## Calculus, Dedekind and Cantor## A three-act play with prologue and epilogue
The field
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:
To complete the picture, let's recall one more theorem:
Let's see what happens with Calculus if it is deprived of Dedekind's
axiom. Take any smaller subfield A ={ x \in Then A is a bounded subset of P, where [a,b]={ x \in _{P}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] , whereas f(b)-f(a)= 2. How was this
counterexample made possible? Obviously, by excluding the number c from
the set _{P}P. This makes clear that
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.
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
The set of all "specific" definitions
being countable, the subfield Let's assume that whenever we speak of a subset of
Let's consider all definable functions in
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.
In particular, theorem 3 remains true for Definable Calculus:
At the beginning of Act III we stated that the field {\bf - 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). - 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. |