We discuss mathematical and physical arguments contrasting continuous and discrete, limitless discretization as arbitrary granularity. In this regard, we focus on Incomputable (lacking an algorithm that computes in finite time) Real Numbers (IRNs). We consider how, for measurements, the usual approach to dealing with IRNs is to approximate to avoid the need for more detailed, unrealistic surveys. In this regard, we contrast effective computation and emergent computation. Furthermore, we consider the alternative option of taking into account the properties of the decimal part of IRNs, such as the occurrence, distribution, combinations, quasi-periodicities, and other contextual properties, e.g., topological. For instance, in correspondence with chaotic behaviors, quasi-periodic solutions, quasi-systems, uniqueness, and singularities, non-computability represents and corresponds to theoretically incomplete properties of the processes of complexity, such as emergence and quantum-like properties. We elaborate upon cases of equivalences and symmetries, characterizing complexity and infiniteness as corresponding to the usage of multiple non-equivalent models that are constructively and theoretically incomplete due to the non-exhaustive nature of the multiplicity of complexity. Finally, we detail alternative computational approaches, such as hypercomputation, natural computing, quantum computing, and analog and hybrid computing. The reality of IRNs is considered to represent the theoretical incompleteness of complex phenomena taking place through collapse from equivalences and symmetries. A world of precise finite values, even if approximated, is assumed to have dynamics that are zippable in analytical formulae and to be computable and symbolically representable in the way it functions. A world of arbitrary precise infinite values with dynamics that are non-zippable in analytical formulae, non-computable, and, for instance, sub-symbolically representable, is assumed to be almost compatible with the coherence of emergence. The real world is assumed to be a continuous combination of the two—functioning and emergent—where the second dominates and is the norm, and the first is the locus of primarily epistemic extracts. Research on IRNs should focus on properties representing and corresponding to those that are detectable in real, even if extreme, phenomena, such as emergence and quantum phenomena.

中文翻译：

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关于不可计算实数的现实性及其系统意义的注记

我们讨论数学和物理参数，将连续和离散、无限离散化作为任意粒度进行对比。在这方面，我们专注于不可计算（缺乏在有限时间内计算的算法）实数 (IRN)。我们考虑，对于测量，处理 IRN 的常用方法如何进行近似以避免需要更详细、不切实际的调查。在这方面，我们对比了有效计算和紧急计算。此外，我们考虑了考虑 IRN 小数部分属性的替代选项，例如出现、分布、组合、准周期性和其他上下文属性，例如拓扑。例如，对应于混沌行为、准周期解、准系统、唯一性和奇点，不可计算性代表并对应于复杂过程的理论上不完整的特性，例如涌现和类似量子的特性。我们详细阐述了等价和对称的情况，将复杂性和无限性表征为对应于多个非等价模型的使用，这些模型由于复杂性多样性的非穷尽性而在建设性和理论上是不完整的。最后，我们详细介绍了替代计算方法，例如超计算、自然计算、量子计算以及模拟和混合计算。IRN 的现实被认为代表了通过等价和对称性崩溃而发生的复杂现象的理论不完整性。一个精确的有限值的世界，即使是近似的，假定具有在分析公式中可压缩的动力学，并且在其运行方式中是可计算和符号表示的。一个由任意精确无限值组成的世界，其动力学在分析公式中是不可压缩的、不可计算的，例如，子符号可表示的，被假定为几乎与出现的连贯性兼容。现实世界被假定为两者的连续组合——功能性和涌现性——其中第二个占主导地位并且是规范，第一个是主要认知提取物的所在地。对 IRN 的研究应侧重于代表和对应于现实中可检测到的特性，即使是极端的现象，例如涌现和量子现象。