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Free Algebras for Gödel-Löb Provability Logic

2014
*
Advances in Modal Logic
*

We give a construction of finitely generated

dblp:conf/aiml/Gool14
fatcat:7uyznivutbcnfe74cybyx5u4uu
*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. ...##
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Incompleteness in a General Setting

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*[8] 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) ...

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

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

arXiv:1602.05728v1
fatcat:tkoejgtcx5gwrcnhd6jvewjc5e
*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. ...##
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Franco Montagna's Work on Provability Logic and Many-valued Logic

2016
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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:

doi:10.1007/s11225-016-9654-3
fatcat:7dhk6qh25fdvfg2q3mydf4qs6m
*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. ...##
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Modal Logic for Induction

2020
*
Advances in Modal Logic
*

axiom of

dblp:conf/aiml/FellinNS20
fatcat:wasffz56b5fxtdhzm7kmnarayy
*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. ...##
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Inconsistency Robustness in Foundations: Mathematics self proves its own Consistency and Other Matters
[article]

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. ...

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Topological interpretations of provability logic
[article]

2012
*
arXiv
*
pre-print

The topic has become a long-term project

arXiv:1210.7317v1
fatcat:3x4kqzfg7jdyheruunf3tpai5q
*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 ...##
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Proof Theory in Philosophy of Mathematics

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 ...

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

2013
*
Journal of Logic and Computation
*

Downloaded from 2 Cut-

doi:10.1093/logcom/ext036
fatcat:wxy26qn65reubasipfkzy2lmq4
*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. ...##
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Explicit Provability and Constructive Semantics

2001
*
Bulletin of Symbolic Logic
*

In 1933

doi:10.2307/2687821
fatcat:twz2ui2ckzabzp6b2f64nlhm6i
*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*( [27] , [29] , [98] , [100] ) with the "nonreflexive"*Löb*principle 2(2F → F ) → 2F . ...##
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CURRENT RESEARCH ON GÖDEL'S INCOMPLETENESS THEOREMS

2021
*
Bulletin of Symbolic Logic
*

*Gödel*comments in a footnote of [46] 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*[46] 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. ...

##
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Reflection principles and provability algebras in formal arithmetic

2005
*
Russian Mathematical Surveys
*

We also consider basic properties of the

doi:10.1070/rm2005v060n02abeh000823
fatcat:vlcfh7twnbfcvmvceieyrlzzem
*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) . ...##
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Current research on Gödel's incompleteness theorems
[article]

2020
*
arXiv
*
pre-print

Thus, G2 may hold

arXiv:2009.04887v2
fatcat:ws3hn7d7rjgbzmayaw2sfqkf6i
*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. ...##
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SELF-REFERENCE IN ARITHMETIC I

2014
*
The Review of Symbolic Logic
*

A

doi:10.1017/s1755020314000288
fatcat:ipmcdt7lgfgmle4rbfd767sqna
*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. ...##
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The Reduction Property Revisited
[article]

2019
*
arXiv
*
pre-print

We answer the question also in a purely

arXiv:1903.03331v1
fatcat:a2wyzjm7gzhetepspokhuxwxlm
*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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