…disallowing these kinds of set like ZFC does, he enlarges ZFC by allowing any set which is a unique solution to a group of equations that may be self-referential. I was introduced to this stuff by a book "Vicious Circles" by Barwise and Moss (http://www.amazon.com/Vicious-Circles-Center-Language-Inform...). The important point is that the Anti-Foundation Axiom lets us model infinite, streaming structures similar to those…
…theories are compatible with standard the set theory, extending it, rather than replacing it, they don't throw out the notion of consistency, but have no problem dealing with self-referential sets and such. "Vicious Circles" [3] is a great exposition of the subject of hypersets and their applications to computer science and logic, among other things and is quote accessible for such a theoretical subject. [1] http://en.wikipedia.org/wiki/Non-well-founded_set_theory…
…programming languages is available in the middle chapters of Practical Foundations for Programming Languages. I originally learned about it from studying Aczel's Antifoundation Axiom from Barwise and Moss' Vicious Circles[0]---in there you can get a much more mathematical foundationalist POV---but Harper's book is more direct. [0]http://www.amazon.com/Vicious-Circles-Center-Language-Inform...
……并不是像 ZFC 那样直接禁止这类集合，而是通过允许任何作为一组可能自指的方程组的唯一解的集合，来对 ZFC 进行扩充。我是通过 Barwise 和 Moss 写的《Vicious Circles》（http://www.amazon.com/Vicious-Circles-Center-Language-Inform...）这本书接触到这些内容的。核心在于，反基础公理（Anti-Foundation Axiom）让我们可以对无限的流式结构进行建模，类似于那些……
…disallowing these kinds of set like ZFC does, he enlarges ZFC by allowing any set which is a unique solution to a group of equations that may be self-referential. I was introduced to this stuff by a book "Vicious Circles" by Barwise and Moss (http://www.amazon.com/Vicious-Circles-Center-Language-Inform...). The important point is that the Anti-Foundation Axiom lets us model infinite, streaming structures similar to those…
……这些理论与标准集合论兼容，是在扩展它而不是取代它，它们并没有抛弃一致性（consistency）的概念，但在处理自指集合这类东西时游刃有余。《Vicious Circles》[3] 对超集合（hypersets）及其在计算机科学和逻辑学等领域的应用做了极佳的阐述，而且对于这么一门理论性极强的主题来说，书写得相当通俗易懂。[1] http://en.wikipedia.org/wiki/Non-well-founded_set_theory…
…theories are compatible with standard the set theory, extending it, rather than replacing it, they don't throw out the notion of consistency, but have no problem dealing with self-referential sets and such. "Vicious Circles" [3] is a great exposition of the subject of hypersets and their applications to computer science and logic, among other things and is quote accessible for such a theoretical subject. [1] http://en.wikipedia.org/wiki/Non-well-founded_set_theory…
……程序设计语言可以在《Practical Foundations for Programming Languages》的中间几章里找到。我最初是通过学习 Barwise 和 Moss 的《Vicious Circles》[0] 中的阿克塞尔反基础公理（Aczel's Antifoundation Axiom）了解它的——在那里你可以获得更加偏向数学基础论（mathematical foundationalist）的视角——不过 Harper 的书更直奔主题。[0]http://www.amazon.com/Vicious-Circles-Center-Language-Inform...
…programming languages is available in the middle chapters of Practical Foundations for Programming Languages. I originally learned about it from studying Aczel's Antifoundation Axiom from Barwise and Moss' Vicious Circles[0]---in there you can get a much more mathematical foundationalist POV---but Harper's book is more direct. [0]http://www.amazon.com/Vicious-Circles-Center-Language-Inform...