I interested in ordinals, and I asked myself: "Why we use \(\omega\), \(\omega2\), \(\omega^2\), \(\omega^\omega\), \(\epsilon_0\), \(\zeta_0\), \(\phi(3,0)\), \(\Gamma_0\) etc. instead of \(\omega\uparrow\uparrow\uparrow\omega\), \(\omega\uparrow\uparrow\uparrow\uparrow\omega\) etc.?"

But I realized this doesn't work beyond \(\epsilon_0\) since \(\omega\uparrow\uparrow(\omega+1) = \omega^{\omega\uparrow\uparrow\omega} = \omega^{\epsilon_0} = \epsilon_0 = \omega\uparrow\uparrow\omega\).

I experimented and created a BEAF-like system, suitable also for ordinals.

It is family of functions [X]a of ordinal a.

## Sequence X and its separators

Let X - sequence of ordinals (or "negative ordinals", i.e. ordinals with "-" sign before them) with separators.

All zeros we may omit.

All left part only with zeros and any separators we also may omit.

A separator may be written two ways: set of (...) or <...>.

### Type and body of separators

A separator has body and type, which are also sequences of ordinals. For example, <\(X_{type}\)|\(X_{body}\)>. If \(X_{type} = 0\), separator may be written only with body: <|\(X_{body}\)> = <\(X_{body}\)>. (For now, we'll consider only zero types).

### (X) separators

(X) is as in BEAF: (1) moves through one row, (2) - through one plane, (3) - through one 3D space etc. (0) should move through one element.

### <X> separators

Easier way to write separators is to use <X>: first right element is number of (0)'s, second right is number of (1)'s, third right is number of (2)'s etc.

In particular,

<1<X>> = (X)

<n<X>> = n (X)'s

<1> = (0) may be written as comma (,):

<1> = , = () = (0)

Examples:

<5> = ,,,,, = ,0,0,0,0, = ()()()()() = (0)(0)(0)(0)(0)

<1,2,3,4,5> = (0)(0)(0)(0)(0)(1)(1)(1)(1)(2)(2)(2)(3)(3)(4)

<1,> = <1,0> = <1<1>> = (1)

<1,2> = ()()(1) = (0)(0)(1) = ,,(1) = ,0,0(1)

<2,> = <2,0> = (1)(1)

<1,,> = <1,0,0> = <1<2>> = (2)

<1,,,> = <1,0,0,0> = <1<3>> = (3)

<2,,3,> = <2,0,3,0> = (1)(1)(1)(3)(3)

<3<1,>> = <3<1,0>> = <3<1<1>>> = <3(1)> = (1,)(1,)(1,) = (1,0)(1,0)(1,0)

### Empty separator <0>

<> = <0> is empty separator, it doesn't add anything:

X<> = X<0> = X

For example,

1,2,3<> = 1,2,3

<0> can only be located at the end of X. If it is located between elements of X, it doesn't separate anything, and these two elements actually should be one. Maybe,

a<0>b = a + b

(But this is not used).

### <X> addition

Adjacent separators <X> may be added:

<1><1> = <2>

<1><1,0> = <1,1>

<1,0><1,0> = <2,0>

etc.

*October 21, 2017 edit: addition of <X> separators is non-commutative:*

*<1><1,0> = <1,1>*

*<1,0><1> = <1,0>*

*<6,7,8,9,1><1,2,3,4,5,0,0,0> = <1,2,3,10,7,8,9,1>*

*<1,2,3,4,5,0,0,0><6,7,8,9,1> = <1,2,3,4,5,0,0,0>*

*Generally, left separator is added to right separator like right ordinal in Cantor normal form is added to left ordinal in Cantor normal form: first from left non-zero ordinal in left separator is added to ordinal in right separator in the same place, all ordinals to right is taken from left separator, and all ordinals to left is taken from right separator.*

