Strong Axioms of Infinity and the search for V

The axioms ZFC do not provide a concise conception of the Universe of Sets. This claim has been well documented in the 50 years since Paul Cohen established that the problem of the Continuum Hypothesis cannot be solved on the basis of these axioms. Gödel's Axiom of Constructibility, V = L, provides a conception of the Universe of Sets which is perfectly concise modulo only large cardinal axioms which are strong axioms of infinity. However the axiom V = L limits the large cardinal axioms which can hold and so the axiom is false. The Inner Model Program which seeks generalizations which are compatible with large cardinal axioms has been extremely successful, but incremental, and therefore by its very nature unable to yield an ultimate enlargement of L. The situation has now changed dramatically and there is, for the first time, a genuine prospect for the construction of an ultimate enlargement of L.