- Not to be confused with Array of.
Ampersand Notation (&N) is a notation for large numbers made by Wikia user CompactStar.[1][2][3] The notation is currently composed of 3 parts, which are as follows:[3]
- Basic Ampersand Notation (B&N)
- Extended Ampersand Notation (X&N)
- Nested Ampersand Notation (N&N)
Ampersand Notation up to Extended Ampersand Notation is well-defined, but unfortunately, several versions of Nested Ampersand Notation are ill-defined, as is explained in #Issues section and #Issues_2 section. Even if the creator will update the definition, this does not automatically mean that it is known to be well-defined, but it would be considered uncertain unless a proof of well-definedness or ill-definedness was found. The creator plans to create the following part in the future:[3]
- Hyper-Nested Ampersand Notation (HN&N)
Basic Ampersand Notation[]
Basic Ampersand Notation (B&N) is the first part of Ampersand Notation.[4]
Definition[]
Let @ represent any sequence of ampersand symbols (&).
A valid expression in B&N is of the form a[@b], where a and b are non-negative integers. b is optional and may be removed.
B&N outputs a non-negative integer for every valid expression, determined using this set of rules:
- a[] = aa
- a[@0]= a[@]
- b > 0: a[@b]= a[@b - 1][@b - 1]...[@b - 1][@b - 1], with a copies of "[@b - 1]"
- a[@&]= a[@a]
Examples[]
2[1]
= 2[0][0]
= 2[][]
= 2[]2[]
= (22)(22)
= 44
= 256
2[&1]
= 2[&0][&0]
= 2[&][&]
= 2[2][&]
= 2[1][1][&]
= 2[0][0][1][&]
= 2[][][1][&]
= (22)(22)[1][&]
= 44[1][&]
= 256[1][&]
= ...
Comparison with fast-growing hierarchy[]
Here, a is a non-negative integer. All inequalities presented only hold true for sufficiently large a, rather than any a.
- a[] ≥ f2(a)
- a[1] ≥ f3(a)
- a[2] ≥ f4(a)
- a[3] ≥ f5(a)
- a[4] ≥ f6(a)
- a[&] ≥ fω(a)
- a[&1] ≥ fω+1(a)
- a[&2] ≥ fω+2(a)
- a[&3] ≥ fω+3(a)
- a[&&] ≥ fω×2(a)
- a[&&1] ≥ fω×2+1(a)
- a[&&&] ≥ fω×3(a)
- a[&&&&] ≥ fω×4(a)
This notation has a limit ordinal of \(\omega^{2}\) in the fast-growing hierarchy.
Extended Ampersand Notation[]
Extended Ampersand Notation (X&N) is the second part of Ampersand Notation.[5]
Definition[]
Define an ampersand-structure as an expression of the form &x, where x is a non-negative integer. x is optional and may be removed.
Let @ represent any sequence of ampersand-structures.
A valid expression in X&N is of the form a[@b], where a and b are non-negative integers. b is optional and may be removed. X&N outputs a non-negative integer for every valid expression, determined using this set of rules:
- a[] = aa
- a[@0]= a[@]
- b > 0: a[@b]= a[@b - 1][@b - 1]...[@b - 1][@b - 1], with a copies of "[@b - 1]"
- a[@&] = a[@a]
- a[@&0]= a[@&]
- x > 0: a[@&x]= a[@&x - 1&x - 1...&x - 1&x - 1] with a copies of "&x - 1"
Examples[]
2[&10]
= 2[&1]
= 2[&0&0]
= 2[&&]
= 2[&2]
= 2[&1][&1]
= 2[&0][&0][&1]
= 2[&][&][&1]
= ...
2[&2&1]
= 2[&2&0][&2&0]
= 2[&2&][&2&]
= 2[&22][&2&]
= 2[&21][&21][&2&]
= 2[&20][&20][&21][&2&]
= 2[&2][&2][&21][&2&]
= 2[&1&1][&2][&21][&2&]
= ...
Comparison with fast-growing hierarchy[]
Here, a is a non-negative integer. All inequalities presented only hold true for sufficiently large a, rather than any a.