*Two previous examples:*

*<6,7,8,9,1><1,2,3,4,5,0,0,0> = ?*

*First from left non-zero ordinal in left separator is 6. Ordinal in right separator in the same place (5th from right) is 4. 4+6 is 10. Four ordinals to right is taken from left separator: 7,8,9,1. All ordinals to left is taken from right separator: 1,2,3.*

*We get*

*<6,7,8,9,1><1,2,3,4,5,0,0,0> = <1,2,3,10,7,8,9,1>*

*<1,2,3,4,5,0,0,0><6,7,8,9,1> = ?*

*First from left non-zero ordinal in left separator is 1. Ordinal in right separator in the same place (8th from right) is 0 (<6,7,8,9,1> = <0,0,0,6,7,8,9,1>). 0+1 is 1. Seven ordinals to right is taken from left separator: 2,3,4,5,0,0,0. All ordinals to left is taken from right separator (all zeros).*

*We get*

*<1,2,3,4,5,0,0,0><6,7,8,9,1> = <1,2,3,4,5,0,0,0>*

### Eraser <-1>

<-1> erases one element:

X,n<-1> = X

For example,

1,2,3<-1> = 1,2

(Also, <-n> should erase n elements, <-n,0> should erase n rows, <-n,0,0> - n planes, but this is not used).

### Last element decreaser X-1

X-1 is X with last element decreased by 1.

For example,

X = 1,2,3,4,5

X-1 = 1,2,3,4,4

X = 1,2,3,4,0

X-1 = 1,2,3,4,-1

X = 0

X-1 = -1

### Last element nullifier X⁰

X⁰ is X with last element set to zero.

For example,

X = 1,2,3,4,5

X⁰ = 1,2,3,4,0

X = 1,2,3,4,0

X⁰ = 1,2,3,4,0

X = 0

X⁰ = 0

*Note: X⁰ was redefined. See last version of this blog with old definition and Notes in Rules section.*

### X*<X'> representation of X

<X'> is a separator at the right end of X. X* is rest of X. So,

X = X*<X'>

For example,

X = 1,2,3 = 1,2,3<>

X* = 1,2,3

X' = 0

X = 1(2)2(1)3,0,0,0,0,0 = 1<1<2>>2<1<1>>3<5> = 1<1,,>2<1,>3<5>

X* = 1(2)2(1)3 = 1<1<2>>2<1<1>>3 = 1<1,,>2<1,>3

X' = 5

X = 1(1)1 = 1<1,0>1<0> = 1<1,>1<>

X* = 1<1,>1

X' = 0

X* and X' are also may be represented in this form:

X* = X**<X*'>

X' = X'*<X">

Always should be

X* = X**<X*'> = X**<0> = X**

X*' = 0

Hence if

X = 0

then

X* = 0

X' = 0

Any X may be uniquely represented in this form X = X*<X'>.

### Identity function [-1]a

[-1] doesn't change anything:

[-1]a = a

### (X; a; b) designation

(X; a; b) = a , if X' ≠ 0

(X; a; b) = b , if X' = 0

### X·a function

X·a is a function mapping X to other "X", depending on ordinal a.

For X = 0

0·a = -1

(in this case X·a doesn't depend on a)

For non-zero X see Rules.

## Rules

If non-zero type separators not used, rules:

1. []а = a + 1

2. [X]a = [(X; -1; X⁰)][X·a]a

3. X·a = X*-1<1<X">>(X'; 1; a)<X'·a>

And, I think, rule for limit ordinals, possibly, may be: if last ordinal in X (including ordinals inside separators) is a limit ordinal \(\alpha\), [X]a is limit of [\(X_i\)]a where \(X_i\) are X with \(\alpha\) replaced by \(\alpha_i\) which limit is \(\alpha\).

