Journal of Symbolic Logic

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

Dan E. Willard

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Abstract

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.

Article information

Source
J. Symbolic Logic Volume 66, Issue 2 (2001), 536-596.

Dates
First available: 6 July 2007

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1183746459

JSTOR
links.jstor.org

Mathematical Reviews number (MathSciNet)
MR1833464

Zentralblatt MATH identifier
0991.03053

Citation

Willard, Dan E. Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles. Journal of Symbolic Logic 66 (2001), no. 2, 536--596. http://projecteuclid.org/euclid.jsl/1183746459.


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