Self-Verifying Axiom Systems, The Incompleteness Theorem and Related Reflection Principles

We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the “Tangibility Reflection Principle”. We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further. NOTE TO THE READER: This article was published in the June 2001 of the Journal of Symbolic Logic, pp. 536-596. Subsequent to this paper, I published extensions of this article in the year 2002, 2005 and 2006 issues of the JSL, as well as in the year 2006 and 2007 issues of APAL and in an year-2009 article in Information and Computation. The text in this pdf file is identical to my JSL 2001 article, except that I have used a larger type faunt for the reader’s convenience. ∗Address: Dep of CS, SUNYA, Albany, NY 12222 or [email protected]. Tel. 518-452-0148. Supported by NSF Grant CCR 99-02726