Must we believe in set theory? (Q2754660)

The author expresses skepticism about the confidence we should have in Zermelo-Fraenkel set theory (ZFC). To make his case clear, he questions the existence of the cardinal \(\kappa\) that is the least \(\lambda\) such that \(\aleph_\lambda =\lambda\), that is, the least ordinal greater than all \(f(i)\), where \(f(0)= \aleph_0\) and \(f(i+1)= \aleph_{f(i)}\) for all natural numbers \(i\). He finds it hard to believe that there is such a large set. He does not claim that he knows that there is no such set, but only that he is agnostic about it. Of course, it is a theorem of ZFC that \(\kappa\) exists, but the author does not feel forced to believe the axioms of ZFC (in spite of what Gödel claimed): ``the burden of proof should be, I think, on one who would adopt a theory so removed from experience and the requirements of the rest of science (including the rest of mathematics)''. His skepticism is not based on constructivist or finitist principles. On the contrary, he asserts his belief in the existence of denumerable sets and he gives a very strong argument for belief in the existence of abstract objects, not just in mathematics but also in everyday life.NEWLINENEWLINENEWLINEThis paper was written near the end of the author's life and, although of a very light nature, shows his erudition, wisdom, and acuteness of mind. He is aleady sorely missed.NEWLINENEWLINENEWLINEThis same paper appears in the author's book: Logic, logic and logic, Cambridge, MA: Harvard Univ. Press (1998; Zbl 0955.03008), pp. 120-132.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00010].