Can We Make the Second Incompleteness Theorem Coordinate Free?
نویسنده
چکیده
Is it possible to give a coordinate free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EA-provable equivalence, (ii) consistency for finitely axiomatized sequential theories can be uniquely characterized modulo EA-provable equivalence. The case of infinitely axiomatized ce theories is more delicate. We carefully discuss this in the paper.
منابع مشابه
On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency
Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem a...
متن کاملSome specially formulated axiomizations for ISigma0 manage to evade the Herbrandized version of the Second Incompleteness Theorem
In 1981, Paris and Wilkie [28] indicated it was an open question whether IΣ0 would satisfy the Second Incompleteness Theorem for Herbrand deduction. We will show that some specially formulated axiomizations for IΣ0 can evade the Herbrandized version of the Second Incompleteness Theorem.
متن کاملThe Surprise Examination Paradox and the Second Incompleteness Theorem
Few theorems in the history of mathematics have inspired mathematicians and philosophers as much as Gödel’s incompleteness theorems. The first incompleteness theorem states that, for any rich enough consistent mathematical theory, there exists a statement that cannot be proved or disproved within the theory. The second incompleteness theorem states that for any rich enough consistent mathematic...
متن کاملA generalization of the Second Incompleteness Theorem and some exceptions to it
This paper will introduce the notion of a naming convention and use this paradigm to both develop a new version of the Second Incompleteness Theorem and to describe when an axiom system can partially evade the Second Incompleteness Theorem.
متن کاملTheorems of Tarski's Undefinability and Godel's Second Incompleteness - Computationally
We show that the existence of a finitely axiomatized theory which can prove all the true Σ1 sentences may imply Gödel’s Second Incompleteness Theorem, by incorporating some bi-theoretic version of the derivability conditions (first discussed by Detlefsen 2001). We also argue that Tarski’s theorem on the undefinability of truth is Gödel’s first incompleteness theorem relativized to definable ora...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Log. Comput.
دوره 21 شماره
صفحات -
تاریخ انتشار 2011