凝聚幂的坦南鲍姆型定理
Tennenbaum-like theorems for cohesive powers
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中文总结 AI 辅助
该研究针对凝聚幂构造，构造了特定图与线性序，分别得到凝聚幂具有度0''、无计算表示的结果，为凝聚幂的编码能力提供了坦南鲍姆型定理相关结论。
中文摘要 AI 辅助
我们研究凝聚幂构造的编码能力。我们构造了一个图$\boldsymbol{\textit{G}}$，使得由任意$\boldsymbol{\textit{Δ}_2}$凝聚集$C$生成的$\boldsymbol{\textit{G}}$的凝聚幂$\boldsymbol{\textit{∏}_C \boldsymbol{\textit{G}}}$具有度$\boldsymbol{0''}$，即$\boldsymbol{0''}$可计算$\boldsymbol{\textit{∏}_C \boldsymbol{\textit{G}}}$的一个表示，且$\boldsymbol{\textit{∏}_C \boldsymbol{\textit{G}}}$的每个表示都可计算$\boldsymbol{0''}$。我们还构造了一个线性序$\boldsymbol{\textit{L}}$，使得$\boldsymbol{\textit{L}}$的任何凝聚幂都没有可计算表示，实现方式是：若$\boldsymbol{\textit{P}}$是$\boldsymbol{\textit{L}}$的凝聚幂的一个表示，则$\boldsymbol{\textit{P}''}$相对于$\boldsymbol{0''}$具有$\boldsymbol{\text{PA}}$-度。
英文摘要
We investigate the encoding ability of the cohesive power construction. We compute a graph $\mathcal{G}$ where the cohesive power $\prod_C \mathcal{G}$ of $\mathcal{G}$ by any $Δ_2$ cohesive set $C$ has degree $0''$. That is, $0''$ computes a presentation of $\prod_C \mathcal{G}$, and every presentation of $\prod_C \mathcal{G}$ computes $0''$. We also compute a linear order $\mathcal{L}$ where no cohesive power of $\mathcal{L}$ has a computable presentation. We accomplish this by ensuring that if $\mathcal{P}$ is a presentation of a cohesive power of $\mathcal{L}$, then $\mathcal{P}''$ has $\mathrm{PA}$-degree relative to $0''$.
