Conceptions of Set and the Foundations of Mathematics

I took my first set theory course, as a philosophy student, at the Department of Mathematics of the University of Rome La Sapienza. It was after taking that course, taught by Professor Claudio Bernardi, that I decided to dedicate more time and energy to the philosophy of set theory if I could. The course was entitled ‘Foundations of Mathematics’, and had one striking feature: it was taught from the ZFC axioms, with almost no mention of naïve set theory and the set-theoretic paradoxes. When I went up to Cambridge in 2005, I began to read some of the standard textbooks on set theory, such as Devlin 1993 and Jech 2003. Naïve set theory was now introduced and the paradoxes were given their due. But after this, ZFC was presented, and one was left with the impression that all roads from naïve set theory lead to the cumulative hierarchy. This book is an attempt to reverse this trend. Perhaps, in the end, we should stick with the iterative conception of set. Indeed, one can read the book as an extended argument to this effect. But the path from the paradoxes to ZFC and cognate systems is much more tortuous than tradition has made it out to be. The book describes and assesses a number of conceptions of set. Being a book on the philosophy of set theory, the focus is very much on the philosophical underpinnings of these conceptions. But history is important, in philosophy as in life, and I have provided some historical background when I deemed it useful–