Self-verifying axiom systems, the incompleteness theorem and related reflection principles
@article{Willard2001SelfverifyingAS,
title={Self-verifying axiom systems, the incompleteness theorem and related reflection principles},
author={Dan E. Willard},
journal={Journal of Symbolic Logic},
year={2001},
volume={66},
pages={536 - 596}
}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…
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References
SHOWING 1-10 OF 98 REFERENCES
Self-Verifying Axiom Systems
- MathematicsKurt Gödel Colloquium
- 1993
We introduce a class of First Order axiom systems which can simultaneously verify their own consistency and prove more Π1 theorems than Peano Arithmetic. Despite these strengths, our axiom systems do…
The Tangibility Reflection Principle for Self-Verifying Axiom Systems
- PhilosophyKurt Gödel Colloquium
- 1997
This work extends some axiom systems that could verify their own consistency and prove all Peano Arithmetic's II1 theorems by introducing some new axiom system which use stronger definitions of self-consistency.
First-Order Logic and Automated Theorem Proving
- Computer ScienceGraduate Texts in Computer Science
- 1996
This monograph on classical logic presents fundamental concepts and results in a rigorous mathematical style and is intended for those interested in computer science and mathematics at the beginning graduate level.
On a generalized logic calculus
- Mathematics
- 1953
In this paper we shall give a generalization of the logic calculus, called LK by Gentzen in his dissertation (1), and prove some metatheorems in th generalized logic calculus, which will be denoted…
Godel's Second Incompleteness Theorem for Q
- Mathematics, PhilosophyJ. Symb. Log.
- 1976
For the first Godel incompleteness theorem, the existence in a formal system of arithmetic L of a sentence which is neither provable nor refutable, all that is required of the formula Th( x ) of L…
Solution of a Problem of Leon Henkin
- Mathematics, PhilosophyJ. Symb. Log.
- 1955
This note presents a solution of the previous problem with respect to the system Z μ, and, more generally, to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function ( k, l ) used below is definable.
The Semantic Tableaux Version of the Second Incompleteness Theorem Extends Almost to Robinson's Arithmetic Q
- MathematicsTABLEAUX
- 2000
The Second Incompleteness Theorem is generalized almost to the level of Robinson’s System Q, such that if α is any finite consistent extension of Q +V then α will be unable to prove its Semantic Tableaux consistency.
Bounded Arithmetic と計算量の根本問題
- Mathematics
- 1996
59.4 T h e C u t Elimination Theorem for Second Order Logic 171 (respectively, v . ) . In particular, (a) guarantees t h a t *"-PIND or c ~ ' ~ I N D , and C ~ ' ~ C R inferences are still valid…