S-σ

S-σ (or Sσ) is a notation introduced by a Japanese Googology Wiki user mrna,The user page of mrna in Japanese Googology Wiki.mrna, Yｶﾞﾁ解析, Google Spreadsheet.mrna, Sσ関数の一覧と展開, Google Spreadsheet., 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.超電磁トワイライトΣ in デュエル・マスターズ Wiki.
== Explanation ==
Let \(T\) denote the recursive set of formal strings given in the recursive way:
# \(0,\Sigma \in T\)
# For any \((i,a) \in \mathbb{N} \times T\), \(\sigma i(a) \in T\).
# 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 ==
== See also ==
* side nesting
Category:Functions
Category:Side nesting
Category:Unformalised
Category:Googology in Asia
Category:Higher computable level