# 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…

## 28 Citations

On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency

- Computer ScienceJournal of Symbolic Logic
- 2006

It is shown that axiomatizations for a computer's floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics.

On the Nature of Godel's Second Incompleteness Theorem

- Philosophy, Mathematics
- 2006

Godel’s “Second” Incompleteness Theorem states that axiom systems of sufficiently great strength are unable to formally verify their own consistency. Let A(x, y, z) denote a 3-way predicate relation…

Passive induction and a solution to a Paris-Wilkie open question

- MathematicsAnn. Pure Appl. Log.
- 2007

The Axiom System ISigma0 Manages to Simultaneously Obey and Evade the Herbrandized Version of the Second Incompleteness Theorem

- EconomicsElectron. Notes Theor. Comput. Sci.
- 2006

About the characterization of a fine line that separates generalizations and boundary-case exceptions for the Second Incompleteness Theorem under semantic tableau deduction

- MathematicsJ. Log. Comput.
- 2021

Our previous research showed that the semantic tableau deductive methodology of Fitting and Smullyan permits boundary-case exceptions to the second incompleteness theorem, if multiplication is…

On the Partial Respects in Which a Real Valued Arithmetic System Can Verify Its Tableaux Consistency

- MathematicsTABLEAUX
- 2005

It is shown Godel's Second Incompleteness Theorem does not preclude axiomizations for a computer’s floating point arithmetic from recognizing their own consistency in certain well defined partial respects.

On the Tender Line Separating Generalizations and Boundary-Case Exceptions for the Second Incompleteness Theorem Under Semantic Tableaux Deduction

- MathematicsLFCS
- 2020

This work shows that if one promotes this schema of theorems into formalized logical axioms, then the meaning of the pronoun “I” in the self-referencing engine changes, and the partial evasions of the Second Incompleteness Theorem come to a complete halt.

A weak set theory that proves its own consistency

- Mathematics, Computer Science
- 2019

This example avoids the strong version of Godel second incompleteness theorem (due to Pudlak) that asserts that no consistent a theory interpreting Robinson's arithmetic $\mathsf{Q}$ proves its own consistency.

The Consistency of Arithmetic

- PhilosophyThe Mathematical Intelligencer
- 2018

In 2010, Vladimir Voevodsky gave a lecture on "What If Current Foundations of Mathematics Are Inconsistent?" Among other things he said that he was seriously suspicious that an inconsistency in PA…

Some New Exceptions for the Semantic Tableaux Version of the Second Incompleteness Theorem

- MathematicsTABLEAUX
- 2002

This article continues our study of axiom systems that can verify their own consistency and prove all Peano Arithmetic's ?1 theorems. We will develop some new types of exceptions for the Semantic…

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