A Special Π^0_1 Class with the Join Property but without Pseudojump Inversion

Abstract

Jananthan and Simpson showed that special Π⁰₁ classes exist in three of the four possible cases determined by whether the Join Property and the Pseudojump Inversion Property hold, leaving open the existence of a special Π⁰₁ class with the Join Property but not the Pseudojump Inversion Property. We construct such a class, thereby answering their question affirmatively and completing the four-way existence pattern. Moreover, the class can be chosen to consist entirely of generalized low reals, with the Join Property holding locally on every nonempty relatively basic open subclass. More generally, every special Π⁰₁ class contained in GL₁ has a special Π⁰₁ extension, still contained in GL₁, on which the Join Property holds locally. We also show that no class contained in GL₁ has the Pseudojump Inversion Property. An independent appendix proves that, for any class with the Join Property, every jump-inversion fibre above 0' is countably infinite.

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BibTeX RIS

BibTeX

@misc{cintioli2026special,
  title = {{A Special $Π^0_1$ Class with the Join Property but without Pseudojump Inversion}},
  author = {Patrizio Cintioli},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37480},
  eprint = {2609.37480},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37480},
}

RIS

TY  - UNPB
TI  - A Special Π^0_1 Class with the Join Property but without Pseudojump Inversion
AU  - Patrizio Cintioli
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.37480
UR  - https://arxiv.org/abs/2609.37480
AB  - Jananthan and Simpson showed that special Π^0_1 classes exist in three of the four possible cases determined by whether the Join Property and the Pseudojump Inversion Property hold, leaving open the existence of a special Π^0_1 class with the Join Property but not the Pseudojump Inversion Property. We construct such a class, thereby answering their question affirmatively and completing the four-way existence pattern. Moreover, the class can be chosen to consist entirely of generalized low reals, with the Join Property holding locally on every nonempty relatively basic open subclass. More generally, every special Π^0_1 class contained in GL_1 has a special Π^0_1 extension, still contained in GL_1, on which the Join Property holds locally. We also show that no class contained in GL_1 has the Pseudojump Inversion Property. An independent appendix proves that, for any class with the Join Property, every jump-inversion fibre above 0' is countably infinite.
ER  -

A Special Π⁰₁ Class with the Join Property but without Pseudojump Inversion

Abstract

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