A new analytic framework is first formalized via the usage of the Monadology (Leibniz 1898), to expand the understanding of Zermelo-Fraenkel-choice set theory (ZFC) and Von Neumann-Bernays-Godel set theory (NBG). Implicitly, the framework levels value, representation and information separately. Given the fact that there exists a coincidental equivalence between Von Neumann universe and originally-formalized motivation in ZFC, this work hypothesizes the essential of ordered values for one monand, to carry out efficient communication with the rest. This work then focuses on the relationship among values, representation and information (and suggests potential methods for quantitative analysis). First, this framework generalizes the definition of values and representations from "Indexes approximate Values" principle by (Peng 2023) via surreal numbers (Knuth 1974). Second, credited to surreal numbers, this work recursively connects representations and information via subsets of sets. Therefore, the definition to metric space(s) is naturally formed by representations, and quantitative methods (e.g., Hausdorff Distance) can be applied for quantitative analysis among (sub)sets. Third, this framework conjectures that: as long as the metric space is (or can be formed as) complete, the existence tests can be performed via Cauchy Sequence (or its generalized methods). This work finally revisits the communication theory, and suggests new perspectives from the new analytic framework. Particularly, this work hypothesizes a (quantitative) relationship between values and representation, and conjectures that: the optimal construction of representations exists, and it can be derived as the core value of one monad via Cauchy Inequality (or its generalized methods).

翻译：首先通过使用《单子论》（莱布尼茨，1898年）形式化了一个新的分析框架，以扩展对Zermelo-Fraenkel选择公理集合论（ZFC）和Von Neumann-Bernays-Gödel集合论（NBG）的理解。该框架含蓄地将价值、表征和信息分别置于不同层级。鉴于冯·诺伊曼宇宙与ZFC中原初形式化的动机存在巧合等价，本研究假设单子需具备有序价值，以与其他单子实现高效通信。随后，本文聚焦于价值、表征与信息之间的关系（并提出了定量分析的潜在方法）。首先，该框架通过超现实数（Knuth，1974年）将“索引近似价值”原则（彭，2023年）中价值与表征的定义泛化。其次，借助超现实数，本文通过集合的子集递归地连接表征与信息。由此，度量空间的定义自然由表征形成，可应用豪斯多夫距离等定量方法对（子）集进行定量分析。第三，该框架推测：只要度量空间是（或可构成为）完备的，便可通过柯西序列（或其泛化方法）进行存在性检验。本文最终重新审视通信理论，并从新分析框架提出新视角。特别地，本文假设价值与表征之间存在（定量）关系，并推测：存在表征的最优构造，且可通过柯西不等式（或其泛化方法）将其推导为单子的核心价值。