Abstract

A formal system of arithmetic in quantified modal logic is defined and investigated. The system is consistent with the philosophic attitudes of nominalism and finitism. The system is obtained from a typical Peano-style axiomatization of arithmetic in first-order logic by the application of a translation function mapping first-order formulas into modal formulas. The axioms obtained via this translation are shown to be true when the primitive terms of the language of arithmetic are interpreted as predicates of physical objects. It is further shown that classical logic is preserved under this translation, and, in particular, that excluded middle holds. ;The consequences of this seeming proof that finitism and classical logic are compatible are explored. Hilbert's philosophy of mathematics is criticized. Under an interpretation of the primitive terms which makes them predicates of mental objects or mental constructions, the modal axioms of the system prove true; this fact is used in a critique of intuitionism. ;The necessity of the truth of arithmetic truths is formally provable in the system; just what necessary truth means is analyzed in terms of how it is proved, and the conclusion is reached that any system of arithmetic which accomplishes philosophic aims similar to those of the system developed in this dissertation--the justification of finitism and nominalism and the preservation of classical logic--must presuppose, in its foundations, the doctrine of essentialism. ;The translation can be applied to first-order theories other than arithmetic. Some general results about its application are proved, and a sketch of the requirements of the modal development of set theory and analysis is offered