PREVIOUS - 60 - Godendekathol regiment (to the limit of Cascading-E notation)

Overview

This is an excerpt of Sbiis Saibian's original ExE page of the definition of extended cascading-E notation.

We have already observed the five common formal rules of the extensible-E system since the extended hyper-E notation as follows (note that I am going to use % instead of & to avoid conflict with the collapsing-E notation):

Let Ea%a2% ... &an be any expression in xE^, where a1 through an are n positive integers, and all % are hyper-products (which may or may not be distinct.) Each individual % may be chosen from the set of legal separators.

Below are the five formal rules of xE^. Let %k be the kth hyper-product and L(%k) be the last cascader of the kth hyper-product.

• [RULE 1]: Base rule — With no hyperions, we have E[b]a = b^a. b is default to 10.

• [RULE 2]: Decomposition rule — If L(%(n-1)) ≠ #^n (the last cascader is not of the form of copies of hyperions):
  E@a%b = E@a%[b]a (@ indicates the unchanged remainder of the expression and %[b] is the fundamental sequence of % (see below))

• [RULE 3]: Termination rule — If the last argument is 1, it can be removed: E@a&1 = E@a

• [RULE 4]: Expansion rule — L(%(n-1)) = #^n and %k ≠ #:
  E@a&*#(b+1) = E@a&a&*#b

• [RULE 5]; Recursion rule — In other words, for the singular hyperions:
  E@a#(b+1) = E@a#(E@a#b)

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

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), if and only if L(%) ≠ #^n.

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 ≥ #):
(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.E.1. &(a){k+c*#}#[1] = &(a){c}#

IV.E.2. &(a){k+c*#}#[n] = &(a){k+c+c*#}#[n-1]

Natural language equivalent of the formal rules:

The formal rules of the notation are very similar to those of Extended Cascading-E and Hyper-Extended Cascading-E notation extensions, except they are just the rewritten versions of the xE^ and #xE^ rules.

Case I. If L is an exponent operator (^), where a, b belong to previous member of %:

When the expression ends with copies of #: Copy the previous delimiter using hyper-product and one hyperion mark less after #. If the latter of the delimiter is not copies of #, or the last hyper-product is not copies of #, consult the rules for decomposing intermediate delimiter structures.

Case II. If L is in the form of a{p}b, where a, b ∈ %, and (p is natural number greater or equal to 2, or p is the successor ordinal):

Let the hyper operator iterate recursively via the up-arrow notation rules. If there are more than one hyperion after carets, or the last hyper-product is in the form of copies of hyperions, decompose the delimiter structure via the caret-tops, and with the identical delimiter structures. If the latter of the delimiter is not copies of #, consult the rules for decomposing intermediate delimiter structures.

Case III. If L in in the form of a{p}b>c, where a, b ∈ %, c ∈ %+, and (p is natural number greater or equal to 2, or p is the successor ordinal):

For each delimiter structures based on the hyper-operators from the least tetrational operator (^^#), with caret-tops, let it decompose into copies of hyperoperators from left to right. If there are plus signs followed by a single hyperion mark, let it decompose in the similar fashion, by removing the plus sign and a hyperion mark. If there are two or more consecutive hyperion marks, or the last hyper-product is in the form of copies of hyperions, use the hyperion-addition rule. If the latter of the delimiter is not copies of #, consult the rules for decomposing intermediate delimiter structures.

Case IV. If p in a{p}b is not a natural number or the successor ordinal

Consult the rule for decomposing delimiter structures inside curly braces, in a similar fashion to the rules on caret-tops.

This is actually an excerpt of the definition of collapsing-E notation page. I actually wrote the definition of the collapsing-E notation after Saibian's existing (hyper-)extended cascading-E notation.

Now let's analyze the ordinal using multiple transfinite ordinal notations as well as Bashicu matrix system (BMS) and Y sequence.

Ordinal analysis

• E100#^^#100

  • Veblen: ε0

  • Buchholz: ψ0(Ω)

  • Madore: ψ(0)

  • BMS: (0,0)(1,1)

  • Y-SEQ: (1,2,4)

• E100#^^#100#100

  • Veblen: ε0+1

  • Buchholz: ψ0(Ω)+1

  • Madore: ψ(0)+1

  • BMS: (0,0)(1,1)(0,0)

  • Y-SEQ: (1,2,4,1)

• E100#^^#100#100#100

  • Veblen: ε0+2

  • Buchholz: ψ0(Ω)+2

  • Madore: ψ(0)+2

  • BMS: (0,0)(1,1)(0,0)(0,0)

  • Y-SEQ: (1,2,4,1,1)

• E100#^^#100##100

  • Veblen: ε0+ω

  • Buchholz: ψ0(Ω)+ω

  • Madore: ψ(0)+ω

  • BMS: (0,0)(1,1)(0,0)(1,0)

  • Y-SEQ: (1,2,4,1,2)

• E100#^^#100###100

  • Veblen: ε0+ω^2

  • Buchholz: ψ0(Ω)+ω^2

  • Madore: ψ(0)+ω^2

  • BMS: (0,0)(1,1)(0,0)(1,0)(1,0)

  • Y-SEQ: (1,2,4,1,2,2)

• E100#^^#100#^#100

  • Veblen: ε0+ω^ω

  • Buchholz: ψ0(Ω)+ω^ω

  • Madore: ψ(0)+ω^ω

  • BMS: (0,0)(1,1)(0,0)(1,0)(2,0)

  • Y-SEQ: (1,2,4,1,2,3)

• E100#^^#100#^#^#100

  • Veblen: ε0+ω^ω^ω

  • Buchholz: ψ0(Ω)+ω^ω^ω

  • Madore: ψ(0)+ω^ω^ω

  • BMS: (0,0)(1,1)(0,0)(1,0)(2,0)(3,0)

  • Y-SEQ: (1,2,4,1,2,3,4)

• E100#^^#100#^^#100

  • Veblen: ε0·2

  • Buchholz: ψ0(Ω)·2

  • Madore: ψ(0)·2

  • BMS: (0,0)(1,1)(0,0)(1,1)

  • Y-SEQ: (1,2,4,1,2,4)

• E100#^^#100#^^#100#^^#100

  • Veblen: ε0·3

  • Buchholz: ψ0(Ω)·3

  • Madore: ψ(0)·3

  • BMS: (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)

  • Y-SEQ: (1,2,4,1,2,4,1,2,4)

• E100#^^#*#100

  • Veblen: ε0·ω

  • Buchholz: ψ0(Ω+1)

  • Madore: ψ(0)·ω

  • BMS: (0,0)(1,1)(1,0)

  • Y-SEQ: (1,2,4,2)

• E100#^^#*##100

  • Veblen: ε0·ω^2

  • Buchholz: ψ0(Ω+2)

  • Madore: ψ(0)·ω^2

  • BMS: (0,0)(1,1)(1,0)(1,0)

  • Y-SEQ: (1,2,4,2,2)

• E100#^^#*#^#100

  • Veblen: ε0·ω^ω

  • Buchholz: ψ0(Ω+ω)

  • Madore: ψ(0)·ω^ω

  • BMS: (0,0)(1,1)(1,0)(2,0)

  • Y-SEQ: (1,2,4,2,3)

• E100#^^#*#^#^#100

  • Veblen: ε0·ω^ω^ω

  • Buchholz: ψ0(Ω+ω^ω)

  • Madore: ψ(0)·ω^ω^ω

  • BMS: (0,0)(1,1)(1,0)(2,0)(3,0)

  • Y-SEQ: (1,2,4,2,3,4)

• E100#^^#*#^^#100

  • Veblen: ε0^2

  • Buchholz: ψ0(Ω+ψ0(Ω))

  • Madore: ψ(0)^2

  • BMS: (0,0)(1,1)(1,0)(2,1)

  • Y-SEQ: (1,2,4,2,4)

• E100#^^#*#^^#*#^^#100

  • Veblen: ε0^3

  • Buchholz: ψ0(Ω+ψ0(Ω)·2)

  • Madore: ψ(0)^3

  • BMS: (0,0)(1,1)(1,0)(2,1)(1,0)(2,1)

  • Y-SEQ: (1,2,4,2,4,2,4)

• E100(#^^#)^#100

  • Veblen: ε0^ω

  • Buchholz: ψ0(Ω+ψ0(Ω+1))

  • Madore: ψ(0)^ω

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)

  • Y-SEQ: (1,2,4,3)

• E100(#^^#)^##100

  • Veblen: ε0^ω^2

  • Buchholz: ψ0(Ω+ψ0(Ω+2))

