The concept of an irrational number may, as is well known, be obtained roughly as follows: The interval from 0 to 1 is divided into two parts, each part again into two parts, etc.; in this way finer and finer subdivisions are progressively obtained. A sequence of such intervals in which each term forms part of the term preceding, tends to contract more and more to a point as the process is continued. This is where the actualists take the plunge into the class of completed infinities by declaring that an infinitely long sequence of this kind is a 'real number'.
From this interpretation some unusual consequences follow, which, apart from the general disputability of the actualist interpretation arising from the antinomies, could be advanced as additional arguments against this view: On the one hand, it may be proved in the usual way that these real numbers form a nondenumerable set. On the other hand, however, all theorems, all definitions, and all proofs which can ever be formulated or carried out are denumerable, since they can always be characterized by finitely many symbols. This leads to the conclusion that there are real numbers which can in no way be individually defined and valid theorems which are unutterable and which no one will ever be able to prove. If Skolem's theorem of relativity is invoked, it follows further that the whole of conventional analysis remains quite generally valid in all of its parts if it is interpreted in a certain denumerable model.
Here we might well be tempted to say: If 'the nondenumerable continuum' in this way succeeds in completely eluding our comprehension, is there then any point at all in speaking of it as something real? In [ the next chapter ], I shall show how, in a restricted sense, we can nevertheless answer this question in the affirmative.
At first it will be examined what the constructivist point of view has to offer as a replacement for the actualist concept of irrational numbers. The sequence of divisions of intervals may be begun as before. The notion of the completed infinite sequence of intervals, however, must be rejected as being without sense. The infinite is after all to be regarded only as a possibility, as an expression for the unsoundness of the finite. We can therefore reasonably say: It is possible to push this subdivision further and further. However, no irrational number is obtained in this way; at each stage of the subdivision we are left with a collection of eventually rather proximate rational numbers. It is in this sense that Kronecker maintained that 'there are no irrational numbers at all'. Even a constructivist, however, does not have to be this narrow-minded; there are possibilities of going further. An irrational number can after all be regarded as given if a rule is available which permits the computation of a sequence of intervals of the mentioned kind arbitrarily far. (Here it is convenient, in order to avoid certain formal difficulties, to base the argument on double dyadic intervals, i.e., those which result from pairing two neighbouring dyadic intervals of the same subdivision.)
Such a rule is easily stated, for example, for \/2, generally for \n/m, but also for transcendental numbers such as pi and e, in fact, quite generally, for practically all numbers which are needed in analysis as individually defined numbers; viz., in all cases in which the number concerned can actually be calculated to any desired degree of accuracy.
In order to remain faithful to the constructivist point of view, however, the 'numbers' defined in this way must be handled with care. The temptation must be resisted of regarding such a number as a completed infinitely long dyadic fraction; what is given in this sense is not really the entire number, but only the rule for its progressive realization; the rule itself is finite and merely a suitable representative, in certain contexts, of an infinitely long number of which it can be said, now as ever, that it actually does not really exist.
In his paper entitled 'Das Kontinuum', Weyl has attempted to construct an analysis on the basis of this kind of number concept. (In this paper, Weyl has not exploited the full consequences of the constructivist attitude toward the natural number sequence; but this was later remedied.) One of the difficulties arising in this connection is that of deciding what techniques should be permitted in the computation and hence the definition of numbers. Initially, Weyl carried out a definite delimitation of techniques; usually, however, intuitionists avoid such a delimitation alltogether, a point of view which is not entirely inappropriate since a universally valid delimitation involves fundamental difficulties analogous to those existing for delimited axiom and inference systems, already discussed above, viz: That such delimitations are always open to and in need of further extensions. This actually represents no serious shortcoming; in cases of immediate practical importance it is perfectly clear how the concept of 'calculability' is intended.
Church [ ... ] made this concept of calculability more precise. Independently of Church, Turing formulated an equivalent concept and also applied it, in particular, to the calculability of real numbers. The greater precision consists here in the formulation of a unified concept embracing all 'calculation procedures': this makes it feasible to carry out an impossibility proof [ ... ]; even at this level of precision, however, it cannot he decided of every procedure falling under this concept whether it is a 'calculation' procedure or not.
An extension of the constructivist concept of a real number was devised by Brouwer with his 'free choice sequences'. These sequences are the logical outcome of an attempt to introduce the concept of a function of real numbers. In actualist mathematics this concept, is, as is well-known, defined simply as a relation which correlates with every arbitrary real number a second real number as its functional value. The concept of a completed infinity is here involved threefold: first in the two real numbers, and second in the universal abstract 'correlation'. This concept is therefore of no use to the constructivist. One of the ways open to him is to define a function as a rule which correlates with every rule defining a real number a second rule defining another real number. It is easily seen, however, that the following less restrictive version, which is closer to the actualist function concept, is still entirely compatible with constructivist principles: In place of the concept of an individual real number given by a rule, the procedure of subdividing intervals is once again taken as a starting point, by defining a function specifically as a rule with the following properties: As the sequence of intervals of the above described kind is chosen in some way, the functional rule correlates with a certain finite initial segment of this sequence a first interval of the 'functional value', and after having continued the sequence up to a certain further point, a second such segment, etc. The correlated intervals are therefore once again designed to form a 'nest of intervals'. - In short: In each case a desired finite number of initial places of the functional value should be calculable by the function rule from a sufficiently large number of initial places of the argument value.
That this function concept is still considerably narrower than the actualist one can already be seen from the fact that such a 'function' is always a continuous function. Brouwer proves moreover the uniform continuity of these functions and, in doing so, he makes a rather extensive use, for a constructivist, of 'transfinite' induction.
The argument values occurring in connection with this function concept are what Brouwer calls 'free choice sequences', viz., sequences of intervals in which successive terms can in each case be freely chosen - subject only to the restriction imposed by the fundamental conditions for nested intervals.
Even this number concept must be handled with care; it really has no independent meaning, but only a meaning within a proper context. After all, the concept of a completed infinite sequence is still entirely without sense; free choice sequences may thus be used only in contexts in which a finite initial segment of them or, at most, the possibility of their arbitrary extension, is involved. This is guaranteed in the case of the stated definition of a function.
By means of Brouwer's function concept the most frequently used functions in analysis can now be given suitable constructivist definitions without difficulty. Most of them are after all such that their functional values can be calculated more and more accurately as the argument value is progressively narrowed down.
Considerable differences between intuitionist and classical analysis nevertheless manifest themselves in the further development of the theory, especially in connection with existence theorems, as already mentioned in [ the first chapter ]. Constructivists must, after all, insist that a calculation rule is specified for the number whose existence is asserted; actualist existence proofs often do not meet this requirement.
Intuitionist analysis thus becomes much more complicated than classical analysis. This may already have been noticed in connection with the definitions of the fundamental concepts. Constructivists require, for example, different concepts of real numbers for different uses, whereas a single simple concept suffices in actualist analysis.