Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Journals
  3. Bulletin of Symbolic Logic
  4. Volume 26
Journal Home
Refine listing

Refine listing

Access:
Publication date:

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×
< Back to all issues

Volume 26 - December 2020

Contents

Page 2 of 3

Communication

  • FINDING THE LIMIT OF INCOMPLETENESS I

  • Part of:
  • YONG CHENG
  • Published online by Cambridge University Press:
    16 April 2021, pp. 268-286

  • In this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

Reviews

Meeting Report

In Memoriam

  • IN MEMORIAM: ANNE SJERP TROELSTRA 1939–2019

  • DICK DE JONGH, JOAN RAND MOSCHOVAKIS
  • Published online by Cambridge University Press:
    16 April 2021, pp. 293-295
    • Article
      • You have access
    • PDF
    • Export citation

Meeting Report

Notices

  • NOTICES

  • Published online by Cambridge University Press:
    28 December 2020, pp. 97-102
    • Article
      • You have access
    • PDF
    • Export citation

Thesis Abstracts

  • Monadic Intuitionistic and Modal Logics Admitting Provability Interpretations

  • Kristina Brantley
  • Published online by Cambridge University Press:
    16 April 2021, p. 296
    • Article
      • You have access
    • PDF
    • Export citation

  • Invariant Types in Model Theory

  • Rosario Mennuni
  • Published online by Cambridge University Press:
    16 April 2021, pp. 296-297
    • Article
      • You have access
    • PDF
    • Export citation

Front Cover (OFC, IFC) and matter

  • BSL volume 26 issue 1 Cover and Front matter

  • Published online by Cambridge University Press:
    28 December 2020, pp. f1-f3
    • Article
      • You have access
    • PDF
    • Export citation

Back Cover (OBC, IBC) and matter

  • BSL volume 26 issue 1 Cover and Back matter

  • Published online by Cambridge University Press:
    28 December 2020, pp. b1-b2
    • Article
      • You have access
    • PDF
    • Export citation

Thesis Abstracts

  • A Categorical Characterization of Accessible Domain

  • Patrick Walsh
  • Published online by Cambridge University Press:
    16 April 2021, pp. 298-299
    • Article
      • You have access
    • PDF
    • Export citation

  • Compact Metrizable Structures via Projective Fraïssé Theory With an Application to the Study of Fences

  • Gianluca Basso
  • Published online by Cambridge University Press:
    16 April 2021, pp. 299-300
    • Article
      • You have access
    • PDF
    • Export citation

  • Filling cages. Reverse mathematics and combinatorial principles

  • Marta Fiori Carones
  • Published online by Cambridge University Press:
    16 April 2021, p. 300
    • Article
      • You have access
    • PDF
    • Export citation

  • Measurable combinatorics and orbit equivalence relations

  • Tomasz Cieśla
  • Published online by Cambridge University Press:
    16 April 2021, pp. 300-301
    • Article
      • You have access
    • PDF
    • Export citation

Meeting Report

Page 2 of 3