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The 'Continuum Problem'

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The 'Continuum Problem' Chapter 5 The ‘continuum problem’ 5.1 Introduction It was in 1900 at the Paris conference that Hilbert presented his list of unsolved mathematical problems; this list can be viewed as an assignment for the math- ematical community for the century ahead. As number one on that list, which was entitled Mathematische Probleme, stands the continuum problem, already conjectured by Cantor, and expressed by Hilbert as follows: Jedes System von unendlich vielen reellen Zahlen, d. h. jede un- endliche Zahlen- (oder Punkt)menge, ist entweder der Menge der ganzen nat¨urlichen Zahlen 1, 2, 3, ... oder der Menge s¨amtlicher reellen Zahlen und mithin dem Kontinuum, d. h. etwa den Punkten einer Strecke, ¨aquivalent; im Sinne der ¨aquivalentz gibt es hiernach nur zwei Zahlenmengen, die abz¨ahlbare Menge und das Kontinuum.1 As a related problem, of which the solution may lead to the solution of the original problem, Hilbert mentioned the well-ordering of the continuum. In fact we have to distinguish between the continuum problem which asks for an answer to the question ‘what is the cardinality of the continuum’, and the continuum hypothesis, which conjectures that this cardinality is the next higher after ℵ0, the cardinality of the set of the natural numbers. Since it was proved already by Cantor that the cardinality of c exceeds ℵ0, the claim of the hypothesis is ℵ0 that c = 2 = ℵ1. In this form it is still an unsolved mathematical problem. Different other questions, related to the continuum hypothesis, have been solved, but the hy- pothesis itself still stands as such. 1Presented at the Paris conference of 1900, and published in the G¨ottinger Nachrichten in 1900. See [Hilbert 1900], see also [Hilbert 1932], vol. III, page 298. In our dissertation it was mentioned earlier on page 9. 165 166 CHAPTER 5. THE ‘CONTINUUM PROBLEM’ For Brouwer, the continuum problem is of a completely different nature. The ℵ0 form c = 2 = ℵ1 is a meaningless one for him since ℵ1 is a non-existing car- dinality. The question that he intends to answer on page 66 of his dissertation is: what are the possible cardinalities of subsets of the continuum. If we take the standard interval (0, 1) (which is, in fact, the intuitive continuum between the first and the second experienced event) as point of departure, then a subset of the continuum cannot be anything else but a point set, constructed on the continuum, or a finite or denumerable set of subsegments of the continuum. The aim of this chapter is an analysis to the solution of the continuum prob- lem, as Brouwer presented it in his dissertation and in the notebooks. Brouwer’s solution is simple and is, as we will argue, a direct corollary of the modes of set construction that he admitted. In fact his solution is almost trivial, since the limited number of possible cardinalities immediately follows from his set constructions. The solution from the notebooks is different, but it is just as well an imme- diate corollary of Brouwer’s constructivism. One can wonder why Brouwer did not present the solution from the note- books in his dissertation; it certainly is a more direct and intuitive (although possibly a less sophisticated) one. On the other hand, the proof from the dis- sertation is more in agreement with, and fits better, the three admitted modes of set construction that Brouwer just had given. 5.2 The solution in the dissertation On page 66 ff. of his dissertation, Brouwer proved that every point set which is defined on the measurable continuum (hence constructed on the intuitive continuum), either has the cardinality of the continuum or is denumerable, thus solving the problem and confirming the hypothesis. The proof proceeds as follows:2 let there be given a set, constructed according to the given rule 2, whether or not in combination with the third rule (see our pages 117, 120 and 122), and a continuum. The set can now be mapped on the continuum in the following way: (...) in both sets (the continuum and the given set) a well-defined, and consequently denumerable, point set is chosen in such a way that all the other points can be obtained by approximation from everywhere dense parts of this set. Then the undefined points are brought into one-to-one correspondence by mapping the everywhere dense parts, with respect to which the infinitely proceeding approxi- mations must be taken, onto each other; then the points which were 2In Brouwer’s presentation of the proof the term ‘group’ is used; as said earlier, Brouwer frequently used the term group when the term set clearly expresses the intention; apparently the terminology in this matter was not yet completely fixed. 