I know that there are different subfields of logic, but I am talking about the kind Kurt Gödel studied, that is,symbolic and mathematical logic. Why is he considered a logician, mathematician and philosopher (according to Wiki) and not only a mathematician ? I don't understand how the philosophical questions raised in such enquiries are not mathematical in nature, why it is separate. Why is George Cantor not considered a philosopher and logician(according to Wiki), but only a mathematician ?
The reason is quite simple.
The impact of Cantor's work on math and phil of math was very high, but he published only mathematical papers, without explicit contribution to the (current in his age) discussion about the philosopical issues of mathematics (due mainly to kantian school).
Godel published (in addition to fundamental mathematical papers) a couple of papers with interesting contribution to the (current) discussion about the foundations of mathematics :
- Russell's mathematical logic (1944)
- What is Cantor's continuum problem? (1947)
My guess is that there's no rationale behind this. Why Kurt Gödel can be called a philosopher is because many of his works deal with problems that are identical to those within the philosophy of mathematics. It doesn't seem to me that calling Cantor a philosopher would be bad either, but I also think it's unimportant.
Adding to what someone mentioned about Gödel writing papers, Gödel collaborated with philosophers in his later life. For example, with Hillary Putnam and Hao Wang.
Cantor can certainly be called a philosopher, whether or not Wikipedia agrees. Cantor published in philosophical journals (Zeitschrift für Philosophie und philosophische Kritik) philosophical and theological questions. Zermelo's edition of Cantor's collected works (Springer, 1932) has the title "Collected works of mathematical and philosophical contents".
Cantor himself mentioned various philosophical roots of his work: Ebenso finde ich für meine Auffassung Berührungspunkte in der Philosophie des Nicolaus Cusanus. Man vgl. R. Zimmermann, Der Cardinal Nicolaus von Cusa als Vorgänger Leibnizens (Sitzungsberichte d. Wiener Akademie d. Wiss. Jahrg. 1852). Dasselbe bemerke ich in Beziehung auf Giordano Bruno, den Nachfolger des Cusaners. Man vgl. Brunnhofer, Giordano Brunos Weltanschauung und Verhängnis. Leipzig 1882.
He had correspondense with many, many philosophers and theologians, see for instance
Georg Cantor: Leben, Werk und Wirkung, Herbert Meschkowski, 2. Aufl. BI, Mannheim (1981)
Georg Cantor 1845 - 1918, W. Purkert, H.J. Ilgauds, Birkhäuser, Basel (1987)
Georg Cantor Briefe, H. Meschkowski, W. Nilson (Herausgeber), Springer, Berlin (1991).
Further he speculated about the real father of Jesus Christ and the real author of Shakespear's works. Of course these topics are not sufficient to make him a great philosopher. But his set theory is enough to make him great - and in fact it is pure philosophy as far as he is concerned: The belief in the possiblility to finish infinite sets, based on the holy bible and the teachings of S. Augustin and other fathers of the church.
"The general set theory [...] definitely belongs to metaphysics. [...] and the fact that my presently written work is issued in mathematical journals does not modify the metaphysical contents and character of this work. [...] By me Christian philosophy is for the first time confronted with the true teachings of the infinite in its beginnings." [G. Cantor, letter to T. Esser, February 1896]