Full Idea
Modern model theory investigates which set theoretic structures are models for which collections of sentences.
Gist of Idea
Model theory studies how set theory can model sets of sentences
Source
William D. Hart (The Evolution of Logic [2010], 4)
Book Reference
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.108
A Reaction
So first you must choose your set theory (see Idea 13497). Then you presumably look at how to formalise sentences, and then look at the really tricky ones, many of which will involve various degrees of infinity.
Related Idea
Idea 13497 Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD]