$\mathcal P(ω) / \mathrm{fin}$ 是否能区分左右手？
Does $\mathcal P(ω) / \mathrm{fin}$ know its right hand from its left?
AI总结：
本文研究了$\mathcal P(ω) / \mathrm{fin}$上位移自同构σ与σ⁻¹的共轭性，证明其在连续假设下共轭，并显示其在ZFC下独立。
摘要：
设σ是$\mathcal{P}(ω) / \mathrm{fin}$上的位移自同构，定义为对所有$A \subseteq ω$，有σ([A]) = [A+1]。我们证明连续假设蕴含位移自同构σ及其逆σ⁻¹在$\mathcal{P}(ω) / \mathrm{fin}$的自同构群中是共轭的。由于van Douwen和Shelah的工作，自1980年代以来已知在ZFC下σ和σ⁻¹可能不共轭。我们的结果表明σ和σ⁻¹是否共轭的问题独立于ZFC。作为主定理的推论，我们得出结构⟨$\mathcal{P}(ω) / \mathrm{fin}$,σ⟩和⟨$\mathcal{P}(ω) / \mathrm{fin}$,σ⁻¹⟩在代数动力系统语言（布尔代数及其自同构）下是元素等价的。此推论不依赖于连续假设。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