Also, it is well to remember that

X = X*<X'>

X* = X**<X*'>

X*' = 0 , that is X* = X**

X' = X'*<X">

X-1 decreases by 1 last element of X

X⁰ sets to zero last element of X

(X; a; b) = a , if X' ≠ 0

(X; a; b) = b , if X' = 0

[-1]a = a

0·a = -1

<-1> erases previous element

Consequence: if

X' = 0 (and hence X = X* , X" = 0)

then from Rule 3

X·a = X*-1<1<0>>a<0·a> = X-1,a<-1> = X-1

(in this case X·a doesn't depend on a, same way as for X = 0)

*Note 1: Rules were changed. Old version of Rule 2 was [X]a = [X⁰][X·a]a , and old version of X⁰ was (X; -1; X⁰) (← here is new X⁰). (Actually, it is only X⁰ definition changed, and Rules did not). See last version of this blog with old Rule 2 and X⁰ definition.*

*Note 2: Even older versions of this blog contain truly different version of Rule 3: X·a = X*-1<1<X">>a<X'·a> . I made this version of Rule 3 to simplify Rules when I noticed that [a<a>]a = [1<a + 1>]a (I tried to simplify calculations, so I tried to make Rules so that X·a contain no more than one a. And it turned out that a<a> actually contains one, not two a's). But when I wrote "Second plane" section of this blog, I realized that it only works for multiple rows, and does not for multiple planes and beyond. (Note in the Note 2: actually, does: [a<a,0>]a = [1<a,1>]a ; [a<a,0,0>]a = [1<a,0,1>]a etc., but it is still more complex, and I decided not to change anything, - [1<a,0,0>]a is better than [1<a,0,1>]a). So, I returned the original version of Rule 3. See last version of this blog with old Rule 3 version.*

## One element

Special case of rules for X consisted of single element b:

[0]а = a + 1

[b]a = [0][b-1]a

or

[0]а = a + 1

[b]a = [0]\(^{1+b}\)a

or

[b]a = a + 1 + b

### Limit ordinals

[ω]a is a limit of [0]a, [1]a, [2]a...

[ω+1]a = [0][ω]a

[ω+2]a = [0][ω+1]a

[ω2]a is a limit of [ω]a, [ω+1]a, [ω+2]a...

[ω^{2}]a is a limit of [ω]a, [ω2]a, [ω3]a...

## One-dimensional array

Special case of rules for one-dimensional array of elements of X = X*<b>:

[]а = a + 1

[X]a = [(X; -1; X⁰)][X·a]a

X·a = X*-1,a<b-1>

or

[]а = a + 1

[X]a = [X⁰][X-1]a , b=0

[X]a = [X*-1,a<b-1>]a , b>0

In particular,

[X,b]a = [X,]\(^{1+b}\)a

[b+1,]a = [b,a]a = [b,]\(^{1+a}\)a (this resembles FGH, but with 1+a instead of a)

Consequence:

1<b>·a = a<b-1>

### Ordinal subtraction

Ordinal subtraction is operation, opposite to ordinal addition. Since ordinal addition is non-commutative, there are two ordinal subtractions (as for exponentiation, which is also non-commutative and also has two opposite operations: root and logarithm):

(a - b) + b = a

b + (-b + a) = a

*October 21, 2017 edit: in older versions of this blog I used designation a - b for both subtractions, later I started to use designation -b + a for the second subtraction. See also Ordinal subtraction and integer extension of ordinals blog.*

We'll use the second one.

-1+a is as for natural numbers, if a is a finite ordinal, and -1+a = a , if a is an infinite ordinal.

Examples:

-1 + 5 = 4

but

-1 + (ω + 5) = ω + 5 (not ω + 4 since 1 + ω + 4 = ω + 4 but 1 + ω + 5 = ω + 5).