  • Madore: ψ(0)^ω^2

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(2,0)

  • Y-SEQ: (1,2,4,3,3)

• E100(#^^#)^#^#100

  • Veblen: ε0^ω^ω

  • Buchholz: ψ0(Ω+ψ0(Ω+ω))

  • Madore: ψ(0)^ω^ω

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,0)

  • Y-SEQ: (1,2,4,3,4)

• E100(#^^#)^#^#^#100

  • Veblen: ε0^ω^ω^ω

  • Buchholz: ψ0(Ω+ψ0(Ω+ω^ω))

  • Madore: ψ(0)^ω^ω^ω

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,0)(4,0)

  • Y-SEQ: (1,2,4,3,4,5)

• E100(#^^#)^(#^^#)100

  • Veblen: ε0^ε0

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω)))

  • Madore: ψ(0)^ψ(0)

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,3,5)

• E100(#^^#)^(#^^#*#^^#)100

  • Veblen: ε0^ε0^2

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω)·2))

  • Madore: ψ(0)^ψ(0)^2

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,3,5,3,5)

• E100(#^^#)^(#^^#)^#100

  • Veblen: ε0^ε0^ω

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω+1)))

  • Madore: ψ(0)^ψ(0)^ω

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)

  • Y-SEQ: (1,2,4,3,5,4)

• E100(#^^#)^(#^^#)^#^#100

  • Veblen: ε0^ε0^ω^ω

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω+ω)))

  • Madore: ψ(0)^ψ(0)^ω^ω

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,0)

  • Y-SEQ: (1,2,4,3,5,4,5)

• E100(#^^#)^(#^^#)^(#^^#)100

  • Veblen: ε0^ε0^ε0

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω))))

  • Madore: ψ(0)^ψ(0)^ψ(0)

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)

  • Y-SEQ: (1,2,4,3,5,4,6)

• E100(#^^#)^(#^^#)^(#^^#)^#100

  • Veblen: ε0^ε0^ε0^ω

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω+1))))

  • Madore: ψ(0)^ψ(0)^ψ(0)^ω

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)

  • Y-SEQ: (1,2,4,3,5,4,6,5)

• E100(#^^#)^(#^^#)^(#^^#)^(#^^#)100

  • Veblen: ε0^ε0^ε0^ε0

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω)))))

  • Madore: ψ(0)^ψ(0)^ψ(0)^ψ(0)

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)

  • Y-SEQ: (1,2,4,3,5,4,6,5,7)

• E100(#^^#)^(#^^#)^(#^^#)^(#^^#)^(#^^#)100

  • Veblen: ε0^ε0^ε0^ε0^ε0

  • Buchholz: ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω))))))

  • Madore: ψ(0)^ψ(0)^ψ(0)^ψ(0)^ψ(0)

  • BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)(5,0)(6,1)

  • Y-SEQ: (1,2,4,3,5,4,6,5,7,6,8)

• E100(#^^#)^^#100

  • Veblen: ε1

  • Buchholz: ψ0(Ω·2)

  • Madore: ψ(1)

  • BMS: (0,0)(1,1)(1,1)

  • Y-SEQ: (1,2,4,4)

• E100(#^^#)^^#100#100

  • Veblen: ε1+1

  • Buchholz: ψ0(Ω·2)+1

  • Madore: ψ(1)+1

  • BMS: (0,0)(1,1)(1,1)(0,0)

  • Y-SEQ: (1,2,4,4,1)

• E100(#^^#)^^#100#^^#100

  • Veblen: ε1+ε0

  • Buchholz: ψ0(Ω·2)+ψ0(Ω)

  • Madore: ψ(1)+ψ(0)

  • BMS: (0,0)(1,1)(1,1)(0,0)(1,1)

  • Y-SEQ: (1,2,4,4,1,2,4)

• E100(#^^#)^^#100(#^^#)^^#100

  • Veblen: ε1·2

  • Buchholz: ψ0(Ω·2)·2

  • Madore: ψ(1)·2

  • BMS: (0,0)(1,1)(1,1)(0,0)(1,1)(1,1)

  • Y-SEQ: (1,2,4,4,1,2,4,4)

• E100(#^^#)^^#*#100

  • Veblen: ε1·ω

  • Buchholz: ψ0(Ω·2+1)

  • Madore: ψ(1)·ω

  • BMS: (0,0)(1,1)(1,1)(1,0)

  • Y-SEQ: (1,2,4,4,2)

• E100(#^^#)^^#*#^^#100

  • Veblen: ε1·ε0

  • Buchholz: ψ0(Ω·2+ψ0(Ω))

  • Madore: ψ(1)·ψ(0)

  • BMS: (0,0)(1,1)(1,1)(1,0)(2,1)

  • Y-SEQ: (1,2,4,4,2,4)

• E100(#^^#)^^#*(#^^#)^^#100

  • Veblen: ε1^2

  • Buchholz: ψ0(Ω·2+ψ0(Ω·2))

  • Madore: ψ(1)^2

  • BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)

  • Y-SEQ: (1,2,4,4,2,4,4)

• E100((#^^#)^^#)^#100

  • Veblen: ε1^ω

  • Buchholz: ψ0(Ω·2+ψ0(Ω·2+1))

  • Madore: ψ(1)^ω

  • BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,4,3)

• E100((#^^#)^^#)^((#^^#)^^#)100

  • Veblen: ε1^ε1

  • Buchholz: ψ0(Ω·2+ψ0(Ω·2+ψ0(Ω·2)))

  • Madore: ψ(1)^ψ(1)

  • BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)

  • Y-SEQ: (1,2,4,4,3,5,5)

• E100((#^^#)^^#)^((#^^#)^^#)^((#^^#)^^#)100

  • Veblen: ε1^ε1^ε1

  • Buchholz: ψ0(Ω·2+ψ0(Ω·2+ψ0(Ω·2+ψ0(Ω·2))))

  • Madore: ψ(1)^ψ(1)^ψ(1)

  • BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)(3,0)(4,1)(4,1)

  • Y-SEQ: (1,2,4,4,3,5,5,4,6,6)

• E100((#^^#)^^#)^^#100

  • Veblen: ε2

  • Buchholz: ψ0(Ω·3)

  • Madore: ψ(2)

  • BMS: (0,0)(1,1)(1,1)(1,1)

  • Y-SEQ: (1,2,4,4,4)

• E100(((#^^#)^^#)^^#)^#100

  • Veblen: ε2^ω

  • Buchholz: ψ0(Ω·3+ψ0(Ω·3+1))

  • Madore: ψ(2)^ω

  • BMS: (0,0)(1,1)(1,1)(1,1)(1,0)(2,1)(2,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,4,4,3)

• E100(((#^^#)^^#)^^#)^(((#^^#)^^#)^^#)100

  • Veblen: ε2^ε2

  • Buchholz: ψ0(Ω·3+ψ0(Ω·3+ψ0(Ω·3)))

  • Madore: ψ(2)^ψ(2)

  • BMS: (0,0)(1,1)(1,1)(1,1)(1,0)(2,1)(2,1)(2,1)(2,0)(3,1)(3,1)(3,1)

  • Y-SEQ: (1,2,4,4,4,3,5,5,5)

• E100(((#^^#)^^#)^^#)^^#100

  • Veblen: ε3

  • Buchholz: ψ0(Ω·4)

  • Madore: ψ(3)

  • BMS: (0,0)(1,1)(1,1)(1,1)(1,1)

  • Y-SEQ: (1,2,4,4,4,4)

• E100((((#^^#)^^#)^^#)^^#)^^#100

  • Veblen: ε4

  • Buchholz: ψ0(Ω·5)

  • Madore: ψ(4)

  • BMS: (0,0)(1,1)(1,1)(1,1)(1,1)(1,1)

  • Y-SEQ: (1,2,4,4,4,4,4)

• E100#^^#>#100

  • Veblen: ε(ω)

  • Buchholz: ψ0(Ω·ω)

  • Madore: ψ(ω)

  • BMS: (0,0)(1,1)(2,0)

  • Y-SEQ: (1,2,4,5)

• E100(#^^#>#)^#100

  • Veblen: ε(ω)^ω

  • Buchholz: ψ0(Ω·ω+ψ0(Ω·ω+1))

  • Madore: ψ(ω)^ω

  • BMS: (0,0)(1,1)(2,0)(1,0)(2,1)(3,0)(2,0)

  • Y-SEQ: (1,2,4,5,3)

