初等代数与初等几何可判定性的若干新结果
Some new results on decidability for elementary algebra and geometry
摘要：
我们进行了一个系统的研究的可判定性理论（a）实向量空间，内积空间，和希尔伯特空间和（b）赋范空间，巴拿赫空间和度量空间，所有正式使用2排序的一阶语言。列表（a）的理论证明是可判定的，而列表（b）的理论甚至不是算术的：例如，二维巴拿赫空间的理论与二阶算术的真值集具有相同的多1度。我们发现赋范空间理论中的纯泛空间和纯存在空间片段是可判定的，度量空间理论中的AE片段也是可判定的。这些结果是其类型的尖锐：Hilbert第10个问题的约化表明，度量和赋范空间的EA片段以及赋范空间的AE片段都是不可判定的。
英文摘要：
We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.
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