Well-orderings and finite quantifiers
Esteban López‐Escobar · Journal of the Mathematical Society of Japan · 1968
\S 0. Introduction.That the class $W$ of all (non-empty) well-orderings cannot be characterized using (finite) first-order sentences is a well known result.Almost as well known is that $W$ can be characterized by an infinitely long sentence involving conjunctions of countably many formulas and quantifications over countable sequences of individual variables (cf.[8]).In [4] and [5] it is shown that in order to characterize $W$ in an infinitary first-order language quantifications over infinitely many individual variables are essential.The aim of this paper is to determine how much can we express, concerning well- orderings, in infinitary languages whose only non-logical constant is a binary relation symbol and which allow the conjunction/disjunction of infinitely many formulas but whose quantifiers bind single individual variables.The results in this note are obtained by an elimination of quantifiers, that is we determine a certain class of sentences, which for lack of a better name we shall call " sentences in normal form " or simply " normal sentences ", such that any other sentence is equivalent (as far as well-orderings are concerned) to a dis- junction of normal sentences.The method of carrying out the elimination of quantifiers is essentially an extension of the combination of the methods used by Ehrenfeucht [2] and Mostowski/Tarski [6] for the finite language.