• E100(#^^#>#)^(#^^#>#)100

  • Veblen: ε(ω)^ε(ω)

  • Buchholz: ψ0(Ω·ω+ψ0(Ω·ω+ψ0(Ω·ω)))

  • Madore: ψ(ω)^ψ(ω)

  • BMS: (0,0)(1,1)(2,0)(1,0)(2,1)(3,0)(2,0)(3,1)(4,0)

  • Y-SEQ: (1,2,4,5,3,5,6)

• E100(#^^#>#)^^#100

  • Veblen: ε(ω+1)

  • Buchholz: ψ0(Ω·(ω+1))

  • Madore: ψ(ω+1)

  • BMS: (0,0)(1,1)(2,0)(1,1)

  • Y-SEQ: (1,2,4,5,4)

• E100((#^^#>#)^^#)^^#100

  • Veblen: ε(ω+2)

  • Buchholz: ψ0(Ω·(ω+2))

  • Madore: ψ(ω+2)

  • BMS: (0,0)(1,1)(2,0)(1,1)(1,1)

  • Y-SEQ: (1,2,4,5,4,4)

• E100#^^#>(#+#)100

  • Veblen: ε(ω·2)

  • Buchholz: ψ0(Ω·ω·2)

  • Madore: ψ(ω·2)

  • BMS: (0,0)(1,1)(2,0)(1,1)(2,0)

  • Y-SEQ: (1,2,4,5,4,5)

• E100(#^^#>(#+#))^^#100

  • Veblen: ε(ω·2+1)

  • Buchholz: ψ0(Ω·(ω·2+1))

  • Madore: ψ(ω·2+1)

  • BMS: (0,0)(1,1)(2,0)(1,1)(2,0)(1,1)

  • Y-SEQ: (1,2,4,5,4,5,4)

• E100#^^#>(#+#+#)100

  • Veblen: ε(ω·3)

  • Buchholz: ψ0(Ω·ω·3)

  • Madore: ψ(ω·3)

  • BMS: (0,0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)

  • Y-SEQ: (1,2,4,5,4,5,4,5)

• E100#^^#>(#+#+#+#)100

  • Veblen: ε(ω·4)

  • Buchholz: ψ0(Ω·ω·4)

  • Madore: ψ(ω·4)

  • BMS: (0,0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)

  • Y-SEQ: (1,2,4,5,4,5,4,5,4,5)

• E100#^^#>##100

  • Veblen: ε(ω^2)

  • Buchholz: ψ0(Ω·ω^2)

  • Madore: ψ(ω^2)

  • BMS: (0,0)(1,1)(2,0)(2,0)

  • Y-SEQ: (1,2,4,5,5)

• E100(#^^#>##)^^#100

  • Veblen: ε(ω^2+1)

  • Buchholz: ψ0(Ω·(ω^2+1))

  • Madore: ψ(ω^2+1)

  • BMS: (0,0)(1,1)(2,0)(2,0)(1,1)

  • Y-SEQ: (1,2,4,5,5,4)

• E100#^^#>(##+#)100

  • Veblen: ε(ω^2+ω)

  • Buchholz: ψ0(Ω·(ω^2+ω))

  • Madore: ψ(ω^2+ω)

  • BMS: (0,0)(1,1)(2,0)(2,0)(1,1)(2,0)

  • Y-SEQ: (1,2,4,5,5,4,5)

• E100#^^#>(##+##)100

  • Veblen: ε(ω^2·2)

  • Buchholz: ψ0(Ω·ω^2·2)

  • Madore: ψ(ω^2·2)

  • BMS: (0,0)(1,1)(2,0)(2,0)(1,1)(2,0)(2,0)

  • Y-SEQ: (1,2,4,5,5,4,5,5)

• E100#^^#>###100

  • Veblen: ε(ω^3)

  • Buchholz: ψ0(Ω·ω^3)

  • Madore: ψ(ω^3)

  • BMS: (0,0)(1,1)(2,0)(2,0)(2,0)

  • Y-SEQ: (1,2,4,5,5,5)

• E100#^^#>####100

  • Veblen: ε(ω^4)

  • Buchholz: ψ0(Ω·ω^4)

  • Madore: ψ(ω^4)

  • BMS: (0,0)(1,1)(2,0)(2,0)(2,0)(2,0)

  • Y-SEQ: (1,2,4,5,5,5,5)

• E100#^^#>#^#100

  • Veblen: ε(ω^ω)

  • Buchholz: ψ0(Ω·ω^ω)

  • Madore: ψ(ω^ω)

  • BMS: (0,0)(1,1)(2,0)(3,0)

  • Y-SEQ: (1,2,4,5,6)

• E100#^^#>(#^#*#^#)100

  • Veblen: ε(ω^(ω·2))

  • Buchholz: ψ0(Ω·ω^(ω·2))

  • Madore: ψ(ω^(ω·2))

  • BMS: (0,0)(1,1)(2,0)(3,0)(2,0)(3,0)

  • Y-SEQ: (1,2,4,5,6,5,6)

• E100#^^#>#^##100

  • Veblen: ε(ω^ω^2)

  • Buchholz: ψ0(Ω·ω^ω^2)

  • Madore: ψ(ω^ω^2)

  • BMS: (0,0)(1,1)(2,0)(3,0)(3,0)

  • Y-SEQ: (1,2,4,5,6,6)

• E100#^^#>#^#^#100

  • Veblen: ε(ω^ω^ω)

  • Buchholz: ψ0(Ω·ω^ω^ω)

  • Madore: ψ(ω^ω^ω)

  • BMS: (0,0)(1,1)(2,0)(3,0)(4,0)

  • Y-SEQ: (1,2,4,5,6,7)

• E100#^^#>#^#^#^#100

  • Veblen: ε(ω^ω^ω^ω)

  • Buchholz: ψ0(Ω·ω^ω^ω^ω)

  • Madore: ψ(ω^ω^ω^ω)

  • BMS: (0,0)(1,1)(2,0)(3,0)(4,0)(5,0)

  • Y-SEQ: (1,2,4,5,6,7,8)

• E100#^^#>#^^#100

  • Veblen: ε(ε0)

  • Buchholz: ψ0(Ω·ψ0(Ω))

  • Madore: ψ(ψ(0))

  • BMS: (0,0)(1,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,5,7)

• E100(#^^#>#^^#)^^#100

  • Veblen: ε(ε0+1)

  • Buchholz: ψ0(Ω·(ψ0(Ω)+1))

  • Madore: ψ(ψ(0)+1)

  • BMS: (0,0)(1,1)(2,0)(3,1)(1,1)

  • Y-SEQ: (1,2,4,5,7,4)

• E100#^^#>(#^^#+#)100

  • Veblen: ε(ε0+ω)

  • Buchholz: ψ0(Ω·(ψ0(Ω)+ω))

  • Madore: ψ(ψ(0)+ω)

  • BMS: (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)

  • Y-SEQ: (1,2,4,5,7,4,5)

• E100#^^#>(#^^#+#^^#)100

  • Veblen: ε(ε0·2)

  • Buchholz: ψ0(Ω·ψ0(Ω)·2)

  • Madore: ψ(ψ(0)·2)

  • BMS: (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,5,7,4,5,7)

• E100#^^#>(#^^#*#)100

  • Veblen: ε(ε0·ω)

  • Buchholz: ψ0(Ω·ψ0(Ω+1))

  • Madore: ψ(ψ(0)·ω)

  • BMS: (0,0)(1,1)(2,0)(3,1)(2,0)

  • Y-SEQ: (1,2,4,5,7,5)

• E100#^^#>(#^^#*#^#)100

  • Veblen: ε(ε0·ω^ω)

  • Buchholz: ψ0(Ω·ψ0(Ω+ω))

  • Madore: ψ(ψ(0)·ω^ω)

  • BMS: (0,0)(1,1)(2,0)(3,1)(2,0)(3,0)

  • Y-SEQ: (1,2,4,5,7,5,6)

• E100#^^#>(#^^#*#^^#)100

  • Veblen: ε(ε0^2)

  • Buchholz: ψ0(Ω·ψ0(Ω+ψ0(Ω)))

  • Madore: ψ(ψ(0)^2)

  • BMS: (0,0)(1,1)(2,0)(3,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,5,7,5,7)

• E100#^^#>(#^^#)^#100

  • Veblen: ε(ε0^ω)

  • Buchholz: ψ0(Ω·ψ0(Ω+ψ0(Ω+1)))

  • Madore: ψ(ψ(0)^ω)

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,0)

  • Y-SEQ: (1,2,4,5,7,6)