### Two elements

[1,0]a = [a]a = a + 1 + a

[1,1]a = [1,0][1,0]a = a + 1 + a + 1 + a + 1 + a

[1,2]a = [1,0][1,1]a = a + 1 + a + 1 + a + 1 + a + 1 + a + 1 + a + 1 + a + 1 + a

[1,b]a = -1 + (1 + a)*2\(^{1+b}\) for finite or countable a

[1,b]a = a*2\(^{1+b}\) for countable a

[1,b]a = a*2\(^b\) for countable a and b

[2,0]a = [1,a]a = -1 + (1 + a)*2\(^{1+a}\) for finite or countable a

[2,0]a = a*2\(^a\) for countable a

[2,1]a = [2,0][2,0]a

[2,2]a = [2,0][2,1]a = [2,0][2,0][2,0]a

[2,3]a = [2,0][2,2]a = [2,0][2,0][2,0][2,0]a

[3,0]a = [2,a]a

[3,1]a = [3,0][3,0]a

[3,2]a = [3,0][3,1]a = [3,0][3,0][3,0]a

[4,0]a = [3,a]a

[5,0]a = [4,a]a

etc.

### Three elements

[1,0,0]a = [1<2>]a = [a,0]a = [a<1>]a

[1,0,1]a = [1,0,0][1,0,0]a

[1,0,2]a = [1,0,0][1,0,1]a = [1,0,0][1,0,0][1,0,0]a

[1,0,3]a = [1,0,0][1,0,2]a = [1,0,0][1,0,0][1,0,0][1,0,0]a

[1,1,0]a = [1,0,a]a

[1,1,1]a = [1,1,0][1,1,0]a

[1,1,2]a = [1,1,0][1,1,1]a = [1,1,0][1,1,0][1,1,0]a

[1,2,0]a = [1,1,a]a

[1,3,0]a = [1,2,a]a

[2,0,0]a = [1,a,0]a

[3,0,0]a = [2,a,0]a

etc.

### Four elements

[1,0,0,0]a = [1<3>]a = [a,0,0]a = [a<2>]a

[1,0,0,1]a = [1,0,0,0][1,0,0,0]a

[1,0,1,0]a = [1,0,0,a]a

[1,1,0,0]a = [1,0,a,0]a

[2,0,0,0]a = [1,a,0,0]a

etc.

### Five and more elements

[1,0,0,0,0]a = [1<4>]a = [a,0,0,0]a = [a<3>]a

[1<5>]a = [a<4>]a

[1<6>]a = [a<5>]a

[1<7>]a = [a<6>]a

etc.

### Limit ordinals

[1,ω]a is a limit of [1,0]a, [1,1]a, [1,2]a...

[2,ω,0]a is a limit of [2,0,0]a, [2,1,0]a, [2,2,0]a...

[ω,ω,0,0]a is a limit of [ω,0,0,0]a, [ω,1,0,0]a, [ω,2,0,0]a...

[ω<ω>]a is a limit of [ω<0>]a, [ω<1>]a, [ω<2>]a...

## Multi-dimensional array

Special case of rules for multi-dimensional array of elements of X = X*<X'*<b>>:

[]а = a + 1

[X]a = [(X; -1; X⁰)][X·a]a

X·a = X*-1<1<b>>(X'; 1; a)<X'·a>

or

[]а = a + 1

[X]a = [X⁰][X-1]a , X' = 0

[X]a = [X*-1,a<X'-1>]a , X' ≠ 0 , b = 0

[X]a = [X*-1<1<b>>1<X'*-1,a<b-1>>]a , X' ≠ 0 , b > 0

or

[]а = a + 1

[X]a = [X⁰][X-1]a , X' = 0

[X]a = [X·a]a , X' ≠ 0

X·a = X*-1,a<X'-1> , X' ≠ 0 , b = 0

X·a = X*-1<1<b>>1<X'·a> , X' ≠ 0 , b > 0

Consequences:

1<X'>·a = a<X'-1> , if b = 0

1<X'>·a = 1<X'·a> , if b ≠ 0

1<1<b>>·a = 1<a<b-1>> , if b ≠ 0

### Second row

One row separator:

<1,0> = <1,> = <1<1>> = (1)

