学科全知书！

“化形，境界就大罗金仙后期了！”

“幸好有系统，可能永远也无法化形成人了。”

“系统，还有什么地方没有签到？”

〖宿主，还有诸天万界没有签到〗

“诸天万界！还有这么多。”

〖任务：去打一顿准提与接引，奖励：学科全知书〗

“系统，我现在才大罗金仙后期怎么打？”

〖宿主，可以使用本系统奖励的世界〗

〖宿主可以把世界扩大一次”便是让世界粒子数个世界粒子数相乘”，每一次扩大都是世界粒子数个世界粒子数相乘！

空间：一维空间包含了无限个零维空间，二维空间包含了无限个一维空间，三维空间包含了无限个二维空间，四维空间包含了无限个三维空间……一直到无限维度。

时间：时间线会分裂，每一个瞬间都可以分裂出无穷个分支，每一个分支都是一个独立的宇宙，每一个分支又会在当前节点上分裂出无穷个分支，如此反复，无穷尽也！

世界：所有空间丶所有时间…………………！

世界粒子数量超过世界！

宿主还可以增加世界扩大的速度，变成一秒扩大无限次，经过无限时间后！

宿主一拳打爆无穷无尽…………………个世界也可以〗

〖因为，宿主在那个世界就是神一样的存在〗

“可以操控世界的一切？”

〖可以〗

“我也太厉害了”

〖宿主可以把接引与准提拉到你那个世界去〗

“好办法！”

学科全知书有所有学科的知识！

以下也是学科全知书里面的东西。

高阶图灵机：

一台预言机，除了拥有与图灵机相同的功能，还可以随时解出任何图灵机的

停机问题(但不能解出预言机本身的停机问题)，也就是说，Σ函数和BB函数在

预言机这里是可解的.所有n状态的预言机busybeaver中，一台预言机能够

“扫过“的格子区域所含的最大格子数记作Σ2(n)，能够进行操作的最多步数记

作BB2(n).这2个函数的增长率远远超过Σ(n)和BB(n).

下面，我们把普通图灵机记作1阶图灵机，把预言机记作2阶图灵机.设想一

下什么是3阶图灵机:它除了拥有与图灵机相同的功能，还可以随时解出任何1

阶或2阶图灵机的停机问题.所有n状态的3阶busybeaver中，一台3阶图

灵机能够“扫过“的格子区域所含的最大格子数记作Σ3(n)，能够进行操作的最

多步数记作BB3(n).这2个函数的增长率远远超过Σ2(n)和BB2(n).

一般地，m阶图灵机除了拥有与图灵机相同的功能，还可以随时解出任何小

于m阶图灵机的停机问题.所有n状态的m阶busybeaver中，一台m阶图灵

机能够“扫过“的格子区域所含的最大格子数记作Σm(n)，能够进行操作的最多步数记作BBm(n).

一台n个状态的图灵机最多可以运行k步记这个东西为BB(n)

图灵机其实是个非常简单的玩意儿，和“掰手指数数”是一个性质的。认真看书就明白了，别胡思乱想。

先问一个问题，你认为“计算”是什么？

比如说，掰手指计数，左手伸4个手指，右手伸3个手指；然后把双手所有伸出的手指数一遍，7个。

很好，这就是个“用‘手指计算器’算3+4=7”的案例。

类似的，算盘每根柱子代表一个十进制位；然后一根柱子被木格分到上边的一个珠子代表5、下边一个珠子代表1。

就好像掰手指数数一样，上边拨下来一颗珠子，这是伸出了5根手指；下面拨上去两个珠子，这是伸出了两根手指——不进位加二对应的口诀是二上二，于是5+2=7。

如果在这个基础上再加8呢？怎么算7+8？

7加8要进位，对应的口诀是八去二进一，翻译过来是“加八相当于先加十后减二”，即当前位拨下两颗珠子，高位拨上一颗珠子，答案就摆在算盘上了（习题：如果是3+3这样超过5但不足10的，你认为应该如何拨珠子？如果前面说的你真理解了，自己三下五除二就能搞出个口诀来）。

珠算口诀_百度百科

嗯，别研究珠算了。也别着急研究机械计算机。它们都不是关键。你越往里钻，反而会离真理越远——能像图灵一样，先钻进最复杂的机械结构（机械加解密机）然后再把它化简成掰手指的，实在太少了。

伟大的图灵已经帮我们抓住了关键，从最容易的来就足够了。

让我们尽量忘掉细节，告诉我，“计算”这个动作究竟是什么？

从掰手指、玩算盘、拨机械计算器的数字轮的经验，我们知道了：计算就是一些机械的操作步骤，按机械的规定机械的执行这些步骤，我们就可以得到某个输入的计算结果。

太啰嗦了。废话能更少点吗？

嗯……计算动作……就是通过一些机械动作，把问题映射到答案的操作过程。

嘶~~这个定义字数倒是少了，但怎么成天书了？

你这样想：3+4是问题集合的一个问题；7是答案集合的一个答案；算3+4就是通过一些机械步骤（1、左右手分别伸出3和4根手指；2、数一数一共伸出了几根手指），然后找到了7，对吧？

你看，完全不需要什么“智能”，按死规矩来就行了。

仔细想一下，看看各种计算，从加减乘除到乘方开方积分求导……甚至加密解密……是不是都可以这么看？

问题/原始数据/密文--->答案/密文/原始数据

很好，既然一切计算归根结底都可以是机械过程，那么有没有什么机械过程可以模拟一切运算呢？

你看，摆手指头、算盘、机械计算机，它们的确可以帮我们做一些运算；但它们只能做特定种类的那些运算：有没有可能弄出一个“可以帮我们做任何运算”的机器呢？

————————————————————————————

还是继续玩简单粗暴的：计算，其实不就是“一系列从问题到答案的映射步骤”吗？

那么，随问题不同而各不相同的“映射步骤”本身，可以用不太多的几条“万能映射步骤”来映射吗？

图灵说了，可以。

我们可以做一个机械装置，它可以：

1、从纸带读入内容或输出信息到纸带上（纸带是无限长的）

2、移动到纸带上的其它位置

3、有一套控制规则和一个内部状态寄存器；可根据状态寄存器内容以及当前纸带格子里内容，决定下一步动作（在纸带上移动、读取/写入内容或停机等）

这就是一台图灵机。

其中，“控制规则”就是“映射步骤”（类似珠算口诀、机械计算器操作动作等），“内部状态”就是黑板/演草纸上面的内容（或者算盘珠的摆放状态、伸出的手指）；而“纸带内容”就是要解决的问题以及输出答案的地方——和人们掰手指或手工演算一模一样。