• E100#^^#>(#^^#)^(#^^#)100

  • Veblen: ε(ε0^ε0)

  • Buchholz: ψ0(Ω·ψ0(Ω+ψ0(Ω+ψ0(Ω))))

  • Madore: ψ(ψ(0)^ψ(0))

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,0)(4,1)

  • Y-SEQ: (1,2,4,5,7,6,8)

• E100#^^#>(#^^#)^(#^^#)^(#^^#)100

  • Veblen: ε(ε0^ε0^ε0)

  • Buchholz: ψ0(Ω·ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω)))))

  • Madore: ψ(ψ(0)^ψ(0)^ψ(0))

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)

  • Y-SEQ: (1,2,4,5,7,6,8)

• E100#^^#>(#^^#)^^#100

  • Veblen: ε(ε1)

  • Buchholz: ψ0(Ω·ψ0(Ω·2))

  • Madore: ψ(ψ(1))

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,1)

  • Y-SEQ: (1,2,4,5,7,7)

• E100#^^#>((#^^#)^^#)^((#^^#)^^#)100

  • Veblen: ε(ε1^ε1)

  • Buchholz: ψ0(Ω·ψ0(Ω·2+ψ0(Ω·2+ψ0(Ω·2))))

  • Madore: ψ(ψ(1)^ψ(1))

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,1)(3,0)(4,1)(4,1)

  • Y-SEQ: (1,2,4,5,7,7,6,8,8)

• E100#^^#>((#^^#)^^#)^^#100

  • Veblen: ε(ε2)

  • Buchholz: ψ0(Ω·ψ0(Ω·3))

  • Madore: ψ(ψ(2))

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,1)(3,1)

  • Y-SEQ: (1,2,4,5,7,7,7)

• E100#^^#>(((#^^#)^^#)^^#)^^#100

  • Veblen: ε(ε3)

  • Buchholz: ψ0(Ω·ψ0(Ω·4))

  • Madore: ψ(ψ(3))

  • BMS: (0,0)(1,1)(2,0)(3,1)(3,1)(3,1)(3,1)

  • Y-SEQ: (1,2,4,5,7,7,7,7)

• E100#^^#>#^^#>#100

  • Veblen: ε(ε(ω))

  • Buchholz: ψ0(Ω·ψ0(Ω·ω))

  • Madore: ψ(ψ(ω))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)

  • Y-SEQ: (1,2,4,5,7,8)

• E100#^^#>(#^^#>#)^^#100

  • Veblen: ε(ε(ω+1))

  • Buchholz: ψ0(Ω·ψ0(Ω·(ω+1)))

  • Madore: ψ(ψ(ω+1))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(3,1)

  • Y-SEQ: (1,2,4,5,7,8,7)

• E100#^^#>#^^#>(#+#)100

  • Veblen: ε(ε(ω·2))

  • Buchholz: ψ0(Ω·ψ0(Ω·ω·2))

  • Madore: ψ(ψ(ω·2))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(3,1)(4,0)

  • Y-SEQ: (1,2,4,5,7,8,7,8)

• E100#^^#>#^^#>##100

  • Veblen: ε(ε(ω^2))

  • Buchholz: ψ0(Ω·ψ0(Ω·ω^2))

  • Madore: ψ(ψ(ω^2))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(4,0)

  • Y-SEQ: (1,2,4,5,7,8,8)

• E100#^^#>#^^#>#^#100

  • Veblen: ε(ε(ω^ω))

  • Buchholz: ψ0(Ω·ψ0(Ω·ω^ω))

  • Madore: ψ(ψ(ω^ω))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,0)

  • Y-SEQ: (1,2,4,5,7,8,9)

• E100#^^#>#^^#>#^#^#100

  • Veblen: ε(ε(ω^ω^ω))

  • Buchholz: ψ0(Ω·ψ0(Ω·ω^ω^ω))

  • Madore: ψ(ψ(ω^ω^ω))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,0)(6,0)

  • Y-SEQ: (1,2,4,5,7,8,9,10)

• E100#^^#>#^^#>#^^#100

  • Veblen: ε(ε(ε0))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω)))

  • Madore: ψ(ψ(ψ(0)))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)

  • Y-SEQ: (1,2,4,5,7,8,10)

• E100#^^#>#^^#>(#^^#)^^#100

  • Veblen: ε(ε(ε1))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·2)))

  • Madore: ψ(ψ(ψ(1)))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(5,1)

  • Y-SEQ: (1,2,4,5,7,8,10,10)

• E100#^^#>#^^#>#^^#>#100

  • Veblen: ε(ε(ε(ω)))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ω)))

  • Madore: ψ(ψ(ψ(ω)))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)

  • Y-SEQ: (1,2,4,5,7,8,10,11)

• E100#^^#>#^^#>#^^#>#^#100

  • Veblen: ε(ε(ε(ω^ω)))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ω^ω)))

  • Madore: ψ(ψ(ψ(ω^ω)))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,0)

  • Y-SEQ: (1,2,4,5,7,8,10,11,12)

• E100#^^#>#^^#>#^^#>#^^#100

  • Veblen: ε(ε(ε(ε0)))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω))))

  • Madore: ψ(ψ(ψ(ψ(0))))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)

  • Y-SEQ: (1,2,4,5,7,8,10,11,13)

• E100#^^#>#^^#>#^^#>#^^#>#100

  • Veblen: ε(ε(ε(ε(ω))))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω·ω))))

  • Madore: ψ(ψ(ψ(ψ(ω))))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)(8,0)

  • Y-SEQ: (1,2,4,5,7,8,10,11,13,14)

• E100#^^#>#^^#>#^^#>#^^#>#^^#100

  • Veblen: ε(ε(ε(ε(ε0))))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω)))))

  • Madore: ψ(ψ(ψ(ψ(ψ(0)))))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)(8,0)(9,1)

  • Y-SEQ: (1,2,4,5,7,8,10,11,13,14,16)

• E100#^^#>#^^#>#^^#>#^^#>#^^#>#^^#100

  • Veblen: ε(ε(ε(ε(ε(ε0)))))

  • Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω))))))

  • Madore: ψ(ψ(ψ(ψ(ψ(ψ(0))))))

  • BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)(8,0)(9,1)(10,0)(11,1)

  • Y-SEQ: (1,2,4,5,7,8,10,11,13,14,16,17,19)

Moving on...

• E100#^^##100

  • Veblen: ζ0

  • Buchholz: ψ0(Ω^2)

  • Madore: ψ(Ω)

  • BMS: (0,0)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6)

• E100#^^##100#100

  • Veblen: ζ0+1

  • Buchholz: ψ0(Ω^2)+1

  • Madore: ψ(Ω)+1

  • BMS: (0,0)(1,1)(2,1)(0,0)

  • Y-SEQ: (1,2,4,6,1)

• E100#^^##100#^^##100

  • Veblen: ζ0·2

  • Buchholz: ψ0(Ω^2)·2

  • Madore: ψ(Ω)·2

  • BMS: (0,0)(1,1)(2,1)(0,0)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,1,2,4,6)

• E100#^^##*#100

  • Veblen: ζ0·ω

  • Buchholz: ψ0(Ω^2+1)

  • Madore: ψ(Ω)·ω

  • BMS: (0,0)(1,1)(2,1)(1,0)

  • Y-SEQ: (1,2,4,6,2)

• E100#^^##*#^^##100

  • Veblen: ζ0^2

  • Buchholz: ψ0(Ω^2+ψ0(Ω^2))

  • Madore: ψ(Ω)^2

  • BMS: (0,0)(1,1)(2,1)(1,0)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,2,4,6)

• E100(#^^##)^#100

  • Veblen: ζ0^ω

  • Buchholz: ψ0(Ω^2+ψ0(Ω^2+1))

  • Madore: ψ(Ω)^ω

  • BMS: (0,0)(1,1)(2,1)(1,0)(2,1)(3,1)(2,0)

  • Y-SEQ: (1,2,4,6,3)

• E100(#^^##)^(#^^##)100

  • Veblen: ζ0^ζ0

  • Buchholz: ψ0(Ω^2+ψ0(Ω^2+ψ0(Ω^2)))

  • Madore: ψ(Ω)^ψ(Ω)

  • BMS: (0,0)(1,1)(2,1)(1,0)(2,1)(3,1)(2,0)(3,1)(4,1)

  • Y-SEQ: (1,2,4,6,3,5,7)

