548 19. LOGIC AND SET THEORYBeyond that point, everything was human-made and therefore had to be finite. If
you spoke of a number or function, you had an obligation to say how it was defined.
His 1845 dissertation, which he was unable to polish to his satisfaction until 1881,
when he published it as "Foundations of an arithmetical theory of algebraic quan-
tities" in honor of his teacher Kummer, shows how careful he was in his definitions.
Instead of an arbitrary field defined axiomatical ly as we would now do, he wrote:A domain of rationality is in general an arbitrarily bounded domain
of magnitudes, but only to the extent that the concept permits.
To be specific, since a domain of rationality can be enlarged only
by the adjoining of arbitrarily chosen elements 9t, each arbitrary
extension of its boundary requires the simultaneous inclusion of all
quantities rationally expressible in terms of the new Element.In this way, while one could enlarge a field to make an equation solvable,
the individual elements of the larger field could still be described constructively.
Kronecker's concept of a general field can be described as "finitistic." It is the
minimal object that contains the necessary elements. Borel took this point of
view in regard to measurable sets, and Hubert was later to take a similar point
of view in describing formal languages, saying that a meaningful formula must be
obtained from a specified list of elements by a finite number of applications of
certain rules of combination. This approach was safer and more explicit than, for
example, Bernoulli's original definition of a function as an expression formed "in
some manner" from variables and constants. The "manner" was limited in a very
definite way.
Cantor believed that Kronecker had delayed the publication of his first paper
on infinite cardinal numbers. Whether that is the case or not, it is clear that
Kronecker would not have approved of some of his principles of inference. As
Grattan-Guinness points out, much of what is believed about the animosity between
Cantor and Kronecker is based on Cantor's own reports, which may be unreliable.
Cantor was subject to periodic bouts of depression, probably caused by metabolic
imbalances having nothing to do with his external circumstances. In fact, he had
little to complain of in terms of the acceptance of his theories. It is true that there
was some resistance to it, notably from Kronecker (until his death in 1891) and
then from Poincare. But there was also a great deal of support, from Weierstrass,
Klein, Hilbert, and many others. In fact, as early as 1892, the journal Bibliotheca
mathematica published a "Notice historique" on set theory by Giulio Vivanti (1859-
1949), who noted that there had already been several expositions of the theory,
and that it was still being developed by mathematicians, applied to the theory of
functions of a real variable, and studied from a philosophical point of view.
2.4. Existence and the axiom of choice. In the early days Cantor's set theory
seemed to allow a remarkable amount of freedom in the "construction" or, rather,
the calling into existence, of new sets. Cantor seems to have been influenced in his
introduction of the term set by an essay that Dedekind began in 1872, but did not
publish until 1887 (see Grattan-Guinness, 2000, p. 104), in which he referred to a
"system" as "various things o, b, c... comprehended from any cause under one point
of view." Dedekind defined a "thing" to be "any object of our thought." Just as
Descartes was able to conceive many things clearly and distinctly, mathematicians
seemed to be able to form many "things" into "systems." For example, given any