118 research outputs found
Incompleteness of boundedly axiomatizable theories
Our main result (Theorem A) shows the incompleteness of any consistent
sequential theory T formulated in a finite language such that T is axiomatized
by a collection of sentences of bounded quantifier-alternation-depth. Our proof
employs an appropriate reduction mechanism to rule out the possibility of
completeness by simply invoking Tarski's Undefinability of Truth theorem. We
also use the proof strategy of Theorem A to obtain other incompleteness results
(as in Theorems A+; B and B+).Comment: 6 pages; in this version reference to MathOverflow work of Emil
Je\v{r}\'{a}bek has been added, and the author list of [AGLRZ] is now
up-to-dat
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
On the relationship between meta-mathematical properties of theories
In this work, we understand incompleteness in an abstract way via
metamathematical properties of formal theories. We systematically examine the
relationship between the following twelve important metamathematical properties
of theories: Rosser, EI (Effectively inseparable), RI (Recursively
inseparable), TP (Turing persistent), EHU (essentially hereditarily
undecidable), EU (essentially undecidable), Creative,
(theories with Turing degree ), REW (all RE sets are
weakly representable), RFD (all recursive functions are definable), RSS (all
recursive sets are strongly representable), RSW (all recursive sets are weakly
representable). Given any two properties and of these properties, we
examine whether the property implies the property , and whether the
property implies the property .Comment: 23 page
How to Say Things with Formalisms
Recent attention to "self-consistent" (Rosser-style) systems raises anew the question of the proper interpretation of the Gödel Second Incompleteness Theorem and its effect on Hilbert's Program. The traditional rendering and consequence is defended with new arguments justifying the intensional correctness of the derivability conditions
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
Max Dehn, Axel Thue, and the Undecidable
This is a short essay on the roles of Max Dehn and Axel Thue in the
formulation of the word problem for (semi)groups, and the story of the proofs
showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve
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