In his book "A practical theory of programming" Eric Hehner proposes and applies a remarkably radical reformulation of set theory, in which the collection and packaging of elements are seen as separate activities. This provides for unpackaged collections, referred to as "bunches". Bunches allow us to reason about non-determinism at the level of terms, and, very remarkably, allow us to reason about the conceptual entity "nothing", which is just an empty bunch (and very different from an empty set). This eliminates mathematical "gaps" caused by undefined terms. We compare the use of bunches with other approaches to this problem, and we illustrate the use of bunch theory in formulating program semantics which combines non-deterministic, preferential, and probabilistic choice. We show how an existing axiomatisation of set theory can be extended to incorporate bunches, and we provide and validate a model. Standard functions are lifted when applied to a bunch of values, but we also define a wholistic function application which allows whole bunches to be accepted as arguments, and we develop its associated fixed point theory.

中文翻译：

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束理论：关于应用、公理和模型的工作笔记

Eric Hehner 在他的《编程的实用理论》一书中提出并应用了集合论的一种非常激进的重构，其中元素的收集和包装被视为单独的活动。这提供了未打包的集合，称为“束”。Bunches 允许我们在术语级别对非确定性进行推理，并且非常显着地，允许我们对概念实体“无”进行推理，它只是一个空束（与空集非常不同）。这消除了由未定义术语引起的数学“差距”。我们将束的使用与解决此问题的其他方法进行比较，并说明束理论在制定程序语义中的使用，该语义结合了非确定性、优先性和概率选择。我们展示了如何扩展集合论的现有公理化以合并束，并且我们提供并验证了一个模型。标准函数在应用于一堆值时会被提升，但我们也定义了一个完整的函数应用程序，它允许整串作为参数被接受，我们开发了它相关的不动点理论。