10,242 research outputs found
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
A Note on the Incompleteness of G\"odel's Incompleteness Theorems
In this note we observe that automated theorem provers (ATPs) that
recursively enumerate theorems in a particular way will fail to identify some
valid theorems for a reason that is analogous to how G\"odel proved the
existence of what are now referred to as G\"odel statements. This observation
has no significant practical or theoretical implications, but it may be of
pedagogical value for honing the intuition of students about recursive
enumeration in the context of ATP.Comment: Converted to a pedagogical focu
Gödel\u27s Incompleteness Theorems
The Incompleteness Theorems of Kurt Godel are very famous both within and outside of mathematics. They focus on independence and consistency within mathematics and hence a more thorough understanding of these is beneficial to their study. The proofs of the theorems involve many ideas which may be unfamiliar to many, including those of formal systems, Godel numbering, and recursive functions and relations. The arguments themselves mirror the Liar’s Paradox in that Godel constructs a statement asserting its own unprovability and then shows that such a statement and its negation must both be independent of the system, otherwise the system is inconsistent. We then proceed to survey various interpretations of the Incompleteness Theorems, focusing on potential misapplications of the theorems
Incompleteness and Computability: An Open Introduction to Gödel's Theorems
Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus
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
On the philosophical relevance of Gödel's incompleteness theorems
A survey of more philosophical applications of Gödel's incompleteness results
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