然后呢，问题虽然是无限的；但解答某个问题需要的“控制规则”的数目是有限的，所以这些“控制规则”总可以编码然后存到纸带上。

我们可以给每个不同的控制规则一个编号，比如M。（标准术语体系里，控制规则M被称为“图灵机M”。个人认为这容易造成混淆，故这里杜撰了个“控制规则M”）

图灵证明了，可以设计一些特殊的控制规则，使得它控制下的图灵机可以从纸带读入任意的控制规则M，然后仿照控制规则M去处理纸带内容。

这种可以模拟任意其他计算规则的图灵机就是“通用图灵机”；我们把“足以模拟任意其他计算规则”这个要求称为“图灵完备”。

图灵完备的“通用控制规则“并不需要多复杂。事实上，最简单的那些……简直简单的令人发指。

——打个比方的话，人+算盘就是一种具有一定通用性的计算机；“人”学会了不同的珠算口诀，就可以按照这些口诀给出的“控制规则”计算加减乘除甚至平方开方运算了。

——说的更具体一点：当我们需要做“加法”时，就把书翻到“珠算加法口诀”那一页；然后被加数是3，于是执行口诀“三下五除二”……如此反复，你看，结果就出来了。

——图灵机也类似：需要做加法时，我们查表然后把纸带移到加法规则那块（相当于需要一个“计算规则映射表”，然后找出其中的加法规则表）；然后读入数据，按加法规则算出结果。

——两者的不同点在于，算盘需要人完成“学习规则-执行规则”这个动作；而图灵机通过机器读入各种映射表自动执行“读入映射规则-执行规则”这个动作。

说的更浅显一些，就是：支持“通用控制规则”的图灵机，不管它看起来有多简单，但它的确就是一种万能机器。

既然组成“通用控制规则”的规则并不需要多少，更不需要很复杂。那么，搞出来一种“万能解题机”就不困难。

很快，在图灵理论的指导下，人类第一台计算机出现了。

——————————————————

现代计算机就是一种高度复杂化了的图灵机——虽然高度复杂化了，但图灵机该有的，它一概不缺：比如纸带（用内存/磁盘模拟）、内部状态寄存器（现代计算机有大量的标志位和大量的寄存器甚至寄存器组）、在纸带上移动（各种跳转指令、各种复杂的寻址操作）、控制规则（CPU指令集），等等。

至于程序员……其实就是借助CPU指令集甚至某种高级语言（通用控制规则），为某个具体问题设计解题步骤的（控制规则M，或者……notepad.exe、qq.exe等等）——然后CPU按照程序员给出的步骤1、2、3、4做下去，问题答案就出来了（当然，要是某些步骤没设计好、某些特殊情况没考虑到……先生，你编的bug诞生了）。

比如说呢，前面提的那个“给算盘设计一条珠算口诀”问题，其实就是程序员日常工作（的一部分）了——都是“为某个问题设计机械解决步骤”嘛。

你可以回想一下乘除法的运算规则，然后仿照我前面提到的“步骤思想”，设计一套珠算口诀来。相信我，你可以的——设计出来后，记得到算盘上Debug哦。

如果你能搞定“自制算盘口诀”，就说明阁下骨骼精奇，定能把码农大业发扬光大：这本祖传的《编译原理》，你掏100，就白送给你了-_-|||……

——当然，如果你连加法口诀都没法补全……那么阁下很可能并不适合吃程序员这碗饭：理所当然的，这篇文章……你应该也看不懂。

总之，图灵机就是现代计算机的终极简化版；除了最核心的计算原理，其它的一概没有。

显然，如果真正理解了图灵机的计算原理、再从它开始来理解现代计算机工作原理，那是最简洁最容易不过的了。

只是各种教材/书籍大概都有点“祖师爷崇拜“情结，这才把它讲的晦涩难懂扑朔迷离……

“图灵机”本质上一种抽象计算模型，其更抽象的意义为一种数学逻辑机，可以看作等价于任何有限逻辑数学过程的终极强大的逻辑机器。

图灵机是一个虚拟的机器，由数学家阿兰·图灵1936年提出来的，尽管这个机器很简单，但它可以模拟计算机的任何算法，无论这个算法有多复杂。

Infinite Time Turing Machines：We extend in a natural way the operation of Turing machines to infinite ordinal time， and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. Every Pi^1_1 set， for example， is decidable by such machines， and the semi-decidable sets form a portion of the Delta^1_2 sets. Our oracle concept leads to a notion of relative computability for reals and sets of reals and a rich degree structure， stratified by two natural jump operators.

Definition

The original model by Hamkins and Lewis has three one-sided， two-color countably infinite tapes， called input， scratch and output tapes， and a single read-write head which reads one cell from each tape.(Models with one tape have been considered as well— surprisingly， they have been shown to be weaker.[3] To our knowledge， tapes with more than two colors have not been explored.) The transition table takes into account (but not necessarily behaves in the same way) all three tapes simultaneously， and writes three at once. The scratch and output tapes are blank initially， and the input tape contains the input to the machine.

