The Surprise Examination Paradox and the Second Incompleteness Theorem

We give a new proof for Godel's second incompleteness theorem, based on Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that resembles the surprise examination paradox. We then go the other way around and suggest that the second incompleteness theorem gives a possible resolution of the surprise examination paradox. Roughly speaking, we argue that the flaw in the derivation of the paradox is that it contains a hidden assumption that one can prove the consistency of the mathematical theory in which the derivation is done; which is impossible by the second incompleteness theorem.Comment: 8 page

Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Godel's Incompleteness Phenomenon - Computationally

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of such theories are undecidable

A MACHINE-ASSISTED PROOF OF GÖDEL'S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS

A formalisation of G\"odel's incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows {\'S}wierczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalisation itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail.This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/S175502031400011

Is Complexity a Source of Incompleteness?

In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is invariant under the change of the G\"odel numbering. For this measure, the theorems of a finitely-specified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length $n$ is provable in the theory tends to zero when $n$ tends to infinity, while the probability that a sentence of length $n$ is true is strictly positive.Comment: 15 pages, improved versio

Reflection using the derivability conditions

Reflection principles are a way to build non-conservative true extensions of a theory. However the application of a reflection principle needs a proof predicate, and the effort needed to provide this is so great as to be not really practical. We look at a possible way to avoid this effort by using, instead of a proof predicate, a predicate defined using only necessary `modal' properties. Surprisingly, we can produce powerful non-conservative extensions this way. But a reflection principle based on such a predicate is essentially weaker, and we also consider its limitations

Current research on G\"odel's incompleteness theorems

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logi