[1<1,0>]a = [1<a>]a

[1<1,0>1]a = [1<1,0>][1<1,0>]a

[1<1,0>2]a = [1<1,0>][1<1,0>1]a = [1<1,0>][1<1,0>][1<1,0>]a

[1<1,0>1,0]a = [1<1,0>a]a

[1<1,0>1,0,0]a = [1<1,0>a,0]a

[1<1,0>1,0,0,0]a = [1<1,0>a,0,0]a

[2<1,0>]a = [1<1,0>1<a>]a

[3<1,0>]a = [2<1,0>1<a>]a

[1,0<1,0>]a = [1<1,1>]a = [a<1,0>]a

[1,1<1,0>]a = [1,0<1,0>1<a>]a = [1<1,1>1<a>]a

[1,2<1,0>]a = [1,1<1,0>1<a>]a

[1,0,0<1,0>]a = [1<1,2>]a = [a<1,1>]a = [a,0<1,0>]a

[1,0,0,0<1,0>]a = [1<1,3>]a = [a<1,2>]a = [a,0,0<1,0>]a

[1<1,4>]a = [a<1,3>]a

[1<1,5>]a = [a<1,4>]a

etc.

### Plane beyond second row

Multiple rows separators:

<2,0> = <2,> = <2<1>> = (1)(1)

<3,0> = <3,> = <3<1>> = (1)(1)(1)

etc.

[1<2,0>]a = [1<a><1,0>]a = [1<1,a>]a

[1<2,0>1]a = [1<2,0>][1<2,0>]a

[1<1,0>1<1,0>]a = [1<1,0><1,0>1<a>]a = [1<2,0>1<a>]a

[1<1,0>2<1,0>]a = [1<1,0>1<1,0>1<a>]a

[1<1,0>3<1,0>]a = [1<1,0>2<1,0>1<a>]a

[1<1,0>1,0<1,0>]a = [1<1,0>1<1,1>]a = [1<1,0>a<1,0>]a

[1<1,0>1,1<1,0>]a = [1<1,0>1,0<1,0>1<a>]a = [1<1,0>1<1,1>1<a>]a

[1<1,0>1,0,0<1,0>]a = [1<1,0>1<1,2>]a = [1<1,0>a<1,1>]a

[2<2,0>]a = [1<1,0>1<a><1,0>]a = [1<1,0>1<1,a>]a

[3<2,0>]a = [2<1,0>1<1,a>]a

[1,0<2,0>]a = [1<2,1>]a = [a<2,0>]a

[1<2,2>]a = [a<2,1>]a

[1<3,0>]a = [1<1,0>1<a><2,0>]a = [1<1,0>1<2,a>]a

[1<4,0>]a = [1<1,0>1<3,a>]a

[1<5,0>]a = [1<1,0>1<4,a>]a

etc.

### Second plane

One plane separator:

<1,0,0> = <1,,> = <1<2>> = (2)

[1<1,0,0>]a = [1<a,0>]a

[1<1,0,0>1]a = [1<1,0,0>][1<1,0,0>]a

[1<1,0,0>1,0]a = [1<1,0,0>a]a

[1<1,0,0>1,0,0]a = [1<1,0,0>a,0]a

[1<1,0,0>1<1,0>]a = [1<1,0,0>1<a>]a

[1<1,0,0>2<1,0>]a = [1<1,0,0>1<1,0>1<a>]a

[1<1,0,0>1<1,1>]a = [1<1,0,0>a<1,0>]a

[1<1,0,0>1<2,0>]a = [1<1,0,0>1<1,a>]a

[1<1,0,0>1<2,1>]a = [1<1,0,0>a<2,0>]a

[1<1,0,0>1<3,0>]a = [1<1,0,0>1<2,a>]a

[2<1,0,0>]a = [1<1,0,0>1<a,0>]a

[3<1,0,0>]a = [2<1,0,0>1<a,0>]a

etc.

### 3D space beyond second plane

Multiple planes separators:

<2,0,0> = <2,,> = <2<2>> = (2)(2)

<3,0,0> = <3,,> = <3<2>> = (2)(2)(2)

etc.

