On the relationships between some meta-mathematical properties of arithmetical theories
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909816983650304 |
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| author | Cheng, Yong |
| author_facet | Cheng, Yong |
| contents | In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (Effectively inseparable), RI (Recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $\mathbf{0}^{\prime}$ (theories with Turing degree $\mathbf{0}^{\prime}$), 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 $P$ and $Q$ of these properties, we examine whether the property $P$ implies $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15285 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the relationships between some meta-mathematical properties of arithmetical theories Cheng, Yong Logic 03F40, 03F30, 03D35 In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (Effectively inseparable), RI (Recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $\mathbf{0}^{\prime}$ (theories with Turing degree $\mathbf{0}^{\prime}$), 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 $P$ and $Q$ of these properties, we examine whether the property $P$ implies $Q$. |
| title | On the relationships between some meta-mathematical properties of arithmetical theories |
| topic | Logic 03F40, 03F30, 03D35 |
| url | https://arxiv.org/abs/2303.15285 |