整数的集合理论表示
Set theoretical Representations of Integers, I
摘要：
本文从Kolmogorov复杂性的角度重新考虑了整数的一些经典自然语义（即函数的迭代器、集合的基数、等价关系的指数）。每个这样的语义可以附加一个简单的整数表示，我们适当地有效化，以开发一个相关的柯尔莫哥洛夫理论。这种有效化是我们在本文中介绍的“自枚举系统”的一般概念的特殊情况。我们的主要结果断言，这样的有效化，Kolmogorov理论允许定量区分潜在的语义。我们的特点得到这样的effectivizations的家庭，并证明了相关的Kolmogorov复杂性构成一个层次结构，这与通过跳跃预言和/或无限计算定义的Kolmogorov复杂性相一致。这与众所周知的事实，即通常的柯尔莫哥洛夫复杂性不依赖于（一个常数）所选择的整数的算术表示，让它在任何基础一元，二进制等。此外，在一个概念的角度来看，我们的结果可以被看作是一种手段来衡量这些不同的语义的抽象程度。
英文摘要：
We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of "self-enumerated system" that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations. This contrasts with the well-known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base unary, binary et so on. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics.
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