- a[&1] ≥ fω2(a)
- a[&11] ≥ fω2+1(a)
- a[&1&] ≥ fω2+ω(a)
- a[&1&1] ≥ fω2×2(a)
- a[&2] ≥ fω3(a)
- a[&2&2] ≥ fω3×2(a)
- a[&3] ≥ fω4(a)
- a[&4] ≥ fω5(a)
Nested Ampersand Notation[]
Nested Ampersand Notation (N&N) is the third part of Ampersand Notation.[6][7]
Although the creator states that it is comparable to \(\varepsilon_0\) with respect to the Wainer hierarchy without a source or a proof, the notation is ill-defined as is explained in #Issues section and #Issues_2 section. The reader should be careful that analysis without a proof might be incorrect, even when the one who writes it regard the result as an obvious fact which does not require a proof.
Old definition[]
This is the original version.[6] Define an ampersand-structure as an expression of the form &@x, where @ is any sequence of ampersand-structures, and x is a non-negative integer. x is optional and may be removed.
The number of layers in an ampersand-structure is the number of nested subscripts, e.g. & has 0 layers, &5 has 1 layer, and &&7 has 2 layers.
A valid expression in N&N is of the form a[@b], where @ is any sequence of ampersand structures, and a and b are non-negative integers. b is optional and may be removed.
N&N is intended to output a non-negative integer for every valid expression, determined using this set of rules:
- a[] = aa
- a[@0] = a[@]
- b > 0: a[@b]= a[@b - 1][@b - 1]...[@b - 1][@b - 1], with a copies of "[@b - 1]"
- # is another ampersand-structure sequence: a[@&&...�] with x layers in the ampersand-structure after @ = a[@&&...&#] with x layers in the ampersand-structure after @ (x - 1 if # is empty and x > 0)
- # is another ampersand-structure sequence: a[@&&...&#&] with x layers in the ampersand-structure after @ = a[@&&...&#a] with x layers in the ampersand-structure after @
- # is another ampersand structure sequence, y > 0: a[@&&...&#x] with y layers in the ampersand-structure after @ = a[@&&...&#x - 1&#x - 1...&#x - 1&#x - 1] with y layers in the ampersand-structure after @ and a copies of "#x - 1"
Issues[]
The notion of "layer" is just described in this ambiguous way using the unformalised word "number of nested subscripts" with finitely many examples,[6] and hence is not formalised. For example, the layer of \(\&_{\& \&}\) and \(\&_{\&_{\&} \&}\) are quite ambiguous.
Moreover, seriously ambiguos ellipses occur in the set of rules. Say, can \(\&_{\&_{\cdots \&}}\) stand for \(\&\) or \(\&_{\& \&}\)? Maybe no, because they are not obtained as \(\&_{\&_{\alpha \&}}\) for some string \(\alpha\).
Further, the set of rules is incomplete, because there is no rule applicable to valid expressions like \(2[\&]\) and \(2[\&_{\& \&}])\).
In addition, there are multiple rules applicable to a single expression. Say, both of the fourth and sixth rules are applicable to \(2[\&_{\&_{\&_{0}}}]\).
- The fourth rule returns the ambiguous expression "\(2[\&_{\cdots_{\&_{\#}}}]\) with \(3\) layers in the ampersand-structure after the empty string (\(2\) if \(\#\) is empty)" applicable to the case where \(\#\) is actually empty. What does "\(2\) if \(\#\) is empty" mean here? Is it an exceptional definition of the return value of the \(3\)-layered expression \(2[\&_{\&_{\&_{\&}}}]\)? Or is it an exceptional redefinition of the layer of the expression in \(2[\&_{\&_{\&_{\&}}}]\), which is originally \(3\)-layered? Or is it an exceptional definition of the layer of the expression in the expansion, i.e. \(2[\&_{\&_{\&}}]\) instead of \(2[\&_{\&_{\&_{\&}}}]\)? Although such a confusion could be easily removed if one defined rules in a recursive way, but since the original definition is defined in terms of natural language instead of recursion, such a serious ambiguity occurs.
- The sixth rule returns the ill-formed expression \(2[\&_{\&_{\&_{-1} \&_{-1}}}]\).
The creator stated that the growth rate was comparable to \(\varepsilon_0\) in Wainer hierarchy without a proof. However, since the notation is unformalised, the creator's statement on the growth rate does not mathematically make sense.
New definition[]
Later the creator updated the definition of the notation to fix issues present in the old definition.[7]
Define an ampersand-structure as an expression of the form &@x, where @ is any sequence of ampersand-structures, and x is a non-negative integer. x is optional and may be removed.
A valid expression in N&N is of the form a[@b], where @ is any sequence of ampersand structures, and a and b are non-negative integers. b is optional and may be removed.
N&N outputs a non-negative integer for every valid expression, determined using this set of rules:
- a[] = aa
- a[@0]= a[@]
- b > 0: a[@b]= a[@b - 1][@b - 1]...[@b - 1][@b - 1], with a copies of "[@b - 1]"
- # is another ampersand-structure sequence: a[@S(x)#0] = a[@S(x)#]
- # is another ampersand-structure sequence: a[@S(x)#&] = a[@S(x)#a]
- # is another ampersand structure sequence, y > 0: a[@S(x)&#y] = a[@S(x)&#y - 1&#y - 1...&#y - 1&#y - 1] with a copies of "&#y - 1"
In the above definition, the string S(x) for a non-negative integer x is defined as follows:
- S(0) = ø (empty string)
- x > 0: S(x) = &S(x - 1)
For example, S(1) = &, S(2) = &&, and S(3) = &&&.
Note: ø@ where @ is an ampersand-structure sequence is simply @.
Issues[]
Since expressions in this notation is not a simply inlined string, i.e. a string possibly admitting a subscript, the concatenation of two strings in a subscript is ambiguous. Namely, \(S(2)_1\) might be either \(\&_{\& 1}\) or \(\&_{\&_1}\).
Similar to the first definition, \(y\) is undefined and unquantified.
Further, the new set of rules is again incomplete. For example, there is no rule applicable to \(\&_{\& \&_{\& \&}}\). Of course, this is not the unique example to which there is no rule applicable to.
The creator again states that the growth rate is comparable to \(\epsilon_0\) in Wainer hierarchy without a proof. However, since the notation is incomplete, the creator's statement on the growth rate does not mathematically make sense.
Examples[]
Since versions of the notation are ill-defined, these results are just expectations by the creator. The reader should be careful that even if the creator will update the definition and will not clarify that results are just expectations, examples without a proof might be incorrect.
2[&&1]
= 2[&&][&&]
= 2[&2][&&]
= 2[&1][&1][&&]
= 2[&0][&0][&1][&&]
= 2[&][&][&1][&&]
= 2[2][&][&1][&&]
= 2[1][1][&][&1][&&]
= 2[][][1][&][&1][&&]
= 4[][1][&][&1][&&]
= 256[1][&][&1][&&]
= ...
Relation to another notation[]
A Japanese Googology Wiki user Nayuta Ito proved that for any non-negative integers a and b, a[b] in this notation is in fact equal to a in the (b+3)-agon in Steinhaus-Moser Notation under the assumption that these two notations are based on the same formulation of \(0^0\). This is easily proven by the mathematical induction on b:
- When b=0, both a[0] and a in the triangle are equal to a^a.
- Suppose a[k] is equal to a in the (k+3)-agon. Then, when b=k+1:
- a[k+1] = a[k][k]...[k] with a copies of [k] = a in a copies of (k+3)-agons = a in the (k+4)-agon
The last equation holds from the definition of Steinhaus-Moser Notation.
This property shows that Mega is equal to 2[2]=256[1].
Sources[]
- ↑ CompactStar. Ampersand Notation. Retrieved Jul 23, 2022.
- ↑ CompactStar. Ampersand Notation. Retrieved Jul 24, 2022.
- ↑ 3.0 3.1 3.2 CompactStar. Ampersand Notation. Retrieved Jun 17, 2023.
- ↑ CompactStar. Basic Ampersand Notation. Retrieved Jul 23, 2022.
- ↑ CompactStar. Extended Ampersand Notation. Retrieved Jul 24, 2022.
- ↑ 6.0 6.1 6.2 CompactStar. Nested Ampersand Notation. Retrieved Jun 17, 2023.
- ↑ 7.0 7.1 CompactStar. Nested Ampersand Notation. Retrieved Jun 20, 2023.
See also[]
Unol group:
Duol group:
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