Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension
Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension
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有理数序列收敛于左 c.e.
DOI：
10.1142/9789811259296_0005
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Yuki Mizusawa and Toshio Suzuki
中科院分区：
文献类型：
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作者：
Hiroyuki Imai;Masahiro Kumabe;Kenshi Miyabe;Yuki Mizusawa and Toshio Suzuki
在我们以前的工作中，我们用Lipschitz条件刻画了Solovay约简，并引入了拟Solovay约简（qS-约简）作为Hölder条件的对应.本文利用收敛速度研究了与拟Solovay约化密切相关的有效维数和理想。我们证明了左实之间的qS-完备性等价于有一个正的有效Hausdorff维数。qS-完全左实的Solovay度构成一个滤子。另一方面，非qS-完备左实的Solovay度不形成理想。基于对有理序列与约简之间关系的观察，我们引入了qS-约简的一个更强的版本。给定这个约化的次数，下锥（包括给定次数）形成一个理想。通过发展这些调查，我们的有效尺寸的收敛速度。我们给出了基于Solovay约化的第一不完备性定理的一个变形。
In our previous work, we characterized Solovay reducibility using Lipschitz condition, and introduced quasi Solovay reducibility (qS-reducibility, for short) as a Hölder condition counterpart. In this paper, we investigate effective dimensions and ideals closely related to quasi Solovay reducibility by means of the rate of convergence. We show that the qS-completeness among left-ce reals is equivalent to having a positive effective Hausdorff dimension. The Solovay degrees of qS-complete left-ce reals form a filter. On the other hand, the Solovay degrees of non-qS-complete left-ce reals do not form an ideal. Based on observations on the relationships between rational sequences and reducibility, we introduce a stronger version of qS-reducibility. Given a degree of this reducibility, the lower cone (including the given degree) forms an ideal. By developing these investigations, we characterize the effective dimensions by means of the rate of convergence. We give a variation of the first incompleteness theorem based on Solovay reducibility.