Basser Seminar Series
On the Class of True Sentences of First Order Number Theory
Dr Stephen Boyce
Health Information Management, The University of Sydney
Wednesday 11 April 2007, 4-5 pm
School of IT Building, Lecture Theatre 123, Level 1
Abstract
This seminar will present a recursive definition of the class of Gödel numbers of the true sentences of first order number theory, where: (i) we view first order number theory as any system equivalent to Mendelson’s theory S; (ii) the class of recursively defined number theoretic functions (and hence relations) is defined as per Gödel; (iii) the Gödel number of an S sentence is a number associated with this sentence via a mapping similar to Gödel’s mapping of system P into the domain of natural numbers; (iv) the true sentences of S are taken to be the S sentences that are true under the standard interpretation of S, where (v) an interpretation is assigned to S using the method set out in Tarski. By well known arguments the result implies that system S is subject to paradox. The result is a surprise from a classical perspective and from some non-classical perspectives as well. Thus assumption (iv) is basically innocuous. In light of results first established by Gödel, it appears likely that for each member of a wide class of formal systems the class of Gödel numbers of the true sentences of the relevant system may also be recursively defined. That is, the result may extend to a number of formal systems routinely used in computer science and hence assumption (i) is also innocuous.
Speaker's biography
Stephen trained in medicine and worked as a doctor prior to working in health administration. He developed an interest in health information management and health accounting systems in particular, which he has pursued through completion of a Master of Professional Accounting at the Australian National University (ANU) and study towards a PhD in Health Information Management at the University of Sydney. Stephen’s PhD looks at the use of formal methods in the production of health classifications and some foundational issues relating to this enterprise.