ITTMs have a particular state marked as the limit state. At each step， the ITTM proceeds normally with a timer \(t\) starting at zero and incrementing at each step. If it does not halt for any \(t <\omega\)， at \(t =\omega\) each cell on each tape is set to the

limit superior

of that cell's symbol for steps \(t = 0， 1， 2，\ldots\). In addition， the head is set to the leftmost position and the state is set to limit. The ITTM continues as before， and if it continues to \(t =\omega2\) without halting， the same thing happens with the limit superior of the tapes for \(t =\omega，\omega + 1，\omega + 2，\ldots\). In general， if the ITTM reaches a countable limit ordinal \(\alpha\) without halting， at time \(t =\alpha\)， it is set to the limit superior of the tapes at \(t =\beta\) for all \(\beta <\alpha\). That is， a cell is set to 1 iff for all \(\beta <\alpha\)， there is some step \(\beta <\gamma <\alpha\) such that the cell was set to 1 at step \(t =\gamma\).

If the ITTM halts， the contents of the output tape at halting time are the output of the machine. If the ITTM does not halt， but after some point the output tape never changes， the output tape after that point is the eventual output of the machine. For both halting and non-halting cases， all of the configurations of the three tapes over the lifetime of the ITTM are the accidental outputs of the machine. It should be noted that outputs and eventual outputs are unique if they exist， but accidental outputs are not.

Formal definition

Formally， a two-color ITTM is a 5-tuple \(M =(Q，\delta， q_0， q_H， q_L)\) with the following components:

\(Q\) is a finite， non-empty set of states.

\(q_0 \in Q\) is the initial state.

\(q_H \in Q\) is the halting state.

\(q_L \in Q\) is the limit state.

\(\delta :(Q \backslash \{q_H\})imes \{0， 1\}^3 \rightarrow Q imes \{0， 1\}^3 imes \{L， R\}\) is the transition table. Each production has a non-halting state and three colors mapping to a state， three colors， and a movement.

Most definitions of the standard TM have a symbol designated as“blank，“ which is the only symbol that can appear infinitely many times on tape. ITTMs have no such restriction， since ITTM inputs and outputs can be infinitely large.

Configurations

This section defines configurations of the machine and how state transitions work. The semantics defined in this section are identical to those of an ordinary machine (albeit with three tapes).

A configuration of \(M\) is a 3-tuple \((p，au， m)\) where \(p \in Q\) is the current state of the machine，\(au :(\mathbb{N}imes\{0，1，2\})\rightarrow \{0， 1\}\) are the contents of tapes， and \(m \in \mathbb{N}\) is the positions of the head. We use \(C(M)\) to denote the set of configurations. We define \(\leadsto\)(a single computation step) as a binary relation over \(C(M)\) as follows. We say that \((p，au， m)\leadsto (p'，au'， m')\) iff the following are true:

\(\delta(p，(au(m，0)，au(m，1)，au(m，2)))=(p'，(c_0，c_1，c_2)，\Delta m)\).

For all \(n \in \mathbb{N}，i\in\{0，1，2\}\)，\(au'(n，i)=\left\{\begin{array}{lr} c_i &: n = m\\au(n，i)&ext{otherwise}\end{array}\right.\).

\(m'=\left\{\begin{array}{lr} m + 1 &:\Delta m = R \\\max\{m - 1， 0\}&:\Delta m = L \end{array}\right.\)

It can be seen that \(\leadsto\) is a function. That is，\((p'，au'， m')\) is unique.

Input， output， and evolution

An input to an ITTM is simply a function \(I:\mathbb{N}\rightarrow \{0， 1\}\)(meant to be the content of the first tape)， and an initial contents for that input is \(au_0:\mathbb{N}imes\{0，1，2\}\rightarrow \{0，1\}\) such that \(au_0(n，0)=I(n)，au_0(n，1)=au_0(n，2)=0\). A computation of \(M\) on an input \(I\) is a function \(U :\mu \rightarrow C(M)\)(where \(\mu\) is a successor ordinal， or the class \(ext{On}\)) that satisfies the following properties:

\(U(0)=(q_0，au_0， 0)\)

For all \(\alpha + 1 \in \mu\)，\(U(\alpha)\leadsto U(\alpha + 1)\).

For all limit ordinals \(\alpha \in \mu\)，\(U(\alpha)=(q_L，au_\alpha， 0)\) where for all \(i \in \mathbb{N}imes\{0，1，2\}\)，\(au_\alpha(i)=ext{lim sup}_{\beta <\alpha}au_\beta(i)\) where \(U(\beta)=(\bullet，au_\beta，\bullet)\).[4]

If \(\mu \neq ext{On}\)，\(U(\mu - 1)=(q_H，\bullet，\bullet)\).

Halting， writing， and ITTM-computability

If \(\mu\neqext{On}\)， then the machine is halting for \(au_0\)， and we call \(\mu\) the halting time. Define the output of the computation as the function \(H:\mathbb{N}\rightarrow \{0，1\}\)， defined as \(H(n)=au_{\mu-1}(n，2)\). It is not hard to show that this output must be unique.

If \(\mu =ext{On}\)， then the machine is nonhalting for \(au_0\). If there is some \(H :\mathbb{N}\rightarrow \{0，1\}\) such that for all sufficiently large \(\alpha\)，\(\forall n :au_\alpha(n，2)= H(n)\)， then we say that the machine eventually writes \(H\). It is not hard to show that eventual output is also unique if it exists.

For all \(H :\mathbb{N}\rightarrow \{0，1\}\)， if there is an \(\alpha \in \mu， i \in \{0， 1， 2\}\) such that \(\forall n:au_\alpha(n， i)= H(n)\)， we say that the machine accidentally writes \(H\) for the input. Accidental outputs are not unique.

The uniqueness of outputs and eventual outputs allows us to define the partial ITTM-computable functions and partial eventually-computable functions. A partial function \(f :\mathbb{N}\rightarrow \mathbb{N}\) is ITTM-computable iff there exists an ITTM \(M\) such that \(f(n)= m\) iff \(M\) produces output \(m\) with input \(n\)(using an encoding scheme for natural numbers such as \(\underbrace{11...1}_{n}00...\)). A partial function \(f :\mathbb{N}\rightarrow \mathbb{N}\) is eventually computable iff there exists an ITTM \(M\) such that \(f(n)= m\) iff \(M\) produces eventual output \(m\) with input \(n\).

