# AN ESCAPE FROM VARDANYAN’S THEOREM

@article{DEALMEIDABORGES2022ANEF, title={AN ESCAPE FROM VARDANYAN’S THEOREM}, author={Ana DE ALMEIDA BORGES and Joost J. Joosten}, journal={The Journal of Symbolic Logic}, year={2022} }

Vardanyan’s Theorems [36, 37] state that QPLpPAq – the quantified provability logic of Peano Arithmetic – is Π02 complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The…

## One Citation

### Towards a Coq formalization of a quantified modal logic

- Computer ScienceArXiv
- 2022

A Coq formalization of the Quantiﬁed Reﬂection Calculus with one modality, or QRC 1, focuses on the design decisions inherent to the formalization and the insights that led to new and simplifying proofs.

## References

SHOWING 1-10 OF 49 REFERENCES

### Calibrating Provability Logic: From Modal Logic to Reflection Calculus

- Computer ScienceAdvances in Modal Logic
- 2012

This talk outlines a consistency proof for Peano arithmetic based on RC and state a simple combinatorial statement, the so-called Worm principle, that was suggested by the use of GLP but is even more directly related to the Reflection Calculus.

### On predicate provability logics and binumerations of fragments of Peano arithmetic

- Computer ScienceArch. Math. Log.
- 2013

Comparing predicate provability logics of I∑n’s is compared and it is proved that for any natural numbers i, j, there exists a ∑1 binumeration α(x) of some recursive axiomatization of I ∑i such that 0 < i < j.

### Finite Kripke models and predicate logics of provability

- PhilosophyJournal of Symbolic Logic
- 1990

Abstract The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite…

### Quantified Reflection Calculus with One Modality

- Philosophy, MathematicsAiML
- 2020

The first steps towards arithmetical completeness are taken by providing relational semantics for QRC$_1$ with a corresponding completeness proof and showing the finite model property, which implies decidability.

### Provability logic and the completeness principle

- Mathematics, PhilosophyAnn. Pure Appl. Log.
- 2019

### Positive provability logic for uniform reflection principles

- Computer ScienceAnn. Pure Appl. Log.
- 2014

### The degree of the set of sentences of predicate provability logic that are true under every interpretation

- Philosophy, MathematicsThe Journal of Symbolic Logic
- 1987

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability”…

### Reflection algebras and conservation results for theories of iterated truth

- MathematicsAnn. Pure Appl. Log.
- 2022

### A Note on the Model Theory for Positive Modal Logic

- MathematicsFundam. Informaticae
- 2012

The notion of positive bisimulation between two models is defined, the notions of m-saturated models and replete models are studied, and two definability theorems by positive modal sequents for classes of pointed models are presented.

### No Escape from Vardanyan's theorem

- MathematicsArch. Math. Log.
- 2006

Vardanyan's theorem states that the set of PA-valid principles of Quantified Modal Logic, QML, is complete Π02. We generalize this result to a wide class of theories. The crucial step in the…