• E100(#^^##)^(#^^##)^(#^^##)100

  • Veblen: ζ0^ζ0^ζ0

  • Buchholz: ψ0(Ω^2+ψ0(Ω^2+ψ0(Ω^2+ψ0(Ω^2))))

  • Madore: ψ(Ω)^ψ(Ω)^ψ(Ω)

  • BMS: (0,0)(1,1)(2,1)(1,0)(2,1)(3,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)

  • Y-SEQ: (1,2,4,6,3,5,7,4,6,8)

• E100(#^^##)^^#100

  • Veblen: ε(ζ0+1)

  • Buchholz: ψ0(Ω^2+Ω)

  • Madore: ψ(Ω+1)

  • BMS: (0,0)(1,1)(2,1)(1,1)

  • Y-SEQ: (1,2,4,6,4)

• E100((#^^##)^^#)^^#100

  • Veblen: ε(ζ0+2)

  • Buchholz: ψ0(Ω^2+Ω·2)

  • Madore: ψ(Ω+2)

  • BMS: (0,0)(1,1)(2,1)(1,1)(1,1)

  • Y-SEQ: (1,2,4,6,4,4)

• E100(#^^##)^^#>#100

  • Veblen: ε(ζ0+ω)

  • Buchholz: ψ0(Ω^2+Ω·ω)

  • Madore: ψ(Ω+ω)

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,0)

  • Y-SEQ: (1,2,4,6,4,5)

• E100(#^^##)^^#>#^^#100

  • Veblen: ε(ζ0+ε0)

  • Buchholz: ψ0(Ω^2+Ω·ψ0(Ω))

  • Madore: ψ(Ω+ψ(0))

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,6,4,5,7)

• E100(#^^##)^^#>#^^##100

  • Veblen: ε(ζ0·2)

  • Buchholz: ψ0(Ω^2+Ω·ψ0(Ω^2))

  • Madore: ψ(Ω+ψ(Ω))

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)

  • Y-SEQ: (1,2,4,6,4,5,7,9)

• E100(#^^##)^^#>(#^^##)^^#100

  • Veblen: ε(ε(ζ0+1))

  • Buchholz: ψ0(Ω^2+Ω·ψ0(Ω^2+Ω))

  • Madore: ψ(Ω+ψ(Ω+1))

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)

  • Y-SEQ: (1,2,4,6,4,5,7,9,7)

• E100(#^^##)^^#>(#^^##)^^#>(#^^##)^^#100

  • Veblen: ε(ε(ε(ζ0+1)))

  • Buchholz: ψ0(Ω^2+Ω·ψ0(Ω^2+Ω·ψ0(Ω^2+Ω)))

  • Madore: ψ(Ω+ψ(Ω+ψ(Ω+1)))

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1)

  • Y-SEQ: (1,2,4,6,4,5,7,9,7,8,10,12,10)

• E100(#^^##)^^##100

  • Veblen: ζ1

  • Buchholz: ψ0(Ω^2·2)

  • Madore: ψ(Ω·2)

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,4,6)

• E100((#^^##)^^##)^^#100

  • Veblen: ε(ζ1)

  • Buchholz: ψ0(Ω^2·2+Ω)

  • Madore: ψ(Ω·2+1)

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)

  • Y-SEQ: (1,2,4,6,4,6,4)

• E100((#^^##)^^##)^^##100

  • Veblen: ζ2

  • Buchholz: ψ0(Ω^2·3)

  • Madore: ψ(Ω·3)

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,4,6,4,6)

• E100(((#^^##)^^##)^^##)^^##100

  • Veblen: ζ3

  • Buchholz: ψ0(Ω^2·4)

  • Madore: ψ(Ω·4)

  • BMS: (0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,4,6,4,6,4,6)

• E100#^^##>#100

  • Veblen: ζ(ω)

  • Buchholz: ψ0(Ω^2·ω)

  • Madore: ψ(Ω·ω)

  • BMS: (0,0)(1,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,6,5)

• E100(#^^##>#)^^##100

  • Veblen: ζ(ω+1)

  • Buchholz: ψ0(Ω^2·(ω+1))

  • Madore: ψ(Ω·(ω+1))

  • BMS: (0,0)(1,1)(2,1)(2,0)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,5,4,6)

• E100#^^##>(#+#)100

  • Veblen: ζ(ω·2)

  • Buchholz: ψ0(Ω^2·ω·2)

  • Madore: ψ(Ω·ω·2)

  • BMS: (0,0)(1,1)(2,1)(2,0)(1,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,6,5,4,6,5)

• E100#^^##>##100

  • Veblen: ζ(ω^2)

  • Buchholz: ψ0(Ω^2·ω^2)

  • Madore: ψ(Ω·ω^2)

  • BMS: (0,0)(1,1)(2,1)(2,0)(2,0)

  • Y-SEQ: (1,2,4,6,5,5)

• E100#^^##>#^#100

  • Veblen: ζ(ω^ω)

  • Buchholz: ψ0(Ω^2·ω^ω)

  • Madore: ψ(Ω·ω^ω)

  • BMS: (0,0)(1,1)(2,1)(2,0)(3,0)

  • Y-SEQ: (1,2,4,6,5,6)

• E100#^^##>#^^#100

  • Veblen: ζ(ε0)

  • Buchholz: ψ0(Ω^2·ψ0(Ω))

  • Madore: ψ(Ω·ψ(0))

  • BMS: (0,0)(1,1)(2,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,6,5,7)

• E100#^^##>#^^#>#100

  • Veblen: ζ(ε(ω))

  • Buchholz: ψ0(Ω^2·ψ0(Ω·ω))

  • Madore: ψ(Ω·ψ(ω))

  • BMS: (0,0)(1,1)(2,1)(2,0)(3,1)(4,0)

  • Y-SEQ: (1,2,4,6,5,7,8)

• E100#^^##>#^^##100

  • Veblen: ζ(ζ0)

  • Buchholz: ψ0(Ω^2·ψ0(Ω^2))

  • Madore: ψ(Ω·ψ(Ω))

  • BMS: (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)

  • Y-SEQ: (1,2,4,6,5,7,9)

• E100#^^##>#^^##>#^^##100

  • Veblen: ζ(ζ(ζ0))

  • Buchholz: ψ0(Ω^2·ψ0(Ω^2·ψ0(Ω^2)))

  • Madore: ψ(Ω·ψ(Ω·ψ(Ω)))

  • BMS: (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)(5,1)(6,1)

  • Y-SEQ: (1,2,4,6,5,7,9,8,10,12)

• E100#^^###100

  • Veblen: η0

  • Buchholz: ψ0(Ω^3)

  • Madore: ψ(Ω^2)

  • BMS: (0,0)(1,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6)

• E100(#^^###)^^#100

  • Veblen: ε(η0+1)

  • Buchholz: ψ0(Ω^3+Ω)

  • Madore: ψ(Ω^2+1)

  • BMS: (0,0)(1,1)(2,1)(2,1)(1,1)

  • Y-SEQ: (1,2,4,6,6,4)

• E100(#^^###)^^##100

  • Veblen: ζ(η0+1)

  • Buchholz: ψ0(Ω^3+Ω^2)

  • Madore: ψ(Ω^2+Ω)

  • BMS: (0,0)(1,1)(2,1)(2,1)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,4,6)

• E100(#^^###)^^###100

  • Veblen: η1

  • Buchholz: ψ0(Ω^3·2)

  • Madore: ψ(Ω^2·2)

  • BMS: (0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,4,6,6)

• E100#^^###>#100

  • Veblen: η(ω)

  • Buchholz: ψ0(Ω^3·ω)

  • Madore: ψ(Ω^2·ω)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,6,6,5)

• E100#^^###>#^^###100

  • Veblen: η(η0)

  • Buchholz: ψ0(Ω^3·ψ0(Ω^3))

  • Madore: ψ(Ω^2·ψ(Ω^2))

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,0)(3,1)(4,1)(4,1)

  • Y-SEQ: (1,2,4,6,6,5,7,9,9)

• E100#^^####100

  • Veblen: φ(4,0)

  • Buchholz: ψ0(Ω^4)

  • Madore: ψ(Ω^3)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,6)

• E100(#^^####)^^####100

  • Veblen: φ(4,1)

  • Buchholz: ψ0(Ω^4·2)

  • Madore: ψ(Ω^3·2)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(1,1)(2,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,6,4,6,6,6)

• E100#^^####>#100

  • Veblen: φ(4,ω)

  • Buchholz: ψ0(Ω^4·ω)

  • Madore: ψ(Ω^3·ω)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,6,6,6,5)

