SET THEORY – A HISTORICAL VIGNETTE
(Trails, Trials and Hopes of Georg Cantor)
by: alna_lou“The time will come when these
things which are now hidden from you will be
brought into the light.”
--1 Corinthians 4:5 Set theory is a very interesting branch of Mathematics. The creation of this field is somewhat controversial. Early references of the concept of infinity can be traced back as early as 400 BC. However, its modern understanding only arrived during early 19th century. This historical vignette is more focused on the trails, trials and hopes that Cantor faced on his creation of set theory.
Around 450 BC, Zeno of Elea posed paradoxes which have have puzzled, challenged, influenced, inspired, infuriated, and amused philosophers, mathematicians, physicists and school children for over two millennia. Some of the paradoxes he posed are trivial and has connection with the concept of infinity. Since then, mathematicians are struggling with the concept of it.
Now, let’s fast forward to 19th century.
The first notable work on the concept of infinity is by Bernard (Bernhard) Placidus Johann Nepomuk Bolzano, a
Bohemian mathematician,
theologian,
philosopher,
logician and antimilitarist of
German mother tongue. His posthumously published work
Paradoxien des Unendlichen (The Paradoxes of the Infinite) was greatly admired by many of the eminent
logicians who came after him, including
Charles Peirce,
Georg Cantor, and
Richard Dedekind. However, the modern understanding on infinity began with the publication of Cantor’s work: “On a Property of the Collection of All Real Algebraic Numbers."
Cantor claimed that there are sets having
cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}. Thus, there exists an infinite set (which he identifies with the set of
real numbers), which has a larger number of elements, or as he puts it, has a greater 'power' , than the infinite set of
finite whole numbers {1, 2, 3, ...}.
Although this work has found near-universal acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers.
Poincare condemned Cantor’s theory as a “disease” from which he was certain mathematics would someday be cured, because of its counter-intuitive nature. Leopold Kronecker one of Cantor's teachers and among the most prominent members of the German mathematics establishment, even attacked Cantor personally, calling him a “scientific charlatan,” a “renegade” and a “corrupter of youth.” Kronecker also claimed: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." Many mathematicians agreed with
Kronecker that the completed infinite may be part of
philosophy or
theology, but that it has no proper place in mathematics.
The bad feelings between Cantor and Kronecker revolved in large part around the two men’s differing conceptions of the infinite. Kronecker had a finitist Pythagorean view of Mathematics and pronounced, “God made the integers and all the rest is the work of man.” Cantor, on the other hand, dealt with all sorts of transcendental collections and constructions. Kronecker’s assault on the brilliant but hypersensitive Cantor may have been a factor in the latter’s breakdown and ultimate commitment to a mental institution.
The illness may, in fact, have supported his belief that the transfinite numbers had been communicated to him from God. In fact, as he noted in the third motto to his last publication, the Beitrage of 1895: The time will come when these things which are now hidden from you will be brought into the light. This is a familiar passage from the Bible, first Corinthians, and reflects Cantor's belief that he was an intermediary serving as the means of revelation. It also may have served to reflect Cantor's faith that despite any prevailing resistance to his work, it would one day enjoy recognition and praise from mathematicians everywhere.
Cantor may have been right after all since his concepts such as
one-to-one correspondence among sets, his proof that there are more
real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") the
power set operation gives rise to, eventually led to the widespread acceptance of Cantorian set theory.
Around 1900, Rusell and Zermelo independently found paradox on Cantorian definition of set. However, it has been reported that Cantor had already come upon his own version of the paradoxes of set theory in the form of contradictions he associated with the idea of a largest ordinal or cardinal number. This was all explained in letters first to Hilbert in 1897, and then to Dedekind in 1899.
This suggests that Cantor may well have been aware of the paradoxes of set theory much earlier, perhaps as early as the 1880's, when his difficulties with Kronecker were weighing on his mind and he was just beginning to experience his first serious technical problems with set theory.
In 1908, two ways of avoiding the paradox were proposed, Russell's
type theory and
Ernst Zermelo's
axiomatic set theory, the first consciously constructed
axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of
extensionality and unlimited
set abstraction, and evolved into the now-canonical
Zermelo–Fraenkel set theory (ZFC).
Elsewhere, Cantor actually described his conviction about the truth of his theory explicitly in quasi-religious terms: My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined it from all sides for many years; because I have examined all objections that have ever been made against the infinite numbers, and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
Later generations might dismiss the philosophy, look askance at his abundant references to St. Thomas or to the Church Fathers, overlook the metaphysical pronouncements and miss entirely the deeply religious roots of Cantor's later faith in the absolute truth of his theory. But all these commitments contributed to Cantor's resolve not to abandon the transfinite numbers. Opposition seems to have strengthened his determination. His forbearance, as much as anything else he might have contributed, ensured that set theory would survive the early years of doubt and denunciation to flourish eventually as a vigorous, revolutionary force in 20th-century mathematics.
(This is my first write-up in LOGIC and SET THEORY - it's an interesting subject folks.)
References and Suggested Readings:
Paulos, John A. Beyond Numeracy, Ruminations of a Numbers Man. Alfred A. Knopf, Inc., New York, 1991.
GEORG CANTOR AND THE BATTLE FOR TRANSFINITE SET THEORY retrieved from http://www.acmsonline.org/Dauben-Cantor.pdf.
http://science.jrank.org/pages/6083/Set-Theory.html
Atom Feed (xml)