S-σ (or Sσ) is a notation introduced by a Japanese Googology Wiki user mrna,[1][2][3], 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.[4]
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[]
- ↑ 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.
- ↑ 超電磁トワイライトΣ in デュエル・マスターズ Wiki.
See also[]
By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea
By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system
By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation
By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud
See also: Template:Googology in Asia