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第129行： − 从上面描述的递归集合和函数的理论开始，可计算性理论领域已经发展到包括许多密切相关的主题的研究。这些都不是独立的研究领域: 每个领域都从其他领域吸取思想和成果，大多数递归理论家对其中的大多数都很熟悉。+ + − + 第433行：
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→Areas of research
==Areas of research==
==Areas of research==
研究领域
Beginning with the theory of recursive sets and functions described above, the field of recursion theory has grown to include the study of many closely related topics. These are not independent areas of research: each of these areas draws ideas and results from the others, and most recursion theorists are familiar with the majority of them.
Beginning with the theory of recursive sets and functions described above, the field of recursion theory has grown to include the study of many closely related topics. These are not independent areas of research: each of these areas draws ideas and results from the others, and most recursion theorists are familiar with the majority of them.
Beginning with the theory of recursive sets and functions described above, the field of recursion theory has grown to include the study of many closely related topics. These are not independent areas of research: each of these areas draws ideas and results from the others, and most recursion theorists are familiar with the majority of them.
Beginning with the theory of recursive sets and functions described above, the field of recursion theory has grown to include the study of many closely related topics. These are not independent areas of research: each of these areas draws ideas and results from the others, and most recursion theorists are familiar with the majority of them.
从上面描述的递归集合和函数的理论开始，可计算性理论领域已经发展到包括许多密切相关的主题的研究。它们都不是彼此独立的研究领域: 每个领域都从其他领域吸取思想和成果，大多数递归理论专家对其中的大部分都很熟悉。
===Relative computability and the Turing degrees===
===Relative computability and the Turing degrees===
相对可计算性与图灵度
{{Main|Turing reduction|Turing degree}}
{{Main|Turing reduction|Turing degree}}
图灵归约 图灵度
用于数字计算的可计算性理论已经发展得很好。可计算性理论在模拟计算方面发展较差，模拟计算发生在模拟计算机、模拟信号处理、模拟电子学、神经网络和连续时间控制理论，由微分方程和连续动力系统建模(Orponen 1997; Moore 1996)。
用于数字计算的可计算性理论已经发展得很好。可计算性理论在模拟计算方面发展较差，模拟计算发生在模拟计算机、模拟信号处理、模拟电子学、神经网络和连续时间控制理论，由微分方程和连续动力系统建模(Orponen 1997; Moore 1996)。
==Relationships between definability, proof and computability==
==Relationships between definability, proof and computability==