• E100#^^####>#^^####100

  • Veblen: φ(4,φ(4,0))

  • Buchholz: ψ0(Ω^4·ψ0(Ω^4))

  • Madore: ψ(Ω^3·ψ(Ω^3))

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,0)(3,1)(3,1)(3,1)

  • Y-SEQ: (1,2,4,6,6,6,5,7,7,7)

• E100#^^#####100

  • Veblen: φ(5,0)

  • Buchholz: ψ0(Ω^5)

  • Madore: ψ(Ω^4)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,6,6)

• E100#^^#####>#100

  • Veblen: φ(5,ω)

  • Buchholz: ψ0(Ω^5·ω)

  • Madore: ψ(Ω^4·ω)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,0)

  • Y-SEQ: (1,2,4,6,6,6,6,5)

• E100#^^######100

  • Veblen: φ(6,0)

  • Buchholz: ψ0(Ω^6)

  • Madore: ψ(Ω^5)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,6,6,6)

• E100#^^#######100

  • Veblen: φ(7,0)

  • Buchholz: ψ0(Ω^7)

  • Madore: ψ(Ω^6)

  • BMS: (0,0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,6,6,6,6,6)

• E100#^^#^#100

  • Veblen: φ(ω,0)

  • Buchholz: ψ0(Ω^ω)

  • Madore: ψ(Ω^ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,7)

• E100(#^^#^#)^^#100

  • Veblen: ε(φ(ω,0)+1)

  • Buchholz: ψ0(Ω^ω+Ω)

  • Madore: ψ(Ω^ω+1)

  • BMS: (0,0)(1,1)(2,1)(3,0)(1,1)

  • Y-SEQ: (1,2,4,6,7,4)

• E100(#^^#^#)^^#>#100

  • Veblen: ε(φ(ω,0)+ω)

  • Buchholz: ψ0(Ω^ω+Ω·ω)

  • Madore: ψ(Ω^ω+ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,0)

  • Y-SEQ: (1,2,4,6,7,4,5)

• E100(#^^#^#)^^##100

  • Veblen: ζ(φ(ω,0)+1)

  • Buchholz: ψ0(Ω^ω+Ω^2)

  • Madore: ψ(Ω^ω+Ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,7,4,6)

• E100(#^^#^#)^^###100

  • Veblen: η(φ(ω,0)+1)

  • Buchholz: ψ0(Ω^ω+Ω^3)

  • Madore: ψ(Ω^ω+Ω^2)

  • BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,7,4,6,6)

• E100(#^^#^#)^^#^#100

  • Veblen: φ(ω,1)

  • Buchholz: ψ0(Ω^ω·2)

  • Madore: ψ(Ω^ω·2)

  • BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,7,4,6,7)

• E100((#^^#^#)^^#^#)^^#^#100

  • Veblen: φ(ω,2)

  • Buchholz: ψ0(Ω^ω·3)

  • Madore: ψ(Ω^ω·3)

  • BMS: (0,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,7,4,6,7,4,6,7)

• E100#^^(#^#)>#100

  • Veblen: φ(ω,ω)

  • Buchholz: ψ0(Ω^ω·ω)

  • Madore: ψ(Ω^ω·ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,0)

  • Y-SEQ: (1,2,4,6,7,5)

• E100#^^(#^#)>#^^#100

  • Veblen: φ(ω,ε0)

  • Buchholz: ψ0(Ω^ω·ψ0(Ω))

  • Madore: ψ(Ω^ω·ψ0(0))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,0)(3,1)

  • Y-SEQ: (1,2,4,6,7,5,7)

• E100#^^(#^#)>#^^##100

  • Veblen: φ(ω,ζ0)

  • Buchholz: ψ0(Ω^ω·ψ0(Ω^2))

  • Madore: ψ(Ω^ω·ψ0(Ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,0)(3,1)(4,1)

  • Y-SEQ: (1,2,4,6,7,5,7,9)

• E100#^^(#^#)>#^^(#^#)100

  • Veblen: φ(ω,φ(ω,0))

  • Buchholz: ψ0(Ω^ω·ψ0(Ω^ω))

  • Madore: ψ(Ω^ω·ψ(Ω^ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,0)(3,1)(4,1)(5,0)

  • Y-SEQ: (1,2,4,6,7,5,7,9,10)

• E100#^^(#^#*#)100

  • Veblen: φ(ω+1,0)

  • Buchholz: ψ0(Ω^(ω+1))

  • Madore: ψ(Ω^(ω+1))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,1)

  • Y-SEQ: (1,2,4,6,7,6)

• E100#^^(#^#*##)100

  • Veblen: φ(ω+2,0)

  • Buchholz: ψ0(Ω^(ω+2))

  • Madore: ψ(Ω^(ω+2))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,7,6,6)

• E100#^^(#^#*#^#)100

  • Veblen: φ(ω·2,0)

  • Buchholz: ψ0(Ω^(ω·2))

  • Madore: ψ(Ω^(ω·2))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,7,6,7)

• E100#^^(#^#*#^#*#^#)100

  • Veblen: φ(ω·3,0)

  • Buchholz: ψ0(Ω^(ω·3))

  • Madore: ψ(Ω^(ω·3))

  • BMS: (0,0)(1,1)(2,1)(3,0)(2,1)(3,0)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,7,6,7,6,7)

• E100#^^#^##100

  • Veblen: φ(ω^2,0)

  • Buchholz: ψ0(Ω^ω^2)

  • Madore: ψ(Ω^ω^2)

  • BMS: (0,0)(1,1)(2,1)(3,0)(3,0)

  • Y-SEQ: (1,2,4,6,7,7)

• E100#^^(#^##*#^##)100

  • Veblen: φ(ω^2·2,0)

  • Buchholz: ψ0(Ω^(ω^2·2))

  • Madore: ψ(Ω^(ω^2·2))

  • BMS: (0,0)(1,1)(2,1)(3,0)(3,0)(2,1)(3,0)(3,0)

  • Y-SEQ: (1,2,4,6,7,7,6,7,7)

• E100#^^#^###100

  • Veblen: φ(ω^3,0)

  • Buchholz: ψ0(Ω^ω^3)

  • Madore: ψ(Ω^ω^3)

  • BMS: (0,0)(1,1)(2,1)(3,0)(3,0)(3,0)

  • Y-SEQ: (1,2,4,6,7,7,7)

• E100#^^#^####100

  • Veblen: φ(ω^4,0)

  • Buchholz: ψ0(Ω^ω^4)

  • Madore: ψ(Ω^ω^4)

  • BMS: (0,0)(1,1)(2,1)(3,0)(3,0)(3,0)(3,0)

  • Y-SEQ: (1,2,4,6,7,7,7,7)

• E100#^^#^#^#100

  • Veblen: φ(ω^ω,0)

  • Buchholz: ψ0(Ω^ω^ω)

  • Madore: ψ(Ω^ω^ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,0)

  • Y-SEQ: (1,2,4,6,7,8)

• E100#^^#^#^##100

  • Veblen: φ(ω^ω^2,0)

  • Buchholz: ψ0(Ω^ω^ω^2)

  • Madore: ψ(Ω^ω^ω^2)

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,0)(4,0)

  • Y-SEQ: (1,2,4,6,7,8,8)

• E100#^^#^#^#^#100

  • Veblen: φ(ω^ω^ω,0)

  • Buchholz: ψ0(Ω^ω^ω^ω)

  • Madore: ψ(Ω^ω^ω^ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,0)(5,0)

  • Y-SEQ: (1,2,4,6,7,8,9)

• E100#^^#^#^#^#^#100

  • Veblen: φ(ω^ω^ω^ω,0)

  • Buchholz: ψ0(Ω^ω^ω^ω^ω)

  • Madore: ψ(Ω^ω^ω^ω^ω)

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(6,0)

  • Y-SEQ: (1,2,4,6,7,8,9,10)

• E100#^^#^^#100

  • Veblen: φ(ε0,0)

  • Buchholz: ψ0(Ω^ψ0(Ω))

  • Madore: ψ(Ω^ψ(0))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)

  • Y-SEQ: (1,2,4,6,7,9)

• E100#^^(#^^#)^^#100

  • Veblen: φ(ε1,0)

  • Buchholz: ψ0(Ω^ψ0(Ω·2))

  • Madore: ψ(Ω^ψ(1))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(4,1)

  • Y-SEQ: (1,2,4,6,7,9,9)

• E100#^^#^^#>#100

  • Veblen: φ(ε(ω),0)

  • Buchholz: ψ0(Ω^ψ0(Ω·ω))

