Collapsing-E Notation

View full site to see MathJax equation

This is an article which refers to a personal web site as the first source of the notion in the title. Please be careful as a personal web site is not usually a peer-reviewed source unlike academic journals, and hence might include wrong information. For more details, see this document.

The Collapsing-E Notation (&E) is a notation created by wiki user DeepLineMadom^[1] that is intended to fix the awkwardness of the fundamental sequences for Saibian's Hyper-Hyper-Extended Cascading-E notation, within the greater Extensible-E System. It introduces a new high-ranking separator, the ampersand (\(\&\)), which diagonalizes over nests of #{#}#.

The notation is currently incomplete, as per the creator.

Definition[]

Formal definition[]

Let E a1 %(1) a2 %(2) a3 %(3) ... a[n-1] %(n-1) an

where a1 ~ an, and n are natural numbers (positive integers) and %(1) ~ %(k-1) are k-1 delimiters from the set x^

and k ∈ {1, 2, 3, 4, 5, 6, ...} be defined as follows.

Let:

m denote a_k-1
n denote a_k
@ denote the unchanged remainder of an expression
% denote any portion of a delimiter we chose to omit (formerly using & since the symbol & is now used as the hyper-delimiter in the curly brackets {} to diagonalize over uncountable limits of the delimiter structures inside {}).
and %[n] denote the nth member of the fundamental sequence defined for the delimiter %.

Base rules[]

Rule 1. Base rule. For k = 1 (with only one argument and no hyperions), we have E[x]n = x^n
Rule 2. Decomposition rule. For L%(n-1) ≠ #^n (the last cascader is not in the form of #^n):
E[x]@a%b = E[x]@a%[b]a (@ indicates the unchanged remainder of the expression and %[b] is the fundamental sequence of %)
Rule 3. Termination rule. For L%(n-1) = #^n (the last cascader is in the form of #^n), and the last argument is 1, it can be removed:
E[x]@a%1 = E[x]@a
Rule 4. Expansion rule. For L%(n-1) = #^n and %k ≠ # (the last cascader is in the form of #^n but not the single hyperion):
E[x]@a%*#b = E[x]@a%a%*#(b-1)
Rule 5. Recursion rule. Otherwise:
E[x]@a#b = E[x]@(E[x]@a#(b-1))

Decomposition rules[]

In addition the set of legal delimiters must be defined. Let & be the set of legal delimiters in xE^. The set is defined recursively:

We regard all elements of % in {n} as "transfinite n" (including countable and uncountable delimiters), for % ≥ #.

I. # is an element of %
II. If a,b are elements of % then a*b is an element of %
III. If a,b are elements of % then (a){n}(b) for "n ≥ 1 or transfinite n" is an element of %
IV. If a,b are elements of % and c is an element of %+, then (a){n}(b)>(c) for "n ≥ 1 or transfinite n" is an element of % for n>1.
V. If a is an element of % then a is an element of %+
VI. If a,b are elements of %+ then a+b is an element of %+

Lastly the decompositions of decomposable-delimiters must be defined. A delimiter, %, is decomposable (% is a member of %decomp) iff L(%) ≠ #^n.

Also, the decompositions of decomposable hyper-delimiters using "&" must be defined. A delimiter containing "&", %, is also decomposable (% is a member of %hdecomp, also known as an alternative of %decomp), if and only if L(%) ≠ #^n in {}.

The decompositions are defined as follows:

Case I. L = a^b, where a, b ∈ %:
- A. When b = #:
  - I.A.1. %(a)^#[1] = %a
  - I.A.2. %(a)^#[n] = %a*(a)^#[n-1]
- B. When b = k*#:
  - I.B.1. %(a)^(k*#)[1] = %(a)^(k)
  - I.B.2. %(a)^(k*#)[n] = %(a)^(k)*(a)^(k*#)[n-1]
- C. When b ∈ %decomp:
  - %(a)^(b)[n] = %(a)^(b[n])
Case II. L = a{p}b, where a, b ∈ %, and (p > 1 or 0 < p < # in m+p, and m ≥ #) (where p is copies of multiple carets or a successor ordinal):
- A. When b = #:
  - II.A.1. %(a){p}#[1] = %a
  - II.A.2. %(a){p}#[n] = %(a){p-1}((a){p}#[n-1])
- B. When b = k*#:
  - II.B.1. %(a){p}(k*#)[1] = %a
  - II.B.2. %(a){p}(k*#)[n] = %(a){p-1}(k)>((a){p}(k*#)[n-1])
- C. When b ∈ %decomp: %(a){p}(b)[n] = %(a){p}(b[n])
Case III. L = a{p}b>c, where a, b ∈ %, c ∈ %+, and (p > 1 or 0 < p < # in m+p, and m ≥ #) (p is copies of multiple carets or a successor ordinal):
- A. When c = #:
  - III.A.1. %(a){p}(b)>#[1] = %(a){p}(b)
  - III.A.2. %(a){p}(b)>#[n] = %((a){p}(b)>#[n-1]){p}(b)
- B. When c = k+#:
  - III.B.1. %(a){p}(b)>(k+#)[1] = %((a){p}(b)>(k)){p}(b)
  - III.B.2. %(a){p}(b)>#[n] = %((a){p}(b)>(k+#)[n-1]){p}(b)
- C. When c ∈ %decomp: %(a){p}(b)>(c)[n] = %(a){p}(b)>(c[n])
- D. When c = k+d where k ∈ &+ and d ∈ %decomp: %(a){p}(b)>(k+d)[n] = %(a){p}(b)>(k+d[n])
- E. When c = d*# where d ∈ %:
  - III.E.1. %(a){p}(b)>(d*#)[1] = %(a){p}(b)>(d)
  - III.E.2. %(a){p}(b)>(d*#)[n] = %(a){p}(b)>(d+d*#)[n-1]
- F. When c = k+d*# where k ∈ %+ and d ∈ %:
  - III.F.1. %(a){p}(b)>(k+d*#)[1] = %(a){p}(b)>(k+d)
  - III.F.2. %(a){p}(b)>(k+d*#)[n] = %(a){p}(b)>(k+d+d*#)[n-1]
Case IV. L = a{p}b, where L(p) ≠ m (L denotes the last sum of delimiters in {}, and m denotes the natural number):
- A. When p = #: &(a){#}#[n] = &(a)^^^^...^^^^# with n ^'s
- B. When p = k+#: &(a){k+#}#[n] = &(a){k+n}#
- C. When p ∈ %decomp: &(a){p}#[n] = &(a){p[n]}#
- D. When p = k+c, k ∈ %+, c ∈ %decomp: &(a){k+c}#[n] = &(a){k+c[n]}#
- E. When p = c*# where c ∈ %+:
  - IV.E.1. &(a){c*#}#[1] = &(a){c}#
  - IV.E.2. &(a){c*#}#[n] = &(a){c+c*#}#[n-1]
- F. When p = k+c*#, k ∈ %+, c ∈ %:
  - IV.F.1. &(a){k+c*#}#[1] = &(a){c}#
  - IV.F.2. &(a){k+c*#}#[n] = &(a){k+c+c*#}#[n-1]

Rules[]

Rule I. When c = &:
- I1. %a{&}#[1] = %a
- I2. %a{&}#[2] = %a{a}#
- I3. %a{&}#[n] = %a{a{&}#[n-1]}# for n > 2
Rule II. When c = d+&:
- II1. %a{d+&}#[1] = %a{d}#
- II2. %a{d+&}#[n] = %a{d+a{d+&}#[n-1]}#
Rule III. When c = d*& where d ∈ %+:
- III1. %a{d*&}#[1] = %a{d}#
- III2. %a{d*&}#[n] = %a{d*a{d*&}#[n-1]}#
Rule IV. When c = k+d*&, k ∈ %+, d ∈ %:
- IV1. %a{k+d*&}#[1] = %a{k+d}#
- IV2. %a{k+d*&}#[n] = %a{k+d*a{k+d*&}#[n-1]}#
Rule V. When c = d^& where d ∈ %+:
- V1. %a{d^&}#[1] = %a{d}#
- V2. %a{d^&}#[n] = %a{d^a{d^&}#[n-1]}#
Rule VI. When c = k+d^&, k ∈ %+, d ∈ %:
- VI1. %a{k+d^&}#[1] = %a{k+d}#
- VI2. %a{k+d^&}#[n] = %a{k+d^a{k+d^&}#[n-1]}#
Rule VII. When d^& where d ∈ %+, and the exponentiation rules of &, based off the Cascading-E notation rules, are applicable, consult the rules for case I in the decomposition rule:
- VII1. %a{k*d}#[n] = %a{k*d[n]}#
- VII2. %a{k^d}#[n] = %a{k^d[n]}#
- VII3. %a{&^^#}#[n] = %a{&^^n}# = %a{&^&^&^...^&^&^&}# with n &'s

Natural language equivalent[]

Extended Collapsing-E Notation[]

Main article: Extended Collapsing-E Notation

The extended collapsing-E notation is an incomplete extension of Collapsing-E notation also developed by the same wiki user (DeepLineMadom). It introduces subscripted ampersands, \(\&_n\), as the fixed point of \(\alpha\) \(\mapsto\) \(\&_{n-1}^{\alpha}\). DeepLineMadom has so far made the limit of that notation to be around order type \(\psi_0(\Omega_{\omega})\) with accordance to Buchholz's function^[2].

The author has also proposed a hyper-extended collapsing-E notation although there is currently no description of such notation.

Issues[]

Examples[]

Sources[]

[1] ttps://sites.google.com/view/intermediate-exe-continued/collapsing-e-notation/sneak-peak-3

[2] ttps://sites.google.com/view/intermediate-exe-continued/extended-collapsing-e-notation/xce-sneak-peak-2

[1]

[2]