Abstract
This paper belongs to the research on the limit of the first incompleteness theorem. Effectively inseparable () theories can be viewed as an effective version of essentially undecidable () theories, and is stronger than . We examine this question: Are there minimal effectively inseparable theories with respect to interpretability? We propose , the theory version of . We first prove that there are no minimal theories with respect to interpretability (i.e., for any theory T, we can effectively find a theory which is and strictly weaker than T with respect to interpretability). By a theorem due to Pour-EI, we have that is equivalent with . Thus, there are no minimal theories with respect to interpretability. Also, we prove that there are no minimal finitely axiomatizable theories with respect to interpretability.
Citation
Yong Cheng. "There Are No Minimal Effectively Inseparable Theories." Notre Dame J. Formal Logic 64 (4) 425 - 439, November 2023. https://doi.org/10.1215/00294527-2023-0017
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