ABSTRACT
One of the most interesting activities of abstract mathematics is finding new mathematical objects. This involves describing the object and convincing others that the object does exist, often by giving a construction. Set theory is very careful about asserting the existence of set objects: some of the axioms assert the existence of sets, and others assert the existence of certain sets on the basis of constructions involving known sets; and the development of the subject includes showing that other, more sophisticated, constructions of sets are legitimate. The axiom of choice, which can be expressed in several provably equivalent ways, likewise asserts the existence of a set – usually in the form of a function. One of the reasons why the axiom is of interest is that for some people it sounds plausible that the existence of this function should already be deducible from the other standard axioms.