ITTM ordinals

ITTMs are capable of producing far more outputs than ordinary TMs， including ordinals. This leads to several classes of ordinals:

An ordinal \(\alpha\) is writable iff an ITTM with empty input (all zeros) has \(\alpha\) as its output.

An ordinal \(\alpha\) is clockable iff an ITTM with empty input halts at time \(\alpha\).

An ordinal \(\alpha\) is eventually writable iff an ITTM with empty input has \(\alpha\) as its eventual output.

An ordinal \(\alpha\) is accidentally writable iff an ITTM with empty input has \(\alpha\) as an accidental output.

(The first and last two definitions can apply to many mathematical objects other than ordinals) These definitions give rise to several large countable ordinals (the infinite time Turing machine ordinals):

\(\lambda\)， the supremum of all writable ordinals

\(\gamma\)， the supremum of all clockable ordinals

\(\zeta\)， the supremum of all eventually writable ordinals

\(\Sigma\)， the supremum of all accidentally writable ordinals

It has been shown that \(\omega_1^ext{CK}\ll \lambda =\gamma <\zeta <\Sigma\).[5]

Encoding ordinals in bitstrings

To be able to read and write ITTM ordinals， we must specify an encoding scheme to represent ordinals with countably infinite bitstrings. Here is one such scheme:

“1000...“ encodes \(0\).

If string S encodes \(\alpha\)， then the string“0“+ S (where + denotes string concatenation) encodes \(\alpha + 1\).

If strings \(S_0，S_1，S_2，\ldots\) encode \(\alpha_0，\alpha_1，\alpha_2，\ldots\)， then we interleave the strings into a new bitstring \(T\) according to the following scheme:

Start with every bit of \(T\) labeled“empty“.

Write the bits of \(S_0\) in every second bit of \(T\).

Write the bits of \(S_1\) in every second remaining empty bit of \(T\).

Write the bits of \(S_2\) in every second remaining empty bit of \(T\).

Repeat for all of the \(S_i\).

Write“1“ in the first bit of \(T\).

\(T\) encodes \(\sup\{\alpha_i\}\).

Using this encoding，“\(\color{red}{01000...}\)“ represents 1，“\(\color{blue}{001000...}\)“ represents 2，“\(\color{green}{0001000...}\)“ represents 3 and so on.

Then \(1\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{1}\)\(\color{green}{0}\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{0}\)\(0\)\(\color{red}{0}\)\(\color{blue}{1}\)\(\color{red}{0}\)\(\color{green}{0}\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{0}\)\(0\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{0}\)\(\color{green}{0}\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{0}\)\(0\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{0}\)\(\color{green}{1}\)\(\color{red}{0}\)\(\color{blue}{0}\)\(\color{red}{0}\)\(0\)\(\ldots\) represents \(\omega\).

This is of course one of many possible encoding schemes， and within this scheme multiple bitstrings represent a single ordinal.

Klev's \(\mathcal{O}^+\)

Ansten Mørch Klev extended Kleene's \(\mathcal{O}\) to the set of all writable ordinals.[6] Let \(f_0， f_1， f_2，\ldots\) be an enumeration of all ITTM-computable partial functions (where the output of an ITTM is based on the tape when it halts). The rules are otherwise identical to Kleene's \(\mathcal{O}\):

\(0 \in K\) and \(\mathcal{O}^+(0)= 0\).

If \(n \in K\) and \(\mathcal{O}^+(n)=\alpha\)， then \(2^n \in K\)，\(\mathcal{O}^+(2^n)=\alpha + 1\)， and \(n <_{\mathcal{O}^+} 2^n\).

If for all natural numbers \(n\)， we have \(f_i(n)\in K\) and \(f_i(n)<_{\mathcal{O}^+} f_i(n + 1)\)， then \(3 \cdot 5^i \in K\)，\(\mathcal{O}^+(3 \cdot 5^i)=\lim_{k \rightarrow \omega}\mathcal{O}^+(f_i(k))\)， and for all \(k\)\(f_i(k)<_{\mathcal{O}^+} 3 \cdot 5^i\).

\(a <_{\mathcal{O}^+} b\) and \(b <_{\mathcal{O}^+} c\) implies \(a <_{\mathcal{O}^+} c\).

\(\mathcal{O}^+\) is capable of expressing all ordinals \(<\lambda\)(all writable ordinals).

Klev's \(\mathcal{O}^{++}\)

In the same paper， Klev introduced \(\mathcal{O}^{++}\)， which has \(f_i\) enumerates all eventually computable partial functions (that is， the output of an ITTM is based on what the tape converges to). The rules are otherwise identical:

\(0 \in K\) and \(\mathcal{O}^{++}(0)= 0\).

If \(n \in K\) and \(\mathcal{O}^{++}(n)=\alpha\)， then \(2^n \in K\)，\(\mathcal{O}^{++}(2^n)=\alpha + 1\)， and \(n <_{\mathcal{O}^{++}} 2^n\).

If for all natural numbers \(n\)， we have \(f_i(n)\in K\) and \(f_i(n)<_{\mathcal{O}^{++}} f_i(n + 1)\)， then \(3 \cdot 5^i \in K\)，\(\mathcal{O}^{++}(3 \cdot 5^i)=\lim_{k \rightarrow \omega}\mathcal{O}^{++}(f_i(k))\)， and for all \(k\)\(f_i(k)<_{\mathcal{O}^{++}} 3 \cdot 5^i\).

\(a <_{\mathcal{O}^{++}} b\) and \(b <_{\mathcal{O}^{++}} c\) implies \(a <_{\mathcal{O}^{++}} c\).

It is capable of expressing all ordinals \(<\zeta\)(all eventually writable ordinals)

It is worth noting that the concepts of \(\mathcal{O}^+\) and \(\mathcal{O}^{++}\) cannot be extended to all accidentally writable ordinals， because many ITTMs accidentally write more than one ordinal. Indeed， there exist“universal“ ITTMs accidentally writing all accidentally writable ordinals. No ordinal notation is known that specifically targets all ordinals \(<\Sigma\).

