Calculus, Dedekind and Cantor

A three-act play with prologue and epilogue

Prologue. To what extent does Calculus need Dedekind?

Act I. Introduction, or what Calculus is and what Dedekind has to do with it. The first appearance of Cantor.

The field R of real numbers, which we are supposed to know from college, is characterized by a system of axioms, which can be classified in two groups. The first contains all of them but one - Dedekind's axiom. The axioms in this class are related to arithmetic and order. They state, for instance, that the arithmetical operations are associative and commutative, that subtraction is defined for any pair of real numbers while division only for divisors other than zero, that multiplication is distributive with respect to addition, that addition and multiplication by positive numbers preserve the order (i.e. the inequalities), and so on. Their common feature consists in that they state properties of real numbers. Dedekind's axiom is quite different: it describes a property of subsets of the real numbers, namely:

A description of Calculus seems to be somewhat more difficult. The best I can think of is the following: Calculus is a collection of theorems deducible from the axioms of the field of reals, which state properties of real-valued functions defined on subsets of the set of all real numbers. To see what sort of properties these theorems refer to let's have a look at two fundamental theorems of Calculus:

Theorem 1 (Darboux). If one of the values assumed by a continuous function f: [a,b] -> R at a and b is positive and the other is negative, then there is a point c within the interval [a,b] such that f(c)= 0.

Theorem 2 (Lagrange). If a continuous function f: [a,b] -> R is differentiable within the interval [a,b], then there is a point c with a < c < b such that f(b)-f(a)=f'(c)(b-a).

To complete the picture, let's recall one more theorem:

Theorem 3 (Cantor). The field of real numbers is not countable.

Act II. Are all real numbers indispensable? The tragic end of Calculus.

Let's see what happens with Calculus if it is deprived of Dedekind's axiom. Take any smaller subfield P of the real numbers, i.e. a subset of the reals which satisfies all the axioms of the first class. Since P is properly contained in R, there is a number c in R which is not in P. Let = a and b be any numbers in P such that c lies within the interval [a,b] (clearly such numbers exist). Let

      A ={ x \in P | a < x < c}

Then A is a bounded subset of P and a counterexample to Dedekind's in this subfield, since supA = c and c not in P imply that there is no least upper bound in P for A. If the theorems of Calculus were deducible from the axioms of the first class alone, they would hold true for all P-valued functions defined in P. This, however, is not so. For instance, define a function f: [a,b]P -> P, where [a,b]P={ x \in P | x \in [a,b]} is a closed interval in P, as follows:

One can easily see that the function satisfies the assumptions of both theorems 1 and 2, but the claims are false for f. Indeed, the definition of f implies that f(x) is equal to zero for no value of x in [a,b]P. Theorem 2 is false too, since f'(x) is equal to zero for all values x within the interval [a,b]P, whereas f(b)-f(a)= 2. How was this counterexample made possible? Obviously, by excluding the number c from the set P. This makes clear that

Corollary 1. Calculus indeed does need all and each of the real numbers.

So much so that it can be shown that without Dedekind's axiom not only theorems 1 and 2, but the entire Calculus would break down to come to a tragic end. What would remain would be ruins with no interest at all.

Act III. The miraculous salvation of Calculus, or definable numbers and functions

Our failure seems to be due to the conflict between our two principal aims: to make the set of real numbers smaller and to keep the set of functions intact. The breakdown can be avoided, however, if the reduction of the set of reals is followed by a corresponding reduction of the set of functions. Let's see how this can be done.

Let D be the subset of those real numbers which can be defined by means of arithmetical operations, natural numbers, the set of natural numbers, induction and the least upper bound operation (i.e. we only include the numbers which can be specifically defined using the above notions). It is not difficult to prove that D is a subfield of the reals. For example, to see that if x is in D, then z=1/x is in D, it is sufficient to observe that z can be defined as follows:

    z is the only number such that z x = 1, where x is defined by ...

The set of all "specific" definitions being countable, the subfield D is countable, too. Thus D is indeed much smaller than the set of all real numbers (see theorem 3).

Let's assume that whenever we speak of a subset of D, we mean subset definable by means of the notions enumerated above together with the notion of "definable number". It can be proved that Dedekind's axiom holds true for the field D and its (definable) subsets, i.e. every bounded subset of D has a least upper bound in D.

Corollary 2. All the axioms of the real field are valid in the field D (if subset=definable subset).

Let's consider all definable functions in D.

Theorem 4. Every definable function maps definable numbers into definable numbers.

Proof: If x is a definable number and f is a definable function, then the definition of z=3Df(x) goes as follows:

      " z is the only real number such that z=f(x), where f is defined by ... and x is defined by ... ".

Thus if we only consider definable numbers, subsets and functions, we get a model for all the axioms of the reals. Hence all the theorems of Calculus are also valid, since they can be derived from the axioms. Let's call this theory Definable Calculus.

Corollary 3. In spite of the reduction of the real field, we managed to save Calculus.

Corollary 4. All the theorems of Calculus are true in Definable Calculus under the following interpretation:

real numbers definable real numbers
subsets of real numbers definable subsets of real numbers
functions definable functions
etc etc

In particular, theorem 3 remains true for Definable Calculus:

Theorem 5 (Cantor, for Definable Calculus). The field D is not countable.

Epilogue. Amazing consequences, or how the notion of being countable (and many more) depends on how we perceive the world

At the beginning of Act III we stated that the field {\bf D} is countable and then at the end of the same Act we quote Cantor's theorem about D{\bf D} not being countable. Contradiction?! In our attempt to save Calculus we destroyed Mathematics. What have we done?! The explanation is quite simple. The property of being or not being countable is not an absolute property of a set. In some cases it may depend on the viewpoint (model) of the "world" we adopt. In our example,

  1. The model for Calculus contains plenty of elements and functions, no wonder then that among them there is a function f mapping the set of all natural numbers onto D. This means that from the point of view of (traditional) Calculus the set D is countable (see the definition of a countable set).
  2. The model for Definable Calculus contains much less elements and functions. Cantor's theorem states now that there are so few functions that no function mapping the set of all natural functions onto {\bf D} can be found. Addition can be performed "effectively" , so in each model of Calculus we have 2+2=4. On the other hand, the property of being countable is not effectively verifiable, which means that there is no finite algorithm to determine whether a set is countable or not. In the case of such noneffective notions it happens quite often that answers to questions depend on the model we choose. This is exactly what happened with the set {\bf D}.

Another noneffective postulate, used in Geometry, is Archimedes' axiom:

    Laying off repeatedly a given segment a on a straight line we can get a segment greater than a given segment b.

Again the validity of this assertion depends on the model of geometry assumed.