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The field of mathematical logic dealing with computability and its generalizations has been called "recursion theory" since its early days. Robert I. Soare, a prominent researcher in the field, has proposed (Soare 1996) that the field should be called "computability theory" instead. He argues that Turing's terminology using the word "computable" is more natural and more widely understood than the terminology using the word "recursive" introduced by Kleene. Many contemporary researchers have begun to use this alternate terminology. These researchers also use terminology such as partial computable function and computably enumerable (c.e.) set instead of partial recursive function and recursively enumerable (r.e.) set. Not all researchers have been convinced, however, as explained by Fortnow and Simpson.

The field of mathematical logic dealing with computability and its generalizations has been called "recursion theory" since its early days. Robert I. Soare, a prominent researcher in the field, has proposed (Soare 1996) that the field should be called "computability theory" instead. He argues that Turing's terminology using the word "computable" is more natural and more widely understood than the terminology using the word "recursive" introduced by Kleene. Many contemporary researchers have begun to use this alternate terminology. These researchers also use terminology such as partial computable function and computably enumerable (c.e.) set instead of partial recursive function and recursively enumerable (r.e.) set. Not all researchers have been convinced, however, as explained by Fortnow and Simpson.

处理可计算性及其推广的数理逻辑领域自早期以来就被称为“可计算性理论”。这一领域的杰出研究者 Robert i. Soare 提出(Soare 1996) ，这一领域应该被称为“可计算性理论”。他认为，图灵使用的术语“可计算”比 Kleene 引入的术语“递归”更自然，也更广泛地被理解。许多当代研究人员已经开始使用这个替代术语。这些研究人员还使用了一些术语，比如部分可计算可计算函数和可计算可枚举(c.e.)设置代替部分递归函数和递归可枚举(r.e.)集。然而，正如 Fortnow 和 Simpson 所解释的，并非所有的研究人员都被说服了。

处理可计算性及其推广的数理逻辑领域自早期就被称为“递归理论”。这一领域的杰出研究者'''<font color="#ff8000">罗伯特·I·索尔 Robert I. Soare</font>'''提出(索尔1996) ，这一领域应该被称为“可计算性理论”。他认为，图灵使用的术语“可计算”比克莱恩引入的术语“递归”更自然，也更广泛地被理解。许多当代研究人员已经开始使用这个替代术语。这些研究人员还使用部分可计算函数和可计算枚举(c.e.)集等术语来代替部分递归函数和递归可枚举(r.e.)集。然而，正如福特诺和辛普森所解释的，并非所有的研究人员都被说服了。

Some commentators argue that both the names ''recursion theory'' and ''computability theory'' fail to convey the fact that most of the objects studied in recursion theory are not computable.<ref>Harvey Friedman, "[http://www.cs.nyu.edu/pipermail/fom/1998-August/002017.html Renaming recursion theory]," FOM email list, 1998-8-28, accessed 2006-1-9.</ref>

Some commentators argue that both the names ''recursion theory'' and ''computability theory'' fail to convey the fact that most of the objects studied in recursion theory are not computable.<ref>Harvey Friedman, "[http://www.cs.nyu.edu/pipermail/fom/1998-August/002017.html Renaming recursion theory]," FOM email list, 1998-8-28, accessed 2006-1-9.</ref>

Some commentators argue that both the names recursion theory and computability theory fail to convey the fact that most of the objects studied in recursion theory are not computable.

Some commentators argue that both the names recursion theory and computability theory fail to convey the fact that most of the objects studied in recursion theory are not computable.

Rogers (1967) has suggested that a key property of recursion theory is that its results and structures should be invariant under computable bijections on the natural numbers (this suggestion draws on the ideas of the Erlangen program in geometry). The idea is that a computable bijection merely renames numbers in a set, rather than indicating any structure in the set, much as a rotation of the Euclidean plane does not change any geometric aspect of lines drawn on it. Since any two infinite computable sets are linked by a computable bijection, this proposal identifies all the infinite computable sets (the finite computable sets are viewed as trivial). According to Rogers, the sets of interest in recursion theory are the noncomputable sets, partitioned into equivalence classes by computable bijections of the natural numbers.

Rogers (1967) has suggested that a key property of recursion theory is that its results and structures should be invariant under computable bijections on the natural numbers (this suggestion draws on the ideas of the Erlangen program in geometry). The idea is that a computable bijection merely renames numbers in a set, rather than indicating any structure in the set, much as a rotation of the Euclidean plane does not change any geometric aspect of lines drawn on it. Since any two infinite computable sets are linked by a computable bijection, this proposal identifies all the infinite computable sets (the finite computable sets are viewed as trivial). According to Rogers, the sets of interest in recursion theory are the noncomputable sets, partitioned into equivalence classes by computable bijections of the natural numbers.

罗杰斯(1967)建议，可计算性理论的一个关键性质是，它的结果和结构应该在自然数的可计算双射下不变(这个建议借鉴了几何学中爱尔兰根纲领的思想)。这个想法是，一个可计算的双射只是重命名一个集合中的数字，而不是指出集合中的任何结构，就像欧几里德平面的旋转不会改变画在它上面的线的任何几何方面一样。由于任意两个无限可计算集合通过一个可计算的双射联系在一起，这个方案确定了所有无限可计算集合(有限可计算集合被视为平凡集合)。根据 Rogers 的说法，可计算性理论中感兴趣的集合是不可计算的集合，通过可计算的自然数的双射划分成等价类。

罗杰斯(1967)建议，递归理论的一个关键性质是，它的结果和结构应该在自然数的可计算'''<font color="#ff8000">双射 bijection</font>'''下不变(这个建议借鉴了几何学中'''<font color="#ff8000">爱尔兰根纲领 Erlangen program</font>'''的思想)。这个想法是，一个可计算的双射只是重命名一个集合中的数字，而不是指出集合中的任何结构，就像欧几里德平面的旋转不会改变画在它上面的线的任何几何方面一样。由于任意两个无限可计算集合通过一个可计算的双射联系在一起，这个方案确定了所有无限可计算集合(有限可计算集合被视为平凡集合)。根据罗杰斯的说法，递归理论中感兴趣的集合是不可计算的集合，通过可计算的自然数的双射划分成等价类。

 

 

== Professional organizations ==

== Professional organizations ==