  • Madore: ψ(Ω^ψ(ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,0)

  • Y-SEQ: (1,2,4,6,7,9,10)

• E100#^^#^^#>#^^#100

  • Veblen: φ(ε(ε0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω·ψ0(Ω)))

  • Madore: ψ(Ω^ψ(ψ(0)))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,0)(6,1)

  • Y-SEQ: (1,2,4,6,7,9,10,12)

• E100#^^#^^##100

  • Veblen: φ(ζ0,0)

  • Buchholz: ψ0(Ω^ψ0(Ω^2))

  • Madore: ψ(Ω^ψ(Ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)

  • Y-SEQ: (1,2,4,6,7,9,11)

• E100#^^#^^##>#100

  • Veblen: φ(ζ(ω),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^2·ω))

  • Madore: ψ(Ω^ψ(Ω·ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(5,0)

  • Y-SEQ: (1,2,4,6,7,9,11,10)

• E100#^^#^^###100

  • Veblen: φ(η0,0)

  • Buchholz: ψ0(Ω^ψ0(Ω^3))

  • Madore: ψ(Ω^ψ(Ω^2))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(5,1)

  • Y-SEQ: (1,2,4,6,7,9,11,11)

• E100#^^#^^####100

  • Veblen: φ(φ(4,0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^4))

  • Madore: ψ(Ω^ψ(Ω^3))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(5,1)(5,1)

  • Y-SEQ: (1,2,4,6,7,9,11,11,11)

• E100#^^#^^#^#100

  • Veblen: φ(φ(ω,0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ω))

  • Madore: ψ(Ω^ψ(Ω^ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)

  • Y-SEQ: (1,2,4,6,7,9,11,12)

• E100#^^#^^#^#^#100

  • Veblen: φ(φ(ω^ω,0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ω^ω))

  • Madore: ψ(Ω^ψ(Ω^ω^ω))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,0)

  • Y-SEQ: (1,2,4,6,7,9,11,12,13)

• E100#^^#^^#^^#100

  • Veblen: φ(φ(ε0,0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ψ0(Ω)))

  • Madore: ψ(Ω^ψ(Ω^ψ(0)))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)

  • Y-SEQ: (1,2,4,6,7,9,11,12,14)

• E100#^^#^^#^^##100

  • Veblen: φ(φ(ζ0,0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ψ0(Ω^2)))

  • Madore: ψ(Ω^ψ(Ω^ψ(Ω)))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)

  • Y-SEQ: (1,2,4,6,7,9,11,12,14,16)

• E100#^^#^^#^^#^#100

  • Veblen: φ(φ(φ(ω,0),0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ψ0(Ω^ω)))

  • Madore: ψ(Ω^ψ(Ω^ψ(Ω^ω)))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)(9,0)

  • Y-SEQ: (1,2,4,6,7,9,11,12,14,16,17)

• E100#^^#^^#^^#^^##100

  • Veblen: φ(φ(φ(ζ0,0),0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ψ0(Ω^ψ0(Ω^2))))

  • Madore: ψ(Ω^ψ(Ω^ψ(Ω^ψ(Ω))))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)(9,0)(10,1)(11,1)

  • Y-SEQ: (1,2,4,6,7,9,11,12,14,16,17,19,21)

• E100#^^#^^#^^#^^#^^##100

  • Veblen: φ(φ(φ(φ(ζ0,0),0),0),0)

  • Buchholz: ψ0(Ω^ψ0(Ω^ψ0(Ω^ψ0(Ω^ψ0(Ω^2)))))

  • Madore: ψ(Ω^ψ(Ω^ψ(Ω^ψ(Ω^ψ(Ω)))))

  • BMS: (0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)(9,0)(10,1)(11,1)(12,0)(13,1)(14,1)

  • Y-SEQ: (1,2,4,6,7,9,11,12,14,16,17,19,21,22,24,26)

And here we go!

• E100#^^^#100

  • Veblen: Γ0

  • Buchholz: ψ0(Ω^Ω)

  • Madore: ψ(Ω^Ω)

  • BMS: (0,0)(1,1)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8)

• E100(#^^^#)^#100

  • Veblen: Γ0^ω

  • Buchholz: ψ0(Ω^Ω+ψ0(Ω^Ω+1))

  • Madore: ψ(Ω^Ω)^ω

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,0)(2,1)(3,1)(4,1)(2,0)

  • Y-SEQ: (1,2,4,6,8,3)

• E100(#^^^#)^(#^^^#)100

  • Veblen: Γ0^Γ0

  • Buchholz: ψ0(Ω^Ω+ψ0(Ω^Ω+ψ0(Ω^Ω)))

  • Madore: ψ(Ω^Ω)^ψ(Ω^Ω)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,0)(2,1)(3,1)(4,1)(2,0)(3,1)(4,1)(5,1)

  • Y-SEQ: (1,2,4,6,8,3,5,7,9)

• E100(#^^^#)^^#100

  • Veblen: ε(Γ0+1)

  • Buchholz: ψ0(Ω^Ω+Ω)

  • Madore: ψ(Ω^Ω+1)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)

  • Y-SEQ: (1,2,4,6,8,4)

• E100(#^^^#)^^#>#100

  • Veblen: ε(Γ0+ω)

  • Buchholz: ψ0(Ω^Ω+Ω·ω)

  • Madore: ψ(Ω^Ω+ω)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,0)

  • Y-SEQ: (1,2,4,6,8,4,5)

• E100(#^^^#)^^##100

  • Veblen: ζ(Γ0+1)

  • Buchholz: ψ0(Ω^Ω+Ω^2)

  • Madore: ψ(Ω^Ω+Ω)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)

  • Y-SEQ: (1,2,4,6,8,4,6)

• E100(#^^^#)^^#^#100

  • Veblen: φ(ω,Γ0+1)

  • Buchholz: ψ0(Ω^Ω+Ω^ω)

  • Madore: ψ(Ω^Ω+Ω^ω)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,8,4,6,7)

• E100(#^^^#)^^(#^^^#)100

  • Veblen: φ(Γ0,1)

  • Buchholz: ψ0(Ω^Ω+Ω^ψ0(Ω^Ω))

  • Madore: ψ(Ω^Ω+Ω^ψ(Ω^Ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)

  • Y-SEQ: (1,2,4,6,8,4,6,7,9,11,13)

• E100(#^^^#)^^(#^^^#)^^#100

  • Veblen: φ(ε(Γ0+1),0)

  • Buchholz: ψ0(Ω^Ω+Ω^ψ0(Ω^Ω+Ω))

  • Madore: ψ(Ω^Ω+Ω^ψ(Ω^Ω+1))

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,1)

  • Y-SEQ: (1,2,4,6,8,4,6,7,9,11,13,9)

• E100(#^^^#)^^(#^^^#)^^##100

  • Veblen: φ(ζ(Γ0+1),0)

  • Buchholz: ψ0(Ω^Ω+Ω^ψ0(Ω^Ω+Ω^2))

  • Madore: ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,1)(5,1)

  • Y-SEQ: (1,2,4,6,8,4,6,7,9,11,13,9,11)

• E100(#^^^#)^^(#^^^#)^^(#^^^#)^^##100

  • Veblen: φ(φ(ζ(Γ0+1),0),0)

  • Buchholz: ψ0(Ω^Ω+Ω^ψ0(Ω^Ω+Ω^ψ0(Ω^Ω+Ω^2)))

  • Madore: ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,1)(5,1)(6,0)(7,1)(8,1)(9,1)(7,1)(8,1)

  • Y-SEQ: (1,2,4,6,8,4,6,7,9,11,13,9,11,12,14,16,18,14,16)

• E100(#^^^#)^^^#100

  • Veblen: Γ1

  • Buchholz: ψ0(Ω^Ω·2)

  • Madore: ψ(Ω^Ω·2)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,4,6,8)

• E100((#^^^#)^^^#)^^^#100

  • Veblen: Γ2

  • Buchholz: ψ0(Ω^Ω·3)

  • Madore: ψ(Ω^Ω·3)

  • BMS: (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,4,6,8,4,6,8)

• E100#^^^#>#100

  • Veblen: Γ(ω)

  • Buchholz: ψ0(Ω^Ω·ω)

  • Madore: ψ(Ω^Ω·ω)

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,0)

  • Y-SEQ: (1,2,4,6,8,5)

• E100#^^^#>#^^#100

  • Veblen: Γ(ε0)

  • Buchholz: ψ0(Ω^Ω·ψ0(Ω))

  • Madore: ψ(Ω^Ω·ψ(0))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)

