What’s new in logic?

Abstract

One of the first delights of logic the average mathematics student encounters is the uncountability of the set of real numbers. There are more real numbers than natural numbers, more sets of real numbers than reals, more sets of sets of r e a l s . . , and so on. The student may or may not wonder how many different cardinalities there are and how large they can become. It is natural for him to give up wondering after a while: Either he decides the number of such cardinalities is infinite and, forgetting momentarily that he has just learned there to be different infinities, the student considers the matter settled, or he realises there to be a dazzling array of infinities and that he has no handle on the problem. Then again, he could go on to become a specialist and create a few handles. The genus of set theory, i.e., the number of the handles I spoke of, is itself almost a large cardinal. There are: weak and strong inaccessibles; Mahlo cardinals of various ranks; measurables; weakly and strongly compact cardinals; supercompact cardinals; indescribable and extendible cardinals; huge cardinals; and now two more due to Woodin. Despite the wide variety of such cardinal numbers and the variety of their names, experienced set theorists can linearly order them according to size and even announce which ones definitely exist, which ones probably exist, which ones might exist . . . . . As the cardinal gets bigger and bigger, the hypothesis that it exists (called a large cardinal hypothesis or an axiom of infinity) gets progressively more quest ionable as an addit ional axiom of set theory.