5.2. THE SOLUTION IN THE DISSERTATION 167 defined can still be brought into correspondence with each other, because they are denumerable in both sets.3 As usual, Brouwer put it briefly worded; the conclusion of the foregoing procedure is drawn without further arguments or explanation: It follows that every set of points on the measurable continuum (and consequently also on the simple intuitive continuum, which in fact we can only handle after having made it measurable – or built up out of individualized measurable parts) which is not denumerable, has the power of the continuum.4 As an elucidation of this very concise argument and its conclusion, the follow- ing discourse expresses, in our opinion, the intentions and content of Brouwer’s solution. Suppose we have constructed a set according to method 2, i.e. every- where dense and ‘completed to a continuum’; or, according to rule 3, deleted from a continuum a constructed dense scale. The result of the (earlier discussed) rules 2 and 3 can be fully mapped one-to-one onto the continuum, hence has the cardinality of the continuum. This ‘mapping’ can easily be understood if we recall to mind the way in which an everywhere dense set was ‘completed to a continuum’. We discussed this on page 116: we have an everywhere dense denumerable set which is ‘covered by a continuum’, and this ‘covering’ can now be identified with the ‘completion to a continuum’ from rule 2. But then, if we take a continuum C and define an everywhere dense scale (e.g. η) on it, by which every point on C, not belonging to that scale, can be approximated, then the proof of the statement that the result of rule 2 is similar to C, is a trivial one, hence the result of rule 2 has the power of the continuum. And if we delete from C the scale η, then the result is similar to that of rule 3. Also the proof of this claim is a trivial one: map the denumerable scales onto each other and map the approximating sequences in the denumerable scale for undefined points onto each other. Hence a set satisfying the first method of construction is denumerable in virtue of its definition, and the set A, constructed according to method 2 and/or 3, has the cardinality of the continuum, since the above sketched mapping be- tween A and C can be completed one-to-one. No other sets can be constructed and therefore no other sets exist. 3(page 66) in beide verzamelingen (het continu¨umen de gegeven puntverzameling) wordt een welgedefinieerde, dus aftelbare puntgroep uitgekozen z´o,dat alle andere punten als be- naderingen ten opzichte van overal dichte delen van die groep kunnen worden beschouwd, en vervolgens worden de ongedefinieerde punten met elkaar ´e´en-´e´enduidig in correspondentie gebracht, door de overal dichte delen, ten opzichte waarvan in beide de oneindig voortlopende benaderingen moeten worden genomen, op elkaar af te beelden; de wel gedefinieerde punten kunnen dan altijd nog daarna in correspondentie met elkaar worden gebracht, daar ze in beide aftelbaar zijn. 4(page 66, 67) Waaruit volgt, dat elke puntverzameling op het meetbaar continu¨um(dus ook op het intu¨ıtief continu¨umzonder meer, waarmee we immers eerst kunnen werken nadat we het meetbaar – of uit ge¨ındividualiseerde meetbare stukken opgebouwd – hebben gemaakt), die niet aftelbaar is, de machtigheid van het continu¨um bezit. 168 CHAPTER 5. THE ‘CONTINUUM PROBLEM’ Brouwer then concluded: Thus the continuum problem, put forward by Cantor in 18735 and mentioned by Hilbert as still open (‘Mathematische Probleme’, Prob- lem no. 1, page 263),6 seems to be solved, in the first place by keeping strictly to the view: a continuum as set of points must be considered with respect to a scale of order type η.7 Brouwer did not claim that the continuum problem was solved, but that it seemed so. But we immediately see that Brouwer’s solution to the continuum problem is a direct consequence of 1) the three methods of set-construction that he admitted, which methods, in turn, are a consequence of the basic intuition of all mathematics which states that the continuum is no point set, and 2) of his point of departure that only constructible objects are acceptable as mathemat- ical entities on the continuum. ‘Undefined points’ exist only on a continuum, or in a denumerable set which is ‘completed to a continuum’, in relation to an (everywhere dense) scale of constructed points. But then the conclusion that every set either is denumerable or has the cardinality of the continuum becomes obvious and almost trivial. Even if we consider (thereby anticipating on future developments) the perfect spread as a representation of the continuum of the reals, then the conclusion should remain the same under this argument, as a result of the requirement that every subset of the continuum needs an algorithm for the construction of its individual elements, and the result of a construction can never surpass a denumerable cardinality.
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