LOGIC WITH DENUMERABLY LONG FORMULAS AND FINITE STRINGS OF QUANTIFIERS

Abstract

This chapter explains logic with denumerably long formulas and finite strings of quantifiers. In making an extension to an infinitary logic, Tarski originally suggested that infinite strings of quantifiers also be allowed. In some ways, in the denumerable case, this plan is not reasonable. The logic discussed in this paper is called Lω1 ω0, where the second subscript indicates that the strings of quantifiers prefixing any subformula of a formula must be finite in length. Lω1 ω0 is the ordinary finitary predicate logic. Among countable systems at least, the axioms (**) do characterize the well-orderings. Non-standard models of (**) must be uncountable. The downward LÖWENHEIM-SKOLEM theorem for Lω1 ω0 is that every countable set of sentences which has a model of infinite cardinality has models of all smaller infinite cardinalities. The proof is immediate when one realizes that the sub-formulas require Skolem functions for their initial quantifiers, which are each functions of only a finite number of arguments. The corresponding upward theorem is not so obvious.