Georg Cantor at the Dawn of Point-Set Topology - The Real Numbers

In order to modify the proof in his 1870 paper, Cantor needed to develop his own theory of the real numbers. Although our purpose in examining his theory of real numbers is to demonstrate the emergence of point-set topology, Cantor's construction is interesting in its own right. Even Bertrand Russell begrudges Cantor a compliment when he writes

The theory of Cantor . . . with all the requisite clearness, lends itself more easily to the interpretation which I advocate, and is specially designed to prove the existence of limits. [22, p. 283, emphasis original] 

Weierstrass apparently had a theory of the reals at this point ([10, footnote 47], [21]). However, none of Weierstrass' writing on the theory of reals seems to have survived, and we only know of it through his students. Since his students all seem to have published their teacher's theory after 1872, it may not have been well known when Cantor wrote his paper. In any case, in his 1872 paper, Cantor cites Euclid as the definitive source for the theory of real numbers, writing

For comparison ... we mention the book “Elements of Euclid” which remains the decisive treatment of the subject. [6, p. 127]

Still, Cantor was not fully satisfied with this theory, and he desired a more solid foundation for standard operations (addition, etc.) performed on the real numbers. To this end, he defined a sequence of rational numbers \(a_1,a_2,\dots,a_n,\dots \) to possess numberness (Zahlengroße) or to be a fundamental sequence if for every \(\epsilon > 0, \) there exists an integer \(N\) such that whenever \(n\ge N,\) \( \vert a_{n+m}- a_n\vert\,<\epsilon\) for any positive integer \( m.\) (In modern terminology, Cantor has defined a Cauchy sequence of rational numbers.) We associate to any rational sequence possessing numberness a symbol \(b,\) which we refer to as a number value. Cantor writes

This property of sequence \(a_n\) I express in the words \(a_n\) has a certain limit \(b.\)

In other words, by the symbol \(\lim a_n,\) Cantor means the number value \(b\) associated to that sequence \(a_n.\) He denotes the collection of all such fundamental sequences by \(B.\) We will see in Section 5 that every real number corresponds to a fundamental sequence of \(B.\) Cantor goes on to show how we may (for now at least) conceptualize \( B\) as the set of real numbers. However, this is only for intuitive purposes. Cantor is explicit about this last point when he writes

Now these words initially have no other meaning except as an expression for those properties of the sequence, and from the fact that we associate to the series \(a_1,a_2,\dots,a_n,\dots \) a special character \(b,\) it follows that with various series, various characters \( b, b^{\prime}, b^{\prime\prime},\dots\) are formed.

Immediately after making this definition, Cantor is quick to note that the number value \(b\) is simply a formal symbol associated to the sequence \(a_1,a_2,\dots,a_n,\dots .\) This is important to note, as Cantor did not wish to fall into the error of assuming the existence of limits of sequences of real numbers. He simply associates a symbol to any fundamental sequence. Next Cantor defines a total ordering on the set \(B.\) Let \( b, b^{\prime}\) be number values with corresponding fundamental sequences \(a_1,a_2,\dots,a_n,\dots \) and \({a_1^{\prime}},{a^{\prime}_2},\dots,{a^{\prime}_n}\dots ,\) respectively. Then one of the following three relations must hold:

1. For every \(\epsilon > 0,\) there exists a positive integer \(N\) such that for every \(n\ge N, \) we have \({a_n}-{a^{\prime}_n} < \epsilon.\)
2. There is a rational \(\epsilon > 0,\) and a positive integer \(N\) such that for every \(n\ge N, \) we have \({a_n}-{a^{\prime}_n} > \epsilon.\)
3. There is a rational \(\epsilon > 0,\) and a positive integer \(N\) such that for every \(n\ge N, \) we have \({a_n}-{a^{\prime}_n} < -\epsilon.\)

In the first case, Cantor defines \( b= b^{\prime},\) in the second \( b> b^{\prime}\) and in the third \( b< b^{\prime}.\) It is important to again stress that Cantor is defining formal symbol relations. The symbol \(=\) is an equivalence relation on \(B\) (although Cantor does not use this terminology), and we continue to write \(B\) for the set of equivalence classes of number values under the relation \(=.\) Technically speaking, then, an element \(b\in B\) is an equivalence class of fundamental sequences.

Now Cantor is ready to define the operations of addition, subtraction, multiplication and division in \(B.\) For conceptual purposes, we may think of this as defining said operations on all real numbers, but technically speaking we are only formally symbol-pushing. Let the number values \( b, b^{\prime}, b^{\prime\prime}\) correspond to the fundamental sequences \[a_1,a_2,\dots, a_n,\dots\] \[{a^{\prime}_1},{a^{\prime}_2},\dots,{a^{\prime}_n},\dots\] \[{a^{\prime\prime}_1},{a^{\prime\prime}_2},\dots,{a^{\prime\prime}_n},\dots,\] respectively. If \[\lim\left(a_n\pm{a^{\prime}_n}-{a^{\prime\prime}_n}\right)=0,\] \[\lim\left(a_n\cdot{a^{\prime}_n}-{a^{\prime\prime}_n}\right)=0,\] \[\lim\left(\frac{a_n}{{a^{\prime}_n}}-{a^{\prime\prime}_n}\right)=0,\,\,\,{\rm for}\,\,\,{a^{\prime}_n} \not=0,\] then we write \( b\pm b^{\prime}=b^{\prime\prime},\) \( b\,b^{\prime}= b^{\prime\prime},\) and \({\frac{b}{b^{\prime}}} = b^{\prime\prime},\) respectively.

After defining operations on \(B,\) Cantor constructs the set \(C\) from \(B\) in much the same way he defined \(B\) from the rationals. That is, he considers all sequences \(b_1,b_2,\dots,b_n,\dots \) of \(B\) such that the limit of \(b_{n+m} -b_n\) equals \(0\) for some fixed value of \(m.\) We associate a symbol \(c\) to such a sequence and define relations, including \(=, <, >, \pm, \cdot,\) and \(÷,\) among such \(c\) as we did in \(B.\) Continuing in this manner, Cantor is able to construct the sets \(D,E,\dots\) consisting of equivalence classes of fundamental sequences made up of members of the previous set. Cantor then remarks that,

It is reserved for me to come back to all these conditions on another occasion in more detail,

a statement that, at least with hindsight, makes his construction here a precursor to his famous theory of transfinite numbers.