Fundamental sequences

It is possible to define a system of fundamental sequences associated to all ordinals \(<\lambda\) and \(<\zeta\):

For each \(\alpha <\lambda\)， if \(\alpha\) is a limit ordinal， then \(\alpha[n]\) is defined as \(\mathcal{O}^+(f_{i_{\alpha}}(n))\)， where \(i_{\alpha}\) denotes the minimum of the non-empty set \(\{i \in K \mid \mathcal{O}^+(i)=\alpha\}\).

Define \(g(0)= 0\) and \(g(n+1)\) as the notation with smallest \(m \in \mathbb{N}\) such that \(m \in K\) and \(\mathcal{O}^+(g(n))<\mathcal{O}^+(m)\). The fundamental sequence for \(\lambda\)， then， is defined as \(\lambda[n]=\mathcal{O}^+(g(n))\).

Analogously， for each \(\alpha <\zeta\)， if \(\alpha\) is a limit ordinal， then \(\alpha[n]\) is defined as \(\mathcal{O}^{++}(f_{i_{\alpha}}(n))\)， where \(i_{\alpha}\) denotes the minimum of the non-empty set \(\{i \in K \mid \mathcal{O}^{++}(i)=\alpha\}\).

Define \(h(0)= 0\) and \(h(n+1)\) as the notation with smallest \(m \in \mathbb{N}\) such that \(m \in K\) and \(\mathcal{O}^{++}(g(n))<\mathcal{O}^{++}(m)\). The fundamental sequence for \(\zeta\)， then， is defined as \(\zeta[n]=\mathcal{O}^{++}(h(n))\).

The growth rates of functions \(f_\lambda(n)\) and \(f_\zeta(n)\) under this system are supposedly slower than Rayo's function. It is not known how they compare with \(\Sigma_\infty\).

LNGNis the abbreviated name of the large number equal to \(f^{10}(10 \uparrow^{10} 10)\)， where \(f\) is the function defined in First Order Theory beyond Higher Order Set Theory

Definition

Theory

First， a language L is defined by adding a 1-ary function symbol U to the language of first order set theory with countably many variable term symbols and the set-membership relation symbol∈. Define ZFL as the set of L-formulae belonging to the axiom of ZF set theory. Here， the axiom schema of comprehension and replacement in ZFL are parametrized by all L-formulae， i.e. formulae which can include U. Define a formal language L of first order logic by adding countably many constant term symbols， countably many function symbols， countably many relation symbols， and a new 1-ary function symbolΘ to an explicit formalization of L using an explicit Gödel correspondence. Then， we denote by ZFCL the set of L-formulae belonging to the axiom of ZFC set theory， Here， the axiom schema of comprehension and replacement in ZFCL are parametrized by all L-formulae， i.e. formulae which can include the formalization of U， additional constant term symbols， additional function symbols， additional relation symbols， andΘ. We explicitly encode ordinals belowε0 and L-formulae into natural numbers in ZFL， and formalize the Henkin axiom“If there exists an x satisfying P， thenΘ(n) satisfies P“， for each variable term symbol x， each L-formula P with code n formalized into ZFCL by the repetition of successor operations，

Denote by ZFCHL the theory ZFCL augmented by the Henkin axiom schema. The new function symbolΘ plays a role of“a family of Henkin constants“. Please do not confound the base theory ZFL and the formalized theory ZFCHL. Denote by U1 the L-formula“For any ordinalα， U(α)⊨ZFCHL“. Under ZFL augmented by {U1}， U(α) forms a model of ZFCL， and hence forms an L-structure for any ordinalα. We denote by UU(α) the interpretation of U in U(α). We denote by U2 the L-formula“For any ordinalα and anyβ∈α， UU(α)(β)=U(β)“， and by U3 the L-formula“For any ordinalα， there exists an ordinalβ such that |U(α)|=Vβ and for any x∈Vβ and any y∈Vβ， x∈U(α)y is equivalent to x∈y“， where Vβ denotes the von Neumann hierarchy. Define T as the set ZFL∪{U1，U2，U3} of L-formulae.

Embedding

By assigning to each atomic formula xi∈xj in ZFC set theory the L-formula (xi∈xj)∧(xj∈U(0))， the theory T can be regarded as an extension of ZFC set theory. In particular， the set N defined in ZFC set theory is interpreted at U(0) as a term of T， which coincides with the term N defined under ZFL because U(0) is a transitive model of ZFCL. Therefore a large number definable in ZFC set theory is also definable in T， and forms a term which is a large number. Moreover， since L admits countably infinitely many constant term symbols， function symbols， and relation symbols， even a closed formula in a theory given by adding countably many constant term symbols， function symbols， and relation symbols to ZFC set theory can be interpreted at U(0) as a closed formula in T. Further， by assigning to each atomic formula xi∈xj in unsorted MK set theory the L-formula (xi∈xj)∧(xj∈U(0))， the theory T can be regarded as an extension of MK set theory. Roughly speaking， U(0) formally plays a role of the universe of the first order set theory， the power set of U(0) formally plays a role of the universe of the second order set theory and the first order class theory， and its power set formally plays a role of the universe of the third order set theory. Since all of them are included in U(1)， U formally plays a role of a strictly increasing sequence of the universes of higher order set theories. Note that the existence of such a strictly increasing sequence can be constructed in ZFC set theory augmented by Grothendieck universe axiom， which appears in usual mathematics.

Large Number

Explicitly define a surjective map: CNF: N→ε0; i↦CNF(i) using Cantor normal form. For an L-formula P， denote by IsDefinition(P) the L-formula“There exists an x such that P and for any i，(P)[i/x] implies i=x“. Denote by Definable(m，i，P) the L-formula“i∈N， P is an L-formula， U(CNF(i))⊨IsDefinition(P)， and U(CNF(i))⊨(P)[m/x]“， where m in (P)[m/x] is regarded as a parameter in an explicit way. For an n∈N， define f(n) as the sum of m∈N satisfying i∈n， P∈n， and Definable(m，i，P)， In this way， there is finally an uncomputable large function f: N→N; n↦f(n). From here， the Large Number Garden Number is \(f^{10}(10 \uparrow^{10} 10)\).

