irrationale Zahlen), namely the property that it has in common with all “perfect” sets.