The Actual Infinite: A Leibnizian Perspective on Cantor's Paradise

Abstract

<p>This thesis is first and foremost an investigation of the actual infinite. It draws on the work of Richard T. W. Arthur in defense of G. W. Leibniz's view that the infinite, while actual, should be understood syncategorematically. The actual infinite has now, due to the work of Georg Cantor (along with Bernard Bolzano and Richard Dedekind), found a permanent home within the foundations of mathematics. This was made possible by the stipulation that the part-whole axiom does not apply to infinite collections in the way it applies to finite ones: an actual infinite set is defined as a collection that can be placed in a one-to-one correspondence with a proper subset of itself. In my view, however, something more than a stipulation is required to guarantee the coherence of an infinite set. It has not been sufficiently demonstrated that an actual infinite multiplicity can be one and whole, fixed and definite-that is, can be categorematic-but this is being assumed. Satisfactory justification is required, I believe, if the actual infinite is to play such a fundamental role in the discipline of mathematics. Georg Cantor does attempt to provide such justification in the form of three philosophical arguments, which I have called the argument from irrationals, the divine intellect argument, and the domain argument. His arguments, however, rely on an equation of the terms "actual" and "potential" with the terms "categorematic" and "syncategorematic" respectively. But based on the work of G. W. Leibniz, such an equation is faulty. It is completely legitimate to maintain that the infinite is both actual and syncategorematic, a possibility not considered by Cantor. Once such a position is on the table-that is, once it is no longer necessary that the actual implies the categorematic-Cantor's arguments are no longer sound and the actual (and categorematic) infinite stands in need of further justification.</p>