Sometimes themes exist only in the eye of the beholder and emerge from the history of mathematics only with the benefit (or otherwise) of hindsight. There is however at least one issue (no doubt there are others) which was clearly present throughout the foundational debates of the early Twentieth century. It formed the core component of one of the major traditions that grew up as responses to the paradoxes, though the intuition behind it has never been fully articulated nor has it been possible to give a fully fledged mathematical articulation to that intuition. Essentially the central idea is that of ‘indefinite extensibility’ generated by an appropriate form of diagonalisation and it has its fundamental source in Richard's paradox and in particular in Poincaré's response to that paradox.

Of the four great epistemological concerns around which much of the work in the foundations of mathematics this century has concentrated viz the notions of demonstrability, definability, set or class and computability only the latter clearly falls outside the scope of the Richard type reasoning.