COMPARING TWO VERSIONS OF THE REALS
COMPARING TWO VERSIONS OF THE REALS
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比较两个版本的实数
DOI：
10.1017/jsl.2015.77
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
J. Knight
中科院分区：
文献类型：
--
作者：
Gregory Igusa;J. Knight
摘要 Schweber [10] 定义了一种可归约性，使我们能够比较任意基数结构的计算能力。这里我们关注实数的有序域 ${\cal R}$ 和仅编码 ω 子集的结构 ${\cal W}$。在[10]中，观察到 ${\cal W}$ 可简化为 ${\cal R}$ 。我们证明 ${\cal R}$ 不能简化为 ${\cal W}$ 。作为证明的一部分，我们证明，对于具有剩余域 k 的可数递归饱和实闭域 ${\cal P}$，${\cal P}$ 的某些副本不会计算 k 的副本。
Abstract Schweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$ . We prove that ${\cal R}$ is not reducible to ${\cal W}$ . As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.