S-σ (or ) is a notation introduced by a Japanese Googology Wiki user mrna,, and is a notation introduced as a system purely based on side nesting, while SSAN also employs another strategy than side nesting. Although S-σ seems to be the simplest system based on side nesting, it is expected to be essentially as strong as other side nesting notations SSAN and Y function according to mrna. It has not been formalised yet.

It uses a constant term symbol $$\Sigma$$, which solely plays a role similar to $$\Omega$$, $$I$$, $$M$$, and so on in an ordinal collapsing function, but the actual behaviour is quite different from each of them. The symbol $$\Sigma$$ comes from a card 超電磁トワイライトΣ in a Japanese famous card game デュエル・マスターズ.

## Explanation

Let $$T$$ denote the recursive set of formal strings given in the recursive way:

1. $$0,\Sigma \in T$$
2. For any $$(i,a) \in \mathbb{N} \times T$$, $$\sigma i(a) \in T$$.
3. For any $$(a,b) \in T \times T$$, $$a + b \in T$$.

A valid expression in S-σ is an element of $$T$$, but the converse does not necessarily hold.

### σ0

The function symbol $$\sigma 0$$, which is often abbreviated to $$\sigma$$, plays a role analogous to the function $$x \mapsto \omega^x$$. The constant term symbol $$\Sigma$$ was originally denoted by $$\sigma(\Omega)$$, and roughly indicates the current level of the nesting. The term $$\sigma 0 (0)$$ plays a role of the successor of $$0$$, and hence is often abbreviated to $$1$$.

The limit of valid expressions constructed from $$0$$, $$+$$, and $$\sigma 0$$ is $$\Sigma$$, and admits a fundamental sequence given as $$\sigma 0(\cdots \sigma 0(0) \cdots)$$. The set of valid expressions below $$\Sigma$$ seems to be expected to be isomorphic to the ordinal notation given by Cantor normal forms. In particular, $$\Sigma$$ is intended to correspond to $$\varepsilon_0$$. In this realm, $$+$$ plays the obvious role of the addition. On the other hand, $$\Sigma + \Sigma$$ is the limit of Sσ, and is intended to be much greater than $$\varepsilon_0 + \varepsilon_0$$.

### σ1

The first occurrence of $$\sigma 1$$ is $$\Sigma + \sigma 1(\Sigma)$$, which is intended to correspond to $$\varepsilon_0 + \varepsilon_0$$, and admits a fundamental sequence given as $$\Sigma + \sigma 0(\cdots \sigma 0(0) \cdots)$$. It is not surprising that $$\Sigma + \sigma 1(\Sigma) + \sigma 1(\Sigma)$$ is intended to correspond to $$\varepsilon_0 + \varepsilon_0 + \varepsilon_0$$, and $$\sigma 1(\Sigma)$$ always works as the limit of valid expressions below $$\Sigma$$.

### σ2

The first occurrence of $$\sigma 2$$ is $$\Sigma + \sigma 2(\Sigma)$$, which seems to be intended to correspond to $$\varepsilon_1$$, and admits a fundamental sequence given as $$\Sigma + \sigma 1(\Sigma + \cdots \sigma 1(\Sigma + \sigma 1(\Sigma))\cdots$$. The function symbol $$\sigma 1$$ restricted to valid expressions below $$\Sigma + \sigma 2(\Sigma)$$ also plays a role analogous to the function $$x \mapsto \omega^x$$. The difference between $$\sigma 0$$ and $$\sigma 1$$ restricted to this realm is that it is not allowed to consider the expression $$\sigma 0(\Sigma)$$ or an expression of the form $$\sigma 0(\Sigma + a)$$. For example, $$\Sigma + \sigma 1(\Sigma + 1)$$ is a valid expression which is intended to correspond to $$\varepsilon_0 \times \omega$$. Similar to Buchholz's function, $$+1$$ in $$\sigma 1$$ is intended to play the role analogous to $$\times \omega$$.

### σ3

The first occurrence of $$\sigma 3$$ is $$\Sigma + \sigma 3(\Sigma)$$, which seems to be intended to correspond to Bachmann-Howard ordinal, and admits a fundamental sequence given as $$\Sigma + \sigma 2(\Sigma + \cdots \sigma 2(\Sigma + \sigma 2(\Sigma))\cdots$$. The function symbol $$\sigma 2$$ restricted to valid expressions below $$\Sigma + \sigma 3(\Sigma)$$ plays a role similar to Buchholz's function restricted to ordinals below $$\Omega_2$$.

It is surprising that $$\Sigma + \sigma 3(\Sigma) + \sigma 3(\Sigma)$$ is intended to correspond to $$\psi(\Omega_3)$$ and $$\Sigma + \sigma 3(\Sigma) + \sigma 3(\Sigma) + \sigma 3(\Sigma)$$ is intended to correspond to $$\psi(\Omega_4)$$ with respect to an undefined ordinal collapsing function $$\psi$$. In other words, the addition of $$\sigma 3(\Sigma)$$ is intended to correspond to the increment of the index $$x$$ in $$\psi(\Omega_x)$$. As a result, $$\Sigma + \sigma 3(\Sigma + 1)$$ is intended to correspond to $$\psi(\Omega_{\omega})$$. Moreover, $$\Sigma + \sigma 3(\Sigma + \sigma 3(\Sigma))$$ is intended to correspond to $$\psi(I)$$, where $$I$$ is the least weakly inaccessible cardinal, and expressions with $$\sigma 4$$ are intended to go beyond $$\psi(\Omega_{M+1})$$, where $$M$$ is the least weakly Mahlo cardinal. The behaviour of $$\sigma 3$$ is intended to be much more complicated than that of $$\sigma 2$$, and is one of the biggest factor which makes S-σ difficult to be formalised. According to mrna, one of the biggest problem to formalise S-σ is called 3(2(3(2(3(3)))))問題 (English: 3(2(3(2(3(3))))) problem).

## Sources

1. The user page of mrna in Japanese Googology Wiki.
4. 超電磁トワイライトΣ in デュエル・マスターズ Wiki.

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