Zero

The Formaterial thesis that Ordinal Numbers are more fundamental than Cardinals, entails that '0' is no more than a conventional device within the Cardinal System, with no substantive experiential correlate. For example, a '0' bank account balance does not refer to something subsisting in the account, but is only an abbreviated caution that one cannot make any further payments at the time. Now, there is no question that the fruitfulness and efficiency of 0 justifies its intra-Systemic status, but there are anomalies involving it even within the construction. For example, every well-formed Number in the System should have a unique value, but what is the value of 0/0? On the basis of 'n/n=1', 0/0=1; on the basis of '0/n=0', it=0; and, on the basis of 'n/0=Infinity', it=Infinity. Any choice would be arbitrary, so would only amount to a gloss over an incompleteness in the System. Still, the treatment of 0 by Frege and Russell suggest that they regard it as sufficiently well-grounded to serve as a foundation of the Number System. They define it as 'the Class of all empty Classes', and they posit it as the first, or perhaps 'zeroth' would be more appropriate, Number. But its presumed well-groundedness implies a derivation from some experiential element, and what a 'Class' might be, outside of an intellectual construction, is unclear. Furthermore, Russell's own assiduous analysis elsewhere, of phrases like 'the present King of France', would seem to commit him to the position that an 'empty Class' is no Class at all, a nonentity, predicating anything of which would constitute a False proposition. Hence, the Cardinal Number System that is generated on the basis of 0, is a construction, one that does not represent a refutation of the thesis of the priority of the Ordinals. Mathematicians need not be concerned about the existential significance of the Ordinal-Cardinal relation, but Philosophers who presume their Logic to be Universalistic, should be.