Definition

Let \([\phi]\) and \([\psi]\) be Gödel-coded formulas and \(s\) and \(t\) be variable assignments. Define \(ext{Sat}([\phi]， s)\) as follows:

∀R {

{

∀[ψ]， s: R([ψ]，t)([ψ]=“xi∈ xj“∧ t(xi)∈ t(xj))

∨([ψ]=“xi = xj“∧ t(xi)= t(xj))

∨([ψ]=“(¬θ)“∧¬R([θ]， t))

∨([ψ]=“(θ∧ξ)“∧ R([θ]， t)∧ R([ξ]， t))

∨([ψ]=“∃xi(θ)“∧∃t′: R([θ]， t′))

(where t′ is a copy of t with xi changed)

}⇒ R([ϕ]，s)

}

Call a natural number \(m\)“Rayo-nameable in \(n\) symbols“ if there is a formula \(\phi(x_1)\) with less than \(n\) symbols and \(x_1\) as its only free variable that satisfies the following properties:

There is a variable assignment \(s\)， assigning \(x_1 := m\)， such that \(ext{Sat}([\phi(x_1)]， s)\).

For any variable assignment \(t\)， if \(ext{Sat}([\phi(x_1)]， t)\)，\(t\) must have \(x_1 = m\).

\(ext{Rayo}(n)\)， then， is the smallest number greater than all numbers Rayo-nameable in \(n\) symbols.

Note that the x_i's in t(xi)∈ t(xj) and t(xi)= t(xj) in the definition of \(ext{Sat}\) above were x_1 in the original definition. Although x_1 is the only free variable allowed to occur in a Rayo-name， the variable assignment for x_i is actually referred to in the satisfaction of∃-fourmulae. Therefore the original definition did not work as Rayo actually intended， and has been updated by Rayo himself.

Uncountable:\(\alpha \geq \omega_1\).

Regular:\(\alpha\) cannot be expressed as the limit of a set \(S\) of smaller ordinals， where the

order type of \(S\) is less than \(\alpha\). This can alternately be phrased as \(\alpha=\)\(extrm{cof}\)\((\alpha)\). From a cardinal perspective， we can informally say that it cannot be divided into smaller set of smaller sets.

Strong limit:\(\alpha =\beth_\gamma\) for a limit ordinal \(\gamma\)， using the following hierarchy of

beth numbers:

\(\beth_0 =\aleph_0\)

\(\beth_{\alpha + 1}= 2^{\beth_\alpha}\)(cardinal exponentiation)

\(\beth_\alpha =\sup\{\beta <\alpha :\beth_\beta\}\)

For a cardinal \(\kappa\)，“\(\kappa\) is a (strong) limit cardinal“ is also equivalent to“\(\kappa\) is closed under the map \(\lambda au.2^au\) with domain and codomain \(extrm{Card}\)“.

If we replace“strong limit cardinal“ with“(weak) limit cardinal“ by replacing“beth numbers“ with“

aleph number

s“)， we get weakly inaccessible cardinals. The distinction between strongly and weakly inaccessible cardinals only matters if we don't assume the

generalized continuum hypothesis

(GCH). Under GCH， all limit cardinals are strong limit cardinals and may just be called limit cardinals.

Let \(A\) be a subclass of On， and \(n\) a natural number. A limit ordinal \(\alpha\) is said to be \(\Pi_n\)-reflecting on \(A\)(resp.\(\Sigma_n\)-reflecting on \(A\)) if for any \(\Pi_n\)(resp.\(\Sigma_n\)) formula \(\phi(x)\) and any \(b\in \)\(L\)\(_\alpha\)， if \((L_\alpha，\in)\models\phi(b)\) there exists a \(\beta \in A \cap \alpha\) such that \(b\in L_\beta\) and \((L_\beta，\in)\models\phi(b)\).[1] A limit ordinal is said to be \(\Pi_n\)-reflecting ordinal (resp.\(\Sigma_n\)-reflecting ordinal) if it is \(\Pi_n\)-reflecting (resp.\(\Sigma_n\)-reflecting) on \(extrm{On}\).

limit ordinal \(\alpha\) is \(\Pi_0\)-reflecting on \(A\) if and only if \(\alpha=\sup(A\cap\alpha)\).[2](this also applies to \(\Pi_1\)-reflecting ordinals by theorems 1.9 (i) and 1.9 (iii) of the same paper)

An ordinal is \(\Pi_n\)-reflecting if and only if it is \(\Sigma_{n+1}\)-reflecting.

A limit ordinal is \(\Pi_2\)-reflecting if and only if it is admissible and greater than \(\omega\).

A limit ordinal is \(\Pi_2\)-reflecting on the class of \(\Pi_2\)-reflecting ordinals if and only if it is recursively Mahlo.

\(\Pi_3\)-reflecting ordinals are also known as recursively weakly compact ordinals.， and in fact the least recursively weakly compact ordinal is much larger than the least recursively Mahlo ordinal， similar to how the least weakly compact cardinal is much larger than the least Mahlo cardinal

\(\Pi_{n+2}\)-reflecting ordinals are considered to be recursive analogues of \(\Pi_n^1\)-indescribable cardinals.

Higher Order Extension

For a natural number \(m\)， an ordinal \(\alpha\) is \(\Pi_n^m\)-reflecting on \(A\) if for every \(\Pi_n^m\) sentence[7]\(\phi\)，\(L_\alpha\models\phi\rightarrow\exists\beta\in A\cap\alpha(L_\beta\models\phi)\).

A limit ordinal is said to be \(\Pi_n^m\)-reflecting ordinal if it is \(\Pi_n^m\)-reflecting on \(extrm{On}\).

Note that this definition involves sentences， i.e. formulae without parameters， unlike the previous definition for \(\Pi_n\)-reflection. The notion of a \(\Pi_n^0\)-reflecting ordinal is equivalent to that of a \(\Pi_n\)-reflecting ordinal.

