We introduce two schemes of quantifiers analogous to the Härtig quantifier I and the Equal Cofinality quantifier Q e.c. , which tell us about regular cardinals of small Cantor-Bendixson rank. We examine how the Löwenheim-Skolem-Tarski numbers of these quantifiers interact with one another, and with those of I and Q e.c. . We then find the exact lower bound for each of the LST numbers, assuming the consistency of supercompacts.

We study principles of the form: if a name σ is forced to have a certain property φ, then there is a ground model filter g such that σg satisfies φ. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are: Any forcing axiom can be expressed as a name principle. For instance, PFA is equivalent to: A principle for rank 1 names (equivalently, nice names) for subsets of ω1. A principle for rank 2 names for sets of reals. λ-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of BFA via generic absoluteness is a corollary. We further systematically study name principles where φ is a notion of largeness for subsets of ω1 (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.