MATHEMATICS AS A QUASI-EMPIRICAL SCIENCE

The present paper aims at showing that there are times when set theoretical knowledge increases in a non-cumulative way. In other words, what we call 'set theory' is not 'one' theory which grows by simple addition of a theorem after the other, but a finite sequence of theories T(1),..., T(n) in which T(i+1) supersedes T(i). This thesis has great philosophical significance because it implies that there is a sense in which mathematical theories, like the theories belonging to the empirical sciences, are fallible and that, consequently, mathematical knowledge has a quasi-empirical nature. The way I have chosen to provide evidence in favour of the correctness of the main thesis of this article consists in arguing that Cantor-Zermelo set theory is a Lakatosian Mathematical Research Programme (MRP).