Properties

An ordinal \(\alpha\) is \(\Sigma_1^1\)-reflecting iff it's both \((\alpha^++1)\)-stable and locally countable.

Properties

If \(\alpha\) is admissible， then there is no \(\Sigma_1\) projection with parameters in \(L_\alpha\) that maps some \(\gamma<\alpha\) to \(\alpha\)[5]

If \(\alpha\) is admissible， then \(L_\alpha\) satisfies a certain \(\Delta_1\)-separation schema[6].

Extension

An extension of the concept of admissibility is \(\Sigma_n\)-admissibility for natural \(n\). An ordinal \(\alpha\) is \(\Sigma_n\)-admissible if \(L_\alpha\) is rudimentary closed and satisfies the \(\Sigma_n\)-collection schema[7]. Note that ordinals which are \(\Sigma_n\)-admissible need not satisfy \(\Sigma_n\)-separation， for example the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation

The present paper studies the relation between admissibility， reflection and partition properties. After introducing basic notions in Section e，Σn admissible ordinals are characterized using reflection properties (Section 2).Σn partition relations are introduced in Section 3. In Sections 3 and 4 connections are explored between partition properties， admissibility and projecta. Several more characterizations of admissibility are given in Section 5 (usingΣn trees) and Section 6 (usingΣn compactness). The ideas developed in Section 5 are used in Section 7 to study the partition relationκ→σn(κ)2.

Hierarchy

For a set \(X\)， we denote by \(\mathcal{P}(X)\) the set of subsets of \(X\)， also called the power set of \(X\). We define the hierarchy \((V_{\alpha})_{\alpha \in extrm{On}}\) of sets in the following transfinite inductive way:\begin{eqnarray*} V_{\alpha}=\bigcup_{\beta <\alpha}\mathcal{P}(V_{\beta})=\bigcup \{\mathcal{P}(V_{\beta})\mid \beta \in \alpha\}\end{eqnarray*} Then the von Neumann universe \(extrm{WF}\) is defined as its union \(\bigcup_{\alpha \in extrm{On}} V_{\alpha}\).

By the definition， for any \(x \in extrm{WF}\)， there exists an ordinal \(\alpha\) such that \(x \in V_{\alpha}\)， and the well-foundedness of \((extrm{On}，\in)\) implies the existence of the minimum \(extrm{rank}(x)\) of such an \(\alpha\)， which is called the rank of \(x\). The assignment of the rank gives a map \(extrm{WF}o extrm{On}\). In \(extrm{ZFC}\) set theory， the equality \(V =extrm{WF}\) implies that the assignment is a map \(extrm{rank}\colon V o extrm{On}\).

The rank is useful because it allows us to convert the universal quantification \(\forall x : P(x)\) for a predicate \(P(x)\) on a set \(x\) into the universal quantification \(\forall \alpha \in extrm{On}:(\forall x :extrm{rank}(x)=\alpha \rightarrow P(x))\) for the predicate \(\forall x :extrm{rank}(x)=\alpha \rightarrow P(x)\) on an ordinal \(\alpha\). Therefore the transfinite induction on \((extrm{On}，\in)\) implies the transfinite induction on \((V，\in)\).

Examples

When \(\alpha = 0 =\emptyset\)， then the set \(\{\mathcal{P}(V_{\beta})\mid \beta \in \alpha\}\) is empty because there is no set \(\beta\) satisfying \(\beta \in \alpha\)， and hence we have \begin{eqnarray*} V_0 =\bigcup \emptyset =\emptyset =\{\}.\end{eqnarray*} When \(\alpha = 1 =\{0\}\)， then the set \(\{\mathcal{P}(V_{\beta})\mid \beta \in \alpha\}\) coincides with the singleton \(\{\mathcal{P}(V_0)\}\)， and hence we have \begin{eqnarray*} V_1 =\bigcup \{\mathcal{P}(V_0)\}=\mathcal{P}(V_0)=\mathcal{P}(\{\})=\{\{\}\}=\{0\}= 1.\end{eqnarray*} When \(\alpha = 2 =\{0，1\}\)， then the set \(\{\mathcal{P}(V_{\beta})\mid \beta \in \alpha\}\) coincides with the pair \(\{\mathcal{P}(V_0)，\mathcal{P}(V_1)\}\)， and hence we have \begin{eqnarray*}&& V_2 =\bigcup \{\mathcal{P}(V_0)，\mathcal{P}(V_1)\}=\mathcal{P}(V_0)\cup \mathcal{P}(V_1)=\mathcal{P}(\{\})\cup \mathcal{P}(\{\{\}\})\\&=&\{\{\}，\{\{\}\}\}=\{0，1\}= 2.\end{eqnarray*} Generally speaking， we have \(\mathcal{P}(V_{\beta})\subsetneq \mathcal{P}(V_{\alpha})\) for any ordinals \(\beta <\alpha\)， and hence \(V_{\alpha+1}=\mathcal{P}(V_{\alpha})\) for any ordinal \(\alpha\). In particular， we have \begin{eqnarray*} V_3 =\mathcal{P}(V_2)=\mathcal{P}(\{\{\}，\{\{\}\}\})=\{\{\}，\{\{\}\}，\{\{\{\}\}\}，\{\{\}，\{\{\}\}\}\}\supsetneq \{\{\}，\{\{\}\}，\{\{\}，\{\{\}\}\}\}= 3.\end{eqnarray*} From this level， the hierarchy contains many sets which are not ordinals.Application

The von Neuman hierarchy is frequently used to define models of set theories. For example， if \(\alpha\) is an inaccessible cardinal， then \(V_{\alpha}\) forms a model of \(extrm{ZFC}\) set theory. By the soundness of first order logic， it implies that \(extrm{ZFC}\) set theory augmented by the existence of an inaccessible cardinal proves the formalised consistency \(extrm{Con}(extrm{ZFC})\) of \(extrm{ZFC}\) set theory itself. Therefore by Goedel's incompleteness theorem， the existence of an inaccessible cardinal is not provable under \(extrm{ZFC}\) set theory as long as it is consistent.

