Side nesting

Side nesting (横ネスト in Japanese) is a method to generate a googological system introduced by a Japanese googologist mrna. It admits a generalisation called shifting definition.

Feature
One of the most characteristic and confusing feature of side nesting is that even if a notation employs \(+\) as a \(2\)-ary function symbol, it does not necessarily work as the addition. Namely, even if a valid expression \(a\) corresponds to a countable ordinal \(\alpha\) and \(a + a\) is also a valid expression, \(a + a\) does not necessarily corresponds to \(\alpha + \alpha\). The characteristic feature of side nesting on \(+\) is inherited by its generalisation shifting definition. If a notation based on side nesting does not employ \(+\), then a separator plays the role analogous to \(+\) in a notation based on side nesting.

Examples
According to mrna, side nesting is a method which can be found in many notations, while there are few notations which intensionally focus on side nesting. The following three systems are intended to be based on side nesting: Although none of them has been formalised, many Japanese googologists pay attention to their intended behaviour, because they are expected to be poweful. For example, one version of Y function is expected to roughly reach (0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1) in Bashicu matrix system version 2.3 under the assumption of its termination.
 * SSAN created by mrna
 * S-σ created by mrna
 * Y function, which has at least 10 versions, created by a Japanese googologist Yukito

S-σ
S-σ is a notation introduced as a system purely based on side nesting, while SSAN also employs another strategy than side nesting. Here is an explanation of how S-σ is intended to work. Let \(T\) denote the recursive set of formal strings given in the recursive way: The set of valid expressions in Sσ is an element of \(T\), but the converse does not necessarily hold.
 * 1) \(0,\Sigma \in T\)
 * 2) For any \((i,a) \in \mathbb{N} \times T\), \(\sigma i(a) \in T\).
 * 3) For any \((a,b) \in T \times T\), \(a + b \in T\).

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 \(\varepsilon_0 + \varepsilon_0\).

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 \(\Sigma\).

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 \(\times \omega\).

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.