ω 生成冯·诺依曼序数Generate von Neumann Ordinals

使用说明与常见问题How-to & FAQ

「生成冯·诺依曼序数」是什么？What is the Generate von Neumann Ordinals tool?

生成冯·诺依曼自然数（序数）的集合写法：0=∅, 1={∅}, 2={∅,{∅}}, …。可指定从 0 到 N。本地处理。Generate von Neumann naturals (ordinals) in set notation: 0 = empty set, 1 = set containing empty, 2 = set of those, and so on, from 0 through N. Processed locally.

如何使用「生成冯·诺依曼序数」？How do I use the Generate von Neumann Ordinals tool?

在「生成冯·诺依曼序数」中，按需设置「最大 N」，结果会实时显示在下方，点击「复制结果」即可复制。Enter "Max N" and the page lists the von Neumann ordinals 0 to N in set notation, where each number is the set of all smaller ones (0 is the empty set). Click "Copy result" to copy the list.

为什么 N 最大 12？Why is Max N only 12?

冯·诺依曼写法的长度随 N 指数增长，再大会让浏览器卡死。von Neumann notation grows exponentially with N; larger values would freeze the browser.

N = 0 会输出什么？What does N = 0 give?

只输出一行「0 = ∅」。结果是纯文字（逐位处理），不涉及浮点计算。Just one line, 0 = ∅. The result is plain text (processed as text) with no floating-point arithmetic.

输入和输出是什么样的？What does the input and output look like?

例如在「生成冯·诺依曼序数」中把「最大 N」设为 2，输出为：0 = ∅ ⏎ 1 = {∅} ⏎ 2 = {∅,{∅}}（⏎ 表示换行）。For example, in the Generate von Neumann Ordinals with "Max N" set to 2, the output is: 0 = ∅ ⏎ 1 = {∅} ⏎ 2 = {∅,{∅}} (⏎ marks a line break).

有哪些选项可以调整？What options can I adjust?

「生成冯·诺依曼序数」的设置：「最大 N」默认 5。改动任一选项后结果会按新设置重新生成。Settings in the Generate von Neumann Ordinals — "Max N" default 5. Changing any option regenerates the result with the new setting.

「生成冯·诺依曼序数」免费吗？需要注册吗？Is the Generate von Neumann Ordinals free? Do I need an account?

「生成冯·诺依曼序数」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Generate von Neumann Ordinals is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.