Elementary embedding：

Let \(L\) be a formal language. An elementary embedding \(N o M\) for \(L\)-structures \(N\) and \(M\) is a homomorphism \(h \colon N o M\) of \(L\)-structures such that \(\phi(a_1，\ldots，a_n)\) is true in \(N\) if and only if \(\phi(h(a_1)，\ldots，h(a_n))\) is true in \(M\) for any \(n \in \mathbb{N}\)， any \(L\)-formula \(\phi(x_1，\ldots，x_n)\) with \(n\) free variables \(x_1，\ldots，x_n\)， and any parameter \((a_1，\ldots，a_n)\in N^n\).[1] More generally， for a set \(\Sigma\) of \(L\)-formulae， a \(\Sigma\)-elementary embedding \(N o M\) is a homomorphism \(h \colon N o M\) of \(L\)-structures such that \(\phi(a_1，\ldots，a_n)\) is true in \(N\) if and only if \(\phi(h(a_1)，\ldots，h(a_n))\) is true in \(M\) for any any \(n \in \mathbb{N}\)， any \(L\)-formula \(\phi(x_1，\ldots，x_n)\in \Sigma\) with \(n\)-free variables \(x_1，\ldots，x_n\)， and any parameter \((a_1，\ldots，a_n)\in N^n\). Kanamori has also formalized elementary embeddings for structures that are inner models based off of proper classes[2].

Structure：Let \(M\) and \(N\) be \(L\)-structures. A homomorphism \(M o N\) of \(L\)-structures is a map \(h \colon M o N\) satisfying the following:

For any constant symbol \(c\) in \(L\)，\(h(c^M)= c^N\).

For any function symbol \(f\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in M^n\)，\(h(f^M(x_1，\ldots，x_n))= f^N(h(x_1)，\ldots，h(x_n))\).

For any relation symbol \(r\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in M^n\)，\(r^M(x_1，\ldots，x_n)\) implies \(r^N(h(x_1)，\ldots，h(x_n))\).

For example， a homomorphism of monoids regarded as \(L_{extrm{Grp}}\)-structures is precisely a monoid homomorphism. Since \(L_{extrm{FOST}}\) does not have constant symbols and function symbols， a homomorphism of \(L_{extrm{FOST}}\)-structures is precisely a membership-preserving map.

GOOGOLOGY WIKI

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GOOGOLOGY WIKI

Structure

Let \(L\) be a formal language. An \(L\)-structure is a set (or a class depending on the context)\(M\) equipped with a map \((\bullet)^M\) which assigns constants， functions， and relations on \(M\) to constant term symbols， function symbols， and relation symbols in \(L\) preserving the arity.[1]

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Example

The language \(L_{extrm{Grp}}\) of group theory consists of a single constant symbol \(e\) called the unit and a single binary function symbol \(*\) called the operator. Some author defines \(L_{extrm{Grp}}\) as the formal language consisting only of the operator \(*\)， because the unit of a group is definable in terms of the operator. An \(L_{extrm{Grp}}\)-structure is a set \(G\) equipped with a constant \(e^G \in G\) and a binary map \(*^G \colon G imes G o G\). Specific classes of algebraic systems such as monoids， groups， and Abelian groups can be formulated in terms of axioms on an \(L_{extrm{Grp}}\)-structure.

In googology， we mainly consider the laguage \(L_{extrm{FOST}}\) of first order set theory consisting of a single binary relation symbol \(\in\) called the membership relation. An \(L_{extrm{FOST}}\)-structure is a set \(M\) equipped with a binary relation \(\in^M =: R \subset M imes M\). We traditionally abbreviate \((x，y)\in R\) to \(x \in^M y\).

Homomorphism

Let \(M\) and \(N\) be \(L\)-structures. A homomorphism \(M o N\) of \(L\)-structures is a map \(h \colon M o N\) satisfying the following:

For any constant symbol \(c\) in \(L\)，\(h(c^M)= c^N\).

For any function symbol \(f\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in M^n\)，\(h(f^M(x_1，\ldots，x_n))= f^N(h(x_1)，\ldots，h(x_n))\).

For any relation symbol \(r\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in M^n\)，\(r^M(x_1，\ldots，x_n)\) implies \(r^N(h(x_1)，\ldots，h(x_n))\).

For example， a homomorphism of monoids regarded as \(L_{extrm{Grp}}\)-structures is precisely a monoid homomorphism. Since \(L_{extrm{FOST}}\) does not have constant symbols and function symbols， a homomorphism of \(L_{extrm{FOST}}\)-structures is precisely a membership-preserving map.

Substructure

Let \(M\) be an \(L\)-structure. An \(L\)-substructure of \(M\) is a subset \(N \subset M\) satisfying the following:

For any constant symbol \(c\) in \(L\)，\(c^M \in N\).

For any function symbol \(f\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in N^n\)，\(f^M(x_1，\ldots，x_n)\in N\).

In particular，\(N\) forms an \(L\)-structure with respect to the assigment \((\bullet)^N\) defined by the following:

For any constant symbol \(c\) in \(L\)，\(c^N := c^M\).

For any function symbol \(f\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in N^n\)，\(f^N(x_1，\ldots，x_n):= f^M(x_1，\ldots，x_n)\).

For any relation symbol \(r\) in \(L\) with arity \(n \in \mathbb{N}\) and any \((x_1，\ldots，x_n)\in M^n\)，\(r^N(x_1，\ldots，x_n)\) is equivalent to \(r^M(x_1，\ldots，x_n)\).

By the definition， the inclusion map \(N \hookrightarrow M\) forms a homomorphism of \(L\)-structures.

For example， an \(L_{extrm{Grp}}\)-substructure of a monoid regarded as an \(L_{extrm{Grp}}\)-structure is precisely a submonoid. Since \(L_{extrm{FOST}}\) does not have constant term symbols and function symbols， an \(L_{extrm{FOST}}\)-substructure of an \(L_{extrm{FOST}}\)-structure \(M\) is precisely a subset of \(M\).