超越Tenenbaum定理和一个强跳跃倒置定理
Escaping Tennenbaum's Theorem and a Strong Jump Inversion Theorem
AI总结：
本文通过构造一系列理论，证明了介于PA强度之间的片段存在可计算的非标准模型，拓展了Tenenbaum定理的脆弱性，并提出一个通用的强跳跃倒置定理。
摘要：
Tenenbaum定理指出，PA不承认任何非标准可计算模型。2022年，Pakhomov证明了该定理在PA如何表达方面是脆弱的，通过构造一个与PA定义上等价的理论（大致："它是PA但选择了不同的签名"）对于其存在可计算的非标准模型。他展示了这种脆弱性不扩展到真正的算术（任何非标准模型的理论定义上等价于Th(ℕ)的都不是可计算的），但关于这种脆弱性是否扩展到介于PA强度之间的PA片段的问题仍未解决。我们证明了这一点，通过构造一系列理论T^n，它们与"PA加上所有Π^0_n真理"定义上等价，所有这些理论都允许可计算的非标准模型。在过程中，我们产生了一个通用的定理用于强跳跃倒置。除了将该定理应用于获得我们的新结果外，我们还展示了文献中几个已知结果可以被视为该定理的直接应用。
英文摘要：
Tennenbaum's theorem states that PA does not admit any nonstandard computable model. In 2022, Pakhomov proved that this theorem is fragile in regards to how PA is expressed, by constructing a theory that is definitionally equivalent to PA (roughly: "it's PA but with a different choice of signature") for which there is a computable nonstandard model. He showed that this fragility does not extend to true arithmetic (any nonstandard model of a theory definitionally equivalent to $\mathrm{Th}(\mathbb{N})$ is not computable), but the question of whether this fragility extends to fragments of PA of intermediate strength was left open. We show that it does, by constructing a sequence of theories $T^n$ which are definitionally equivalent to: "PA plus all $Π^0_n$ truths", all of which admit computable nonstandard models. In the process, we produce a general-purpose theorem for strong jump inversion. Besides applying this theorem to obtain our novel result, we show that several known results from the literature can be seen as direct applications of our theorem.
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。
