Changing the Order of Quantifiers
3.1.1 Löwenheim begins the proof of his theorem by showing that every formula is logically equivalent to a formula which has a certain normal form. A formula is in (Löwenheim) normal form if it is in prenex form and every existential quantifier precedes every universal quantifier.1 Obviously, the central step in obtaining the normal form of a formula involves moving the existential quantifiers in front of the universal quantifiers, preserving logical equivalence. Löwenheim takes this step by generalizing and applying a transformation introduced by Schröder in the third volume of Vorlesungen.
Schröder only considers the problem of changing the order of quantifiers for the ...