Can you explain Edward Nelson's "Internal set theory" (IST)?

Question

I've already read the Wikipedia post, but I'm looking for more perspectives on it. What does it really mean in plain language? Do you understand it well enough to talk about it in intuitive/philosophical terms?

Answer 1

Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. As a mathematical theory, Set Theory possesses a rich internal structure, and its methods serve as a powerful tool for applications in many other fields of Mathematics. Set Theory, with its emphasis on consistency and independence proofs, provides a gauge for measuring the consistency strength of various mathematical statements. There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe. They are: inner models, independence proofs, large cardinals, and descriptive set theory.