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## 定義

は可算な0でない極限順序数（従って）とする．このとき，の共終数はである．すなわち，以下を満たす順序数の列が存在する．

1. 順序数列は狭義単調増加．
すなわち，任意の自然数に対して
2. の極限は
すなわち，

このような順序数列を，に収束する単調-列と呼ぶ．

を適当な大きさの可算な順序数とおく．いま，何らかの方法でである極限順序数に対して，それぞれに収束する単調-列を1つ定めたとする．このとき，それぞれの数列をと書き，の基本列と呼ぶ．

## 拡張

• 任意のに対しである．
• を含む最小の順序数である．

この定義に則ると，の基本列は空写像である恒等写像に限られ，後続順序数の基本列は写像（つまりを定義域としを終域とする写像であって定義域の唯一の元に送る唯一のもの）に限られる．

### 例

カッコの中はその定義で基本列が与えられた可算順序数の上限と基本列が与えられた順序数全体の集合を表す．

## 展開規則

As we clarified above, the notion of an expansion rule of a general notation does not admit an agreed-upon mathematical formulation. One typical mistake in googology is the confounding of a notation with an expansion rule with an ordinal notation. As a Japanese Googology Wiki user p進大好きbot has pointed out, those two notions are not equivalent at any rate.[1]

One important aspect of an ordinal notation is that every term canonically corresponds to the ordinal given as the order type. On the other hand, a term in a notation with an expansion rule does not necessarily correspond to an ordinal in a way compatible with the expansion. However, the correspondence is at least unique under a suitable formulation.

For example, the following is a formulation given by p進大好きbot.[2] Let $$T$$ be a set. An expansion rule on $$T$$ is a map $$[] \colon T \times \mathbb{N} \to T$$. Then the map $$o \colon T \to \omega_1$$ satisfying the following is unique under mild assumptions:

1. If $$t = t[n]$$ for any $$n \in \mathbb{N}$$, then $$o(t) = 0$$.
2. If $$t \neq t[0] = t[n]$$ for any $$n \in \mathbb{N}$$, then $$o(t) = o(t[0])+1$$.
3. Otherwise, then $$o(t) = \sup_{n \in \mathbb{N}} o(t[n])$$.

We note that $$o$$ is not even unique if we do not assume any mild assumptions. A similar chracterisation works also when we deal with an expansion rule formulated as a partial function $$[] \colon T \times \mathbb{N} \to T$$. For example, replace the conditions in the following way:

1. If $$(t,0)$$ does not belong to the domain of $$[]$$, then $$o(t) = 0$$.
2. If $$(t,n)$$ belongs to the domain of $$[]$$ for any $$n \in \mathbb{N}$$ and $$t[0] \neq t[1]$$, then $$o(t) = \sup_{n \in \mathbb{N}} o(t[n])$$.
3. Otherwise, then $$o(t) = o(t[0])+1$$.

It is also a typical mistake that $$o$$ always exists, and there are many typical circular logics in googology on the termination of a given notation based on the wrong belief.[2][3]

Assuming the existence of $$o$$ for a given notation, we can consider that a term "expresses" an ordinal. In that sense, we refer to "the corresponding ordinal" for a term in a notation which is not an ordinal notation, e.g. バシク行列システム.