Does $\mathcal P(ω) / \mathrm{fin}$ know its right hand from its left?
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918284537888768 |
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| author | Brian, Will |
| author_facet | Brian, Will |
| contents | Let $σ$ denote the shift automorphism on $\mathcal{P}(ω) / \mathrm{fin}$, defined by setting $σ([A]) = [A+1]$ for all $A \subseteq ω$. We show that the Continuum Hypothesis implies the shift automorphism $σ$ and its inverse $σ^{-1}$ are conjugate in the automorphism group of $\mathcal{P}(ω) / \mathrm{fin}$.
Due to work of van Douwen and Shelah, it has been known since the 1980's that it is consistent with $\mathsf{ZFC}$ that $σ$ and $σ^{-1}$ are not conjugate. Our result shows that the question of whether $σ$ and $σ^{-1}$ are conjugate is independent of $\mathsf{ZFC}$.
As a corollary to the main theorem, we deduce that the structures $\langle \mathcal{P}(ω) / \mathrm{fin},σ\rangle$ and $\langle \mathcal{P}(ω) / \mathrm{fin},σ^{-1} \rangle$ are elementarily equivalent in the language of algebraic dynamical systems (Boolean algebras together with an automorphism). This corollary does not depend on the Continuum Hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Does $\mathcal P(ω) / \mathrm{fin}$ know its right hand from its left? Brian, Will Logic Dynamical Systems General Topology Let $σ$ denote the shift automorphism on $\mathcal{P}(ω) / \mathrm{fin}$, defined by setting $σ([A]) = [A+1]$ for all $A \subseteq ω$. We show that the Continuum Hypothesis implies the shift automorphism $σ$ and its inverse $σ^{-1}$ are conjugate in the automorphism group of $\mathcal{P}(ω) / \mathrm{fin}$. Due to work of van Douwen and Shelah, it has been known since the 1980's that it is consistent with $\mathsf{ZFC}$ that $σ$ and $σ^{-1}$ are not conjugate. Our result shows that the question of whether $σ$ and $σ^{-1}$ are conjugate is independent of $\mathsf{ZFC}$. As a corollary to the main theorem, we deduce that the structures $\langle \mathcal{P}(ω) / \mathrm{fin},σ\rangle$ and $\langle \mathcal{P}(ω) / \mathrm{fin},σ^{-1} \rangle$ are elementarily equivalent in the language of algebraic dynamical systems (Boolean algebras together with an automorphism). This corollary does not depend on the Continuum Hypothesis. |
| title | Does $\mathcal P(ω) / \mathrm{fin}$ know its right hand from its left? |
| topic | Logic Dynamical Systems General Topology |
| url | https://arxiv.org/abs/2402.04358 |