[1<2,0,0>]a = [1<1,a,0>]a

[1<2,0,0>1]a = [1<2,0,0>][1<2,0,0>]a

[2<2,0,0>]a = [1<1,0,0>1<1,a,0>]a

[1<2,0,1>]a = [a<2,0,0>]a

[2<2,0,1>]a = [1,a<2,0,0>]a

[1<2,1,0>]a = [1<2,0,a>]a

[2<2,1,0>]a = [1<1,0>1<2,0,a>]a

[1<3,0,0>]a = [1<2,a,0>]a

[1<4,0,0>]a = [1<3,a,0>]a

[1<5,0,0>]a = [1<4,a,0>]a

[1<6,0,0>]a = [1<5,a,0>]a

etc.

### Four and more dimensions

Separators:

<1<3>> = (3)

<1<4>> = (4)

<1<5>> = (5)

etc.

[1<1<3>>]a = [1<a<2>>]a

[1<1<4>>]a = [1<a<3>>]a

[1<1<5>>]a = [1<a<4>>]a

etc.

## Nested array

Special case of rules for nested array of elements of X = X*<X'*<X">> is also general case of these rules:

[]а = a + 1

[X]a = [(X; -1; X⁰)][X·a]a

X·a = X*-1<1<X">>(X'; 1; a)<X'·a>

or

[]а = a + 1

[X]a = [X⁰][X-1]a , X' = 0

[X]a = [X·a]a , X' ≠ 0

X·a = X*-1,a<X'-1> , X' ≠ 0 , X" = 0

X·a = X*-1<1<X">>1<X'·a> , X' ≠ 0 , X" ≠ 0

Consequences.

If X' = 0:

1<X>·a = a<X-1>

1<1<X>>·a = 1<a<X-1>>

1<1<1<X>>>·a = 1<1<a<X-1>>>

1<1<1<1<X>>>>·a = 1<1<1<a<X-1>>>>

1<1<1<1<1<X>>>>>·a = 1<1<1<1<a<X-1>>>>>

etc.

If X' ≠ 0:

1<X>·a = 1<X'·a>

1<1<X>>·a = 1<1<X'·a>>

1<1<1<X>>>·a = 1<1<1<X'·a>>>

1<1<1<1<X>>>>·a = 1<1<1<1<X'·a>>>>

1<1<1<1<1<X>>>>>·a = 1<1<1<1<1<X'·a>>>>>

etc.

### Second row of dimensions

[1<1<1<1>>>]a = [1<1<a>>]a

[1<1<1<1>>>1]a = [1<1<1<1>>>][1<1<1<1>>>]a

[2<1<1<1>>>]a = [1<1<1<1>>>1<1<a>>]a

[3<1<1<1>>>]a = [2<1<1<1>>>1<1<a>>]a

[1<1<1<1>>1>]a = [a<1<1<1>>>]a

[1<1<1<1>>2>]a = [a<1<1<1>>1>]a

[1<2<1<1>>>]a = [1<1<1<1>>1<a>>]a

[1<1<1<1>1>>]a = [1<a<1<1>>>]a

[1<1<1<1>2>>]a = [1<a<1<1>1>>]a

etc.

### Plane of dimensions beyond second row

[1<1<2<1>>>]a = [1<1<1,a>>]a

[1<1<3<1>>>]a = [1<1<2,a>>]a

etc.

### Three and more dimensions of dimensions

[1<1<1<2>>>]a = [1<1<a<1>>>]a

[1<1<1<3>>>]a = [1<1<a<2>>>]a

etc.

### More nesting levels

[1<1<1<1<1>>>>]a = [1<1<1<a>>>]a

[1<1<1<1<1<1>>>>>]a = [1<1<1<1<a>>>>]a

[1<1<1<1<1<1<1>>>>>>]a = [1<1<1<1<1<a>>>>>]a

etc.

