论文标题
$δ_2$粘合套件和无限$π_1$ sets的有效功率$δ_2$
Effective powers of $ω$ over $Δ_2$ cohesive sets and infinite $Π_1$ sets without $Δ_2$ cohesive subsets
论文作者
论文摘要
可计算结构的凝聚力是一种有效的超能，其中粘性集作为超滤器。令$ω$，$ζ$和$η$表示自然数，整数和理性的订单类型。我们研究了$δ_2$凝聚力集的$ω$的可计算副本的凝聚力。我们表明，有一个可计算的副本$ \ MATHCAL {l} $ $ω$，这样，对于每一个$Δ_2$凝聚套件$ c $，$ \ mathcal {l} $ over $ c $ over $ c $的内聚力都有订单式式$ type $ω+η$。这改善了Dimitrov，Harizanov，Morozov，Shafer，A。Soskova和Vatev的较早结果，并通过将$σ_1$凝聚的套装推广到$δ_2$粘性集，并通过计算$δ_2$的$ω$的$δ$，而单个副本具有比所有$δ_2$ COHESIVE套装的所需的凝聚力。此外，我们的结果是最佳的，因为$Δ_2$不能被$π_2$取代。 More generally, we show that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $Σ_2$ sets, thought of as a set of finite order-types, then there is a computable copy $\mathcal{L}$ of $ω$ where the cohesive power of $\mathcal{L}$ over any $Δ_2$内聚力集具有订单型$ω+σ（x \ cup \ {ω+ζη+ω^*\}）$。如果$ x $是有限的和非空的，那么在任何$ \ mathcal {l} $的凝聚力上，在任何$Δ_2$凝聚组上的凝聚力均具有订单型$ω+σ（x）$。我们工作的意外副产品是一种构造没有$Δ_2$内聚力子集的无限$π_1$集的新方法。实际上，我们构建了一个没有$δ_2$ p-cohesive子集的无限$π_1$。无限$π_1$ sets nover $δ_2$ r-cohesive子集概括了D. Martin的经典共同限制C.E.设置的没有最大的超集，并且出现在Lerman，Shore和Soare的工作中。
A cohesive power of a computable structure is an effective ultrapower where a cohesive set acts as an ultrafilter. Let $ω$, $ζ$, and $η$ denote the respective order-types of the natural numbers, the integers, and the rationals. We study cohesive powers of computable copies of $ω$ over $Δ_2$ cohesive sets. We show that there is a computable copy $\mathcal{L}$ of $ω$ such that, for every $Δ_2$ cohesive set $C$, the cohesive power of $\mathcal{L}$ over $C$ has order-type $ω+ η$. This improves an earlier result of Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, and Vatev by generalizing from $Σ_1$ cohesive sets to $Δ_2$ cohesive sets and by computing a single copy of $ω$ that has the desired cohesive power over all $Δ_2$ cohesive sets. Furthermore, our result is optimal in the sense that $Δ_2$ cannot be replaced by $Π_2$. More generally, we show that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $Σ_2$ sets, thought of as a set of finite order-types, then there is a computable copy $\mathcal{L}$ of $ω$ where the cohesive power of $\mathcal{L}$ over any $Δ_2$ cohesive set has order-type $ω+ σ(X \cup \{ω+ ζη+ ω^*\})$. If $X$ is finite and non-empty, then there is also a computable copy $\mathcal{L}$ of $ω$ where the cohesive power of $\mathcal{L}$ over any $Δ_2$ cohesive set has order-type $ω+ σ(X)$. An unexpected byproduct of our work is a new method for constructing infinite $Π_1$ sets that do not have $Δ_2$ cohesive subsets. In fact, we construct an infinite $Π_1$ set that does not have a $Δ_2$ p-cohesive subset. Infinite $Π_1$ sets without $Δ_2$ r-cohesive subsets generalize D. Martin's classic co-infinite c.e. set with no maximal superset and have appeared in the work of Lerman, Shore, and Soare.
