The next levels are given by finding an equivalent formula in Prenex normal form, and counting the number of changes of quantifiers:
12.
As a formula might have several different equivalent formulas in Prenex normal form, it might belong to several different levels of the hierarchy.
13.
Because every formula is equivalent to a formula in prenex normal form, every formula with no set quantifiers is assigned at least one classification.
14.
Together, these restrictions give " predicative prenex polymorphism ", which is essentially the form of polymorphism found in ML and early versions of Haskell.
15.
Tarski's system has the unusual property that all sentences can be written in universal-existential form, a special case of the prenex normal form.
16.
Two popular restrictions of this form are restricted rank polymorphism ( for example, rank-1 or " prenex " polymorphism ) and predicative polymorphism.
17.
It is the set of sentences that, when written in prenex normal form, have an \ exists ^ * \ forall ^ * quantifier prefix and do not contain any function symbols.
18.
Because every formula has a prenex normal form, every formula in the language of second-order arithmetic is \ Sigma ^ 1 _ n or \ Pi ^ 1 _ n for some n.
19.
The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by Herbrandization.
20.
The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by Herbrandization.