  • Y-SEQ: (1,2,4,6,8,5,7)

• E100#^^^#>#^^##100

  • Veblen: Γ(ζ0)

  • Buchholz: ψ0(Ω^Ω·ψ0(Ω^2))

  • Madore: ψ(Ω^Ω·ψ(Ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)

  • Y-SEQ: (1,2,4,6,8,5,7,9)

• E100#^^^#>#^^^#100

  • Veblen: Γ(Γ0)

  • Buchholz: ψ0(Ω^Ω·ψ0(Ω^Ω))

  • Madore: ψ(Ω^Ω·ψ(Ω^Ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)

  • Y-SEQ: (1,2,4,6,8,5,7,9,11)

• E100#^^^#>#^^^#>#^^^#100

  • Veblen: Γ(Γ(Γ0))

  • Buchholz: ψ0(Ω^Ω·ψ0(Ω^Ω·ψ0(Ω^Ω)))

  • Madore: ψ(Ω^Ω·ψ(Ω^Ω·ψ(Ω^Ω)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)(4,0)(5,1)(6,1)(7,1)

  • Y-SEQ: (1,2,4,6,8,5,7,9,11,8,10,12,14)

• E100#^^^##100

  • Veblen: φ(1,1,0)

  • Buchholz: ψ0(Ω^(Ω+1))

  • Madore: ψ(Ω^(Ω+1))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)

  • Y-SEQ: (1,2,4,6,8,6)

• E100#^^^###100

  • Veblen: φ(1,2,0)

  • Buchholz: ψ0(Ω^(Ω+2))

  • Madore: ψ(Ω^(Ω+2))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(2,1)

  • Y-SEQ: (1,2,4,6,8,6,6)

• E100#^^^#^#100

  • Veblen: φ(1,ω,0)

  • Buchholz: ψ0(Ω^(Ω+ω))

  • Madore: ψ(Ω^(Ω+ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,0)

  • Y-SEQ: (1,2,4,6,8,6,7)

• E100#^^^#^^^#100

  • Veblen: φ(1,φ(1,0,0),0)

  • Buchholz: ψ0(Ω^(Ω+ψ0(Ω^Ω)))

  • Madore: ψ(Ω^(Ω+ψ(Ω^Ω)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)

  • Y-SEQ: (1,2,4,6,8,6,7,9,11,13)

• E100#^^^#^^^##100

  • Veblen: φ(1,φ(1,1,0),0)

  • Buchholz: ψ0(Ω^(Ω+ψ0(Ω^(Ω+1))))

  • Madore: ψ(Ω^(Ω+ψ(Ω^(Ω+1))))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(5,1)

  • Y-SEQ: (1,2,4,6,8,6,7,9,11,13,11)

• E100#^^^^#100

  • Veblen: φ(2,0,0)

  • Buchholz: ψ0(Ω^(Ω·2))

  • Madore: ψ(Ω^(Ω·2))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,6,8)

• E100#^^^^##100

  • Veblen: φ(2,1,0)

  • Buchholz: ψ0(Ω^(Ω·2+1))

  • Madore: ψ(Ω^(Ω·2+1))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)

  • Y-SEQ: (1,2,4,6,8,6,8,6)

• E100#^^^^^#100

  • Veblen: φ(3,0,0)

  • Buchholz: ψ0(Ω^(Ω·3))

  • Madore: ψ(Ω^(Ω·3))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,6,8,6,8)

• E100#^^^^^^#100

  • Veblen: φ(4,0,0)

  • Buchholz: ψ0(Ω^(Ω·4))

  • Madore: ψ(Ω^(Ω·4))

  • BMS: (0,0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,6,8,6,8,6,8)

• E100#{#}#100

  • Veblen: φ(ω,0,0)

  • Buchholz: ψ0(Ω^(Ω·ω))

  • Madore: ψ(Ω^(Ω·ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)

  • Y-SEQ: (1,2,4,6,8,7)

• E100#{#}##100

  • Veblen: φ(ω,1,0)

  • Buchholz: ψ0(Ω^(Ω·ω+1))

  • Madore: ψ(Ω^(Ω·ω+1))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(2,1)

  • Y-SEQ: (1,2,4,6,8,7,6)

• E100#{#+1}#100

  • Veblen: φ(ω+1,0,0)

  • Buchholz: ψ0(Ω^(Ω·(ω+1)))

  • Madore: ψ(Ω^(Ω·(ω+1)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(2,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,7,6,8)

• E100#{#+#}#100

  • Veblen: φ(ω·2,0,0)

  • Buchholz: ψ0(Ω^(Ω·ω·2))

  • Madore: ψ(Ω^(Ω·ω·2))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(2,1)(3,1)(3,0)

  • Y-SEQ: (1,2,4,6,8,7,6,8,7)

• E100#{##}#100

  • Veblen: φ(ω^2,0,0)

  • Buchholz: ψ0(Ω^(Ω·ω^2))

  • Madore: ψ(Ω^(Ω·ω^2))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(3,0)

  • Y-SEQ: (1,2,4,6,8,7,7)

• E100#{#^#}#100

  • Veblen: φ(ω^ω,0,0)

  • Buchholz: ψ0(Ω^(Ω·ω^ω))

  • Madore: ψ(Ω^(Ω·ω^ω))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,0)

  • Y-SEQ: (1,2,4,6,8,7,8)

• E100#{#^^#}#100

  • Veblen: φ(ε0,0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω)))

  • Madore: ψ(Ω^(Ω·ψ(0)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)

  • Y-SEQ: (1,2,4,6,8,7,9)

• E100#{#^^##}#100

  • Veblen: φ(ζ0,0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^2)))

  • Madore: ψ(Ω^(Ω·ψ(Ω)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)

  • Y-SEQ: (1,2,4,6,8,7,9,11)

• E100#{#^^#^#}#100

  • Veblen: φ(φ(ω,0),0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^ω)))

  • Madore: ψ(Ω^(Ω·ψ(Ω^ω)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,0)

  • Y-SEQ: (1,2,4,6,8,7,9,11,12)

• E100#{#^^^#}#100

  • Veblen: φ(Γ0,0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^Ω)))

  • Madore: ψ(Ω^(Ω·ψ(Ω^Ω)))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)

  • Y-SEQ: (1,2,4,6,8,7,9,11,13)

• E100#{#^^^^#}#100

  • Veblen: φ(φ(2,0,0),0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^(Ω·2))))

  • Madore: ψ(Ω^(Ω·ψ(Ω^(Ω·2))))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)(5,1)(6,1)

  • Y-SEQ: (1,2,4,6,8,7,9,11,13,11,13)

• E100#{#{#}#}#100

  • Veblen: φ(φ(ω,0,0),0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^(Ω·ω))))

  • Madore: ψ(Ω^(Ω·ψ(Ω^(Ω·ω))))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)(6,0)

  • Y-SEQ: (1,2,4,6,8,7,9,11,13,12)

• E100#{#{#^^^#}#}#100

  • Veblen: φ(φ(Γ0,0,0),0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^(Ω·ψ0(Ω^Ω)))))

  • Madore: ψ(Ω^(Ω·ψ(Ω^(Ω·ψ(Ω^Ω)))))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)(6,0)(7,1)(8,1)(9,1)

  • Y-SEQ: (1,2,4,6,8,7,9,11,13,12,14,16,18)

• E100#{#{#{#^^^#}#}#}#100

  • Veblen: φ(φ(φ(Γ0,0,0),0,0),0,0)

  • Buchholz: ψ0(Ω^(Ω·ψ0(Ω^(Ω·ψ0(Ω^(Ω·ψ0(Ω^Ω)))))))

  • Madore: ψ(Ω^(Ω·ψ(Ω^(Ω·ψ(Ω^(Ω·ψ(Ω^Ω)))))))

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)(6,0)(7,1)(8,1)(9,1)(9,0)(10,1)(11,1)(12,1)

  • Y-SEQ: (1,2,4,6,8,7,9,11,13,12,14,16,18,17,19,21,23)

• E100#{&}#100 (limit of (hyper-)extended cascading-E, entering collapsing-E)

  • Veblen: φ(1,0,0,0)

  • Buchholz: ψ0(Ω^Ω^2)

  • Madore: ψ(Ω^Ω^2)

  • BMS: (0,0)(1,1)(2,1)(3,1)(3,1)

  • Y-SEQ: (1,2,4,6,8,8)

So the limit of extended cascading-E notation is φ(1,0,0,0), also known as Ackermann ordinal.

Without further ado, let's go ahead and coin some new numbers from tethrathoth!

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