A hierarchy of languages with infinitely long expressions

This chapter discusses the hierarchy of languages with infinitely long expressions. Many authors have defined notions of formulas with constructive infinitely long expressions and established some relations between such formulas and the predicates belonging to Kleene's analytic hierarchy. The chapter explores a hierarchy of subsystems of Lω₁ ω₁ by extending the restriction recursive in the formation rules of formulas with constructive infinitely long expressions to definable in the previous system and studies ordinals definable in the hierarchy. The chapter discusses the main theorem, which states that every formula with rank has a Godel number and that there are only Godel numbers, the hierarchy cannot go all the way up to Ω; however, it does not seem to stop very easily. The chapter explains that any ordinal α, definable in terms of Kleene's function quantifiers, is definable at level 0.