Tightness and solidity in fragments of Peano Arithmetic
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909953174798336 |
|---|---|
| author | Gruza, Piotr Kołodziejczyk, Leszek Aleksander Łełyk, Mateusz |
| author_facet | Gruza, Piotr Kołodziejczyk, Leszek Aleksander Łełyk, Mateusz |
| contents | It was shown by Visser that Peano Arithmetic has the property that any two bi-interpretable extensions of it (in the same language) are equivalent. Enayat proposed to refer to this property of a theory as tightness and to carry out a more systematic study of tightness and its stronger variants that he called neatness and solidity. Enayat proved that not only $\mathrm{PA}$, but also $\mathrm{ZF}$ and $\mathrm{Z}_2$ are solid. On the other hand, it was shown in later work by a number of authors that many natural proper fragments of those theories are not even tight. Enayat asked whether there is a proper solid subtheory of the theories listed above. We answer that question in the case of $\mathrm{PA}$ by proving that for every $n$, there exist both a solid theory and a tight but not neat theory strictly between $\mathrm{I}Σ_{n}$ and $\mathrm{PA}$. Moreover, the solid subtheories of $\mathrm{PA}$ can be required to be unable to interpret $\mathrm{PA}$. We also obtain some other separations between properties related to tightness, for example by giving an example of a sequential theory that is neat but not semantically tight in the sense of Freire and Hamkins. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tightness and solidity in fragments of Peano Arithmetic Gruza, Piotr Kołodziejczyk, Leszek Aleksander Łełyk, Mateusz Logic 03H15, 03F30, 03F25 It was shown by Visser that Peano Arithmetic has the property that any two bi-interpretable extensions of it (in the same language) are equivalent. Enayat proposed to refer to this property of a theory as tightness and to carry out a more systematic study of tightness and its stronger variants that he called neatness and solidity. Enayat proved that not only $\mathrm{PA}$, but also $\mathrm{ZF}$ and $\mathrm{Z}_2$ are solid. On the other hand, it was shown in later work by a number of authors that many natural proper fragments of those theories are not even tight. Enayat asked whether there is a proper solid subtheory of the theories listed above. We answer that question in the case of $\mathrm{PA}$ by proving that for every $n$, there exist both a solid theory and a tight but not neat theory strictly between $\mathrm{I}Σ_{n}$ and $\mathrm{PA}$. Moreover, the solid subtheories of $\mathrm{PA}$ can be required to be unable to interpret $\mathrm{PA}$. We also obtain some other separations between properties related to tightness, for example by giving an example of a sequential theory that is neat but not semantically tight in the sense of Freire and Hamkins. |
| title | Tightness and solidity in fragments of Peano Arithmetic |
| topic | Logic 03H15, 03F30, 03F25 |
| url | https://arxiv.org/abs/2512.09120 |