Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 GL and other systems of propositional modal logic
- 2 Peano arithmetic
- 3 The box as Bew(x)
- 4 Semantics for GL and other modal logics
- 5 Completeness and decidability of GL and K, K4, T, B, S4, and S5
- 6 Canonical models
- 7 On GL
- 8 The fixed point theorem
- 9 The arithmetical completeness theorems for GL and GLS
- 10 Trees for GL
- 11 An incomplete system of modal logic
- 12 An S4-preserving proof-theoretical treatment of modality
- 13 Modal logic within set theory
- 14 Modal logic within analysis
- 15 The joint provability logic of consistency and ω-consistency
- 16 On GLB: The fixed point theorem, letterless sentences, and analysis
- 17 Quantified provability logic
- 18 Quantified provability logic with one one-place predicate letter
- Notes
- Bibliography
- Index
- Notation and symbols
15 - The joint provability logic of consistency and ω-consistency
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- Contents
- Preface
- Introduction
- 1 GL and other systems of propositional modal logic
- 2 Peano arithmetic
- 3 The box as Bew(x)
- 4 Semantics for GL and other modal logics
- 5 Completeness and decidability of GL and K, K4, T, B, S4, and S5
- 6 Canonical models
- 7 On GL
- 8 The fixed point theorem
- 9 The arithmetical completeness theorems for GL and GLS
- 10 Trees for GL
- 11 An incomplete system of modal logic
- 12 An S4-preserving proof-theoretical treatment of modality
- 13 Modal logic within set theory
- 14 Modal logic within analysis
- 15 The joint provability logic of consistency and ω-consistency
- 16 On GLB: The fixed point theorem, letterless sentences, and analysis
- 17 Quantified provability logic
- 18 Quantified provability logic with one one-place predicate letter
- Notes
- Bibliography
- Index
- Notation and symbols
Summary
Introduction
We recall from Chapter 3 the definition of the ω-inconsistency of a theory T (whose language contains 0 and s): T is ω-inconsistent iff for some formula A(x), T⊢∃xA(x), and for every natural number n, T⊢¬A(n). T is ω-consistent iff it is not ω-inconsistent. If T is ω-consistent, then T⊬∃xx ≠ x, and therefore T is consistent.
It is easy to show, however, that the converse does not hold: Let T be the theory that results when Bew(┌⊥┐) is added to PA. Since PA does not prove ¬Bew(┌⊥┐), T is consistent and for every n, the Δ sentence ¬Pf(n, ┌⊥┐) is true. Thus for every n, PA⊢¬Pf(n, ┌⊥┐), and so for every n, T⊢¬Pf(n, ┌⊥┐) (T extends PA). But T⊢Bew(┌⊥┐), that is, T⊢∃yPf(y, ┌⊥┐). So, despite its consistency, T is ω-inconsistent.
As a sentence S is said to be inconsistent with T if the theory whose axioms are those of T together with S itself is inconsistent, so S is ω-inconsistent (with T) if the theory whose axioms are those of T together with S is ω-inconsistent. S is ω-consistent iff not ω-inconsistent.
We call a sentence S ω-provable in T iff ¬S is ω-inconsistent with T. So if S is provable in T, S is ω-provable in T.
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- Information
- The Logic of Provability , pp. 187 - 207Publisher: Cambridge University PressPrint publication year: 1994