Self-verifying theories
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Self-verifying theories are consistent
Gödel's incompleteness theorem, these systems cannot contain the theory of Peano arithmetic nor its weak fragment Robinson arithmetic
; nonetheless, they can contain strong theorems.
In outline, the key to Willard's construction of his system is to formalise enough of the
Gödel machinery to talk about provability internally without being able to formalise diagonalisation
. Diagonalisation depends upon being able to prove that multiplication is a total function (and in the earlier versions of the result, addition also). Addition and multiplication are not function symbols of Willard's language; instead, subtraction and division are, with the addition and multiplication predicates being defined in terms of these. Here, one cannot prove the sentence expressing totality of multiplication:
where is the three-place predicate which stands for
When the operations are expressed in this way, provability of a given sentence can be encoded as an arithmetic sentence describing termination of an relative consistency
argument with respect to ordinary arithmetic.
One can further add any true sentence of arithmetic to the theory while still retaining consistency of the theory.
References
- Solovay, Robert M. (9 October 1989). "Injecting Inconsistencies into Models of PA". Annals of Pure and Applied Logic. 44 (1–2): 101–132. .
- Willard, Dan E. (Jun 2001). "Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles". The Journal of Symbolic Logic. 66 (2): 536–596. S2CID 2822314.
- Willard, Dan E. (Mar 2002). "How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem almost to Robinson's Arithmetic Q". The Journal of Symbolic Logic. 67 (1): 465–496. S2CID 8311827.
External links
- Dan Willard's home page. Archived 2018-08-18 at the Wayback Machine