This brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of (...) ways, including topologically, type-theoretically, and through variation. Many previous results from topos theory involving realizability, permutation, and sheaf models of set theory are subsumed, and the prospects for further such unification seem bright. (shrink)

This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed ‘niceness’ by the mathematician Michiel Hazewinkel, a concept we compare to the ‘robustness’ described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but (...) rather whether the choice of our concepts is forced upon us. (shrink)