The Basement of Mathematics

The object of these pages is to give a common framework for discussing mathematics; it is not to specify a rigorous foundation (analogous to the Zermelo-Fraenkel axiomatisation of set theory, for example). My intent is that the concepts and denotations I introduce should be usable by contexts building on such a foundation and, for the most part, expressible in terms of any reasonable foundation. It is to be expected that any such foundation shall impose some restrictions on what entities may be entertained as values, which shall cause some of the entities I discuss to be inexpressible – indeed, not only expected but hoped, as the toolkit I develop here is eminently capable of expressing entities that, if accepted as values, would yield paradoxes. To help ensure at least enough survives that filter to leave us with some useful mathematics, I delineate a core of constructible entities which, I hope, all foundations should be able to allow as values.

My goal, in packaging the foundations, is to provide a general notation and language for the exposition, that captures the essence without the encumbrance of the pedantic details needed by any particular foundation.

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