This chapter presents several constructions of ω ₁ -like models with interesting second order properties. The list includes: recursively saturated rather classless models, rigid recursively saturated models, models whose additive reducts are isomorphic but whose multiplicative reducts are not, and very similar but nonisomorphic models. The chapter also presents two constructions used in completeness theorems for Peano Arithmetic with the Ramsey quantifier and the stationary quantifier.
Keywords: rather classless model, rigid model, additive reduct, multiplicative reduct, Ramsey quantifier, stationary logic