The irrational arrow notation (無理矢印表記 in Japanese) is a variant of the arrow notation defined by a Japanese Googology Wiki user Jason.[1][2] It is an irrational analogue of another variant called the rational arrow notation defined by Jason.[1]
Definition[]
We denote by \(\mathbb{N}_{>0}\) the set of positive integers, and by \(\mathbb{R}_{>0}\) the set of positive numbers. We define a map \begin{eqnarray*} \mathbb{N}_{>0}^2 \times \mathbb{R}_{>0}^2 & \to & \mathbb{N}_{>0} \\ ((a,b),(c,\Delta)) & \mapsto & a \uparrow^{c(\Delta)} b \end{eqnarray*} as the unique one satisfying the following relations:
- If \(b = 1\) or \(c \leq 1\), then \(a \uparrow^{c(\Delta)} b := a^b\).
- If \(b > 1\) and \(c > 1\), then \(a \uparrow^{c(\Delta)} b := a \uparrow^{c'(\Delta')} (a \uparrow^{c(\Delta)} (b-1))\), where \(c'\) and \(\Delta'\) are defined in the following way:
\begin{eqnarray*} c' & := & c \log_{\Delta'+1}(\Delta') \\ \Delta' & := & \left\{ \begin{array}{ll} a \uparrow^{c-1(\Delta)} b & (\Delta \notin (1,a \uparrow^{c-1(\Delta)} b)) \\ \Delta & (\Delta \in (1,a \uparrow^{c-1(\Delta)} b)) \end{array} \right. \end{eqnarray*} Since it is defined on an uncountable set, it is uncomputable by the definition of the computability.
Analysis[]
The order type of the structural ordering of the irrational arrow notation is \(\omega 2\).[3] In order words, there exists a well-founded strict partial ordering \(<\) on \(\mathbb{N}_{>0}^2 \times \mathbb{R}_{>0}^2\) such that the two characteristic relations above refer only to the values of the irrational arrow notation whose imput is smaller than \(((a,b),(c,\Delta))\) with respect to \(<\), and \(\omega 2\) is the least ordinal type of such a well-founded strict partial ordering. In particular, the totality of the irrational arrow notation is provable under a weak arithmetic which can prove the well-foundedness of \(\omega 2\).
On the other hand, the growth rates of the \(1\)-ary function \(x \uparrow^{x(x)} x\) and \(x \uparrow^{x(0)} x\) on \(\mathbb{N}\) are bounded by \(\omega+1\) in Wainer hierarchy.[4][5]
Significance[]
By the argument above, the irrational arrow notation gives a non-trivial example of a large function which can be approximated to an ordinal in FGH with respect to a canonical system of fundamental sequences smaller than the ordinal type of the structural ordering. It is significant because many googologists believe that large functions can be approximated to the ordinal type of the structural ordering in FGH with respect to a canonical system of fundamental sequences. For example, many googologists state that TREE grows as fast as \(f_{\vartheta(\Omega^{\omega} \omega)}(n)\) as if it had already been verified, but what has been actually verified is that the ordinal type of its structural ordering. As the analysis of the irrational arrow notation implies, such a reasoning is critically incorrect.
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
Sources[]
- ↑ 1.0 1.1 Jason, 有理矢印表記と無理矢印表記, Google Document.
- ↑ The user page of Jason in Japanese Googology Wiki.
- ↑ p進大好きbot, 無理矢印表記の全域性, a Japanese Googology Wiki user blog.
- ↑ p進大好きbot, 無理矢印表記の評価その１, a Japanese Googology Wiki user blog.
- ↑ p進大好きbot, 無理矢印表記の評価その２, a Japanese Googology Wiki user blog.