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### Free Algebras for Gödel-Löb Provability Logic

Samuel J. van Gool
2014 Advances in Modal Logic
We give a construction of finitely generated free algebras for Gödel-Löb provability logic, GL.  ...  Our two main techniques are incremental constructions of free algebras and finite duality for partial modal algebras.  ...  Introduction The provability logic GL is the axiomatic extension of the basic modal logic K by the Gödel-Löb axiom 2(2p → p) → 2p.  ...

### Incompleteness in a General Setting

John L. Bell
2007 Bulletin of Symbolic Logic
Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel's second  ...  One of the most important of these efforts was made by Löb  in connection with his analysis of sentences asserting their own provability.  ...  Therefore 0  a , so that, by (***), a = a = 0.  From (i) of Löb's Theorem we see that  satisfies the so-called GL (Gödel-Löb) axiom 13 for a normal modal logic, i.e. the scheme (A  A)    ...

### Some abstract versions of Gödel's second incompleteness theorem based on non-classical logics [article]

Lev Beklemishev, Daniyar Shamkanov
2016 arXiv   pre-print
We study abstract versions of G\"odel's second incompleteness theorem and formulate generalizations of L\"ob's derivability conditions that work for logics weaker than the classical one.  ...  We isolate the role of contraction rule in G\"odel's theorem and give a (toy) example of a system based on modal logic without contraction invalidating G\"odel's argument.  ...  This work is supported by the Russian Foundation for Basic Research, grant 15-01-09218a, and by the Presidential council for support of leading scientific schools.  ...

### Franco Montagna's Work on Provability Logic and Many-valued Logic

Lev Beklemishev, Tommaso Flaminio
2016 Studia Logica: An International Journal for Symbolic Logic
We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.  ...  Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015.  ...  We would like to thank the anonymous referees for careful reading and precious suggestions.  ...

### Modal Logic for Induction

Giulio Fellin, Sara Negri, Peter Schuster
2020 Advances in Modal Logic
axiom of provability logic.  ...  Among other things, we give a direct and elementary syntactical proof of Segerberg's theorem that the Gödel-Löb axiom characterises precisely the (converse) well-founded and transitive Kripke frames.  ...  , logic, combinatorics -REDCOM" belonging to the programme "Ricerca Scientifica di Eccellenza 2018" of the Fondazione Cariverona.  ...

### Inconsistency Robustness in Foundations: Mathematics self proves its own Consistency and Other Matters [article]

Carl Hewitt
2015 arXiv   pre-print
for foundations.  ...  This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis  ...  Provability Logic "The Stanford Encyclopedia of Philosophy" 2010. Albert Visser. Can We Make the Second Incompleteness Theorem Coordinate Free Journal on Logic and Computation. 21(4) 2011.  ...

### Topological interpretations of provability logic [article]

Lev Beklemishev, David Gabelaia
2012 arXiv   pre-print
The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere.  ...  Provability logic concerns the study of modality as provability in formal systems such as Peano arithmetic.  ...  Acknowledgement The first author was supported by the Russian Foundation for Basic Research (RFBR), Russian Presidential Council for Support of Leading Scientific Schools, and the Swiss-Russian cooperation  ...

### Proof Theory in Philosophy of Mathematics

Andrew Arana
2010 Philosophy Compass
logics.  ...  philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability  ...  In [Göd33a] Gödel gave what he took to be basic laws of provability (for instance, the law that if p is provable, then p), noting that these laws were simply the axioms of the modal logic known as S4  ...

### A cut-free sequent system for Grzegorczyk logic, with an application to the Gödel–McKinsey–Tarski embedding

Roy Dyckhoff, Sara Negri
2013 Journal of Logic and Computation
Downloaded from 2 Cut-free sequent system for Grzegorczyk logic of Int into what were later called the provability logics GL and Grz of Gödel-Löb and (resp.) of Grzegorczyk.  ...  It is well-known that intuitionistic propositional logic Int may be faithfully embedded not just into the modal logic S4 but also into the provability logics GL and Grz of Gödel-Löb and Grzegorczyk, and  ...  Int into the Gödel-Löb provability logic GL.  ...

### Explicit Provability and Constructive Semantics

Sergei N. Artemov
2001 Bulletin of Symbolic Logic
In 1933 Gödel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem.  ...  This also achieves Gödel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted  ...  The logic L of formal provability (  ,  ,  ,  ) with the "nonreflexive" Löb principle 2(2F → F ) → 2F .  ...

### CURRENT RESEARCH ON GÖDEL'S INCOMPLETENESS THEOREMS

YONG CHENG
2021 Bulletin of Symbolic Logic
Gödel comments in a footnote of  that G2 is corollary of G1 (and in fact a formalized version of G1): if T is consistent, then the consistency of T is not provable in T where the consistency of T is  ...  Gödel  proves his first incompleteness theorem (G1) for a certain formal system P related to Russell-Whitehead's Principia Mathematica based on the simple theory of types over the natural number series  ...  Thus, G2 may hold for a provability predicate which does not satisfy the Löb condition D3.  ...

### Reflection principles and provability algebras in formal arithmetic

L D Beklemishev
2005 Russian Mathematical Surveys
We also consider basic properties of the provability formula and give short proofs of theorems due to Gödel, Rosser and Löb.  ...  Graded provability algebras and Japaridze's logic Gödel's provability formula has arithmetical complexity Σ 1 .  ...  If T is an elementary presented Π n+1 -extension of EA + , then, provably in EA + , ∀α ∈ S n , T + α * T ≡ n T n on(α) .  ...

### Current research on Gödel's incompleteness theorems [article]

Yong Cheng
2020 arXiv   pre-print
Thus, G2 may hold for a provability predicate which does not satisfy the Löb condition D3.  ...  Provability logic is an important tool for the study of incompleteness and meta-mathematics of arithmetic. The origins of provability logic (e.g.  ...

### SELF-REFERENCE IN ARITHMETIC I

VOLKER HALBACH, ALBERT VISSER
2014 The Review of Symbolic Logic
A Gödel sentence is o en described as a sentence saying about itself that it is not provable and a Henkin sentence as a sentence stating its own provability.  ...  We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. e starting point will be the answer Kreisel gave to Henkin's problem.  ...  For Löb's theorem, see Löb ( ) or, e.g.  ...

### The Reduction Property Revisited [article]

Nika Pona, Joost J. Joosten
2019 arXiv   pre-print
We answer the question also in a purely algebraic setting where Π^0_j-conservation will be suitably defined. The algebraic turn allows for generalizations to transfinite consistency notions.  ...  Both theories essentially speak about the fundamental principle of reflection: if a sentence is provable then it is true.  ...  , write a provability predicate T for the theory T so that N |= T (x) ⇐⇒ x is the Gödel number of a formula that has a proof in T .  ...
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