g-rh@cca.CCA.COM (Richard Harter) (07/06/87)
In my last article I said the universe of logic and Mathematics was subset-countable. Let me begin with the obvious. The distinct symbols that we have available to us is finite in number. The strings that can be built up from these symbols are finite in length (but unbounded) and can be arranged lexicographically; they are Cantor countable. Regardless of which rules we use to determine which strings are statements, the list of legal statements is subset-countable (but not necessarily cantor-countable.) Among the statements there will be those that assert the existence of formal entities. These definition statements are also subset-countable. Let us call such definition statements which can actually be written out, specifiable definition statements. There are two questions that immediately come to mind. These are the decision question and the ontological question. The decision question is whether and how one can determine which statements, if any, define something. The onotological question is whether the specifiable defining statements are sufficient. The general possibilities are: (a) We can demonstrate that there must be formal objects which cannot be explicitly specified. (b) We can demonstrate that there cannot be formal objects which do not have an explicit definition. (c) Neither (a) nor (b) can be demonstrated. Case (c) splits into two sub cases: (c1) We may assume the existence of formal objects which cannot be explicitly specified. The consequences of this assumption differ from those entailed by not making the assumption. (c2) We may assume the existence of formal objects which cannot be explicitly specified. However this assumption has no observable consequences. The conventional exegenesis of Set theory holds that (a) is the case -- that the Ontology of Mathematics is essentially larger than the explicitly definable. The view I wish to take is that case (c2) holds. To be continued. -- Richard Harter, SMDS Inc. [Disclaimers not permitted by company policy.] [I set company policy.]