## Generalized Veblen function

<X> is suitable not only for designation of [X]a functions, but also for designation of other functions of multi-dimentional arrays of variables. Particularly, for multi-dimentional generalization of Veblen function.

Veblen function has one row of variables, but it may be generalized to possess larger arrays of variables.

Generalized Veblen function with <X> designations:

φ(1,0) = φ(1,) = φ(1<1>) = \(\epsilon_0\)

φ(2,0) = φ(2,) = φ(2<1>) = \(\zeta_0\) - Cantor's ordinal

φ(3,0) = φ(3,) = φ(3<1>)

φ(1,0,0) = φ(1,,) = φ(1<2>) = \(\Gamma_0\) - Feferman–Schütte ordinal

φ(1,0,0,0) = φ(1,,,) = φ(1<3>) - Ackermann ordinal

φ(1,0,0,0,0) = φ(1<4>)

φ(1,0,0,0,0,0) = φ(1<5>)

φ(1<ω>) - small Veblen ordinal

φ(1<1,>) - large Veblen ordinal, least ordinal using second row of variables.

Go further.

Let \(V_i\) - ordinals such as \(V_i\) = φ(1<\(V_i\)>), numbered from smaller to larger.

\(V_0\) = φ(1<1,>) - large Veblen ordinal.

\(V_0\) + 1

\(V_0\) + 2

\(V_0\) + 3

...

\(V_0\)2

\(V_0\)3

...

\(V_0^2\)

\(V_0^3\)

...

\(V_0\uparrow\uparrow 2\)

\(V_0\uparrow\uparrow 3\)

...

\(\epsilon_{V_0 + 1}\) (since \(\epsilon_{V_0} = V_0\))

\(\epsilon_{\epsilon_{V_0 + 1}}\)

\(\epsilon_{\epsilon_{\epsilon_{V_0 + 1}}}\)

...

\(\zeta_{V_0 + 1}\) (since \(\zeta_{V_0} = V_0\))

\(\zeta_{\zeta_{V_0 + 1}}\)

\(\zeta_{\zeta_{\zeta_{V_0 + 1}}}\)

...

φ(3,\(V_0\) + 1)

φ(3,φ(3,\(V_0\) + 1))

φ(3,φ(3,φ(3,\(V_0\) + 1)))

...

φ(4,\(V_0\) + 1)

φ(5,\(V_0\) + 1)

...

φ(\(V_0\),0)

φ(φ(\(V_0\),0),0)

φ(φ(φ(\(V_0\),0),0),0)

...

\(\Gamma_{V_0 + 1}\)

\(\Gamma_{\Gamma_{V_0 + 1}}\)

\(\Gamma_{\Gamma_{\Gamma_{V_0 + 1}}}\)

...

φ(1,1,\(V_0\) + 1)

φ(1,1,φ(1,0,\(V_0\) + 1))

φ(1,1,φ(1,0,φ(1,0,\(V_0\) + 1)))

...

φ(1,2,\(V_0\) + 1)

φ(1,3,\(V_0\) + 1)

...

φ(1,\(V_0\),0)

φ(1,φ(1,\(V_0\),0),0)

φ(1,φ(1,φ(1,\(V_0\),0),0),0)

...

φ(2,0,\(V_0\) + 1)

φ(3,0,\(V_0\) + 1)

...

φ(\(V_0\),0,0)

φ(φ(\(V_0\),0,0),0,0)

φ(φ(φ(\(V_0\),0,0),0,0),0,0)

...

φ(1,0,0,\(V_0\) + 1) = φ(1<3>\(V_0\) + 1)

φ(1,0,0,0,\(V_0\) + 1) = φ(1<4>\(V_0\) + 1)

φ(1,0,0,0,0,\(V_0\) + 1) = φ(1<5>\(V_0\) + 1)

...

φ(2<\(V_0\)>)

φ(3<\(V_0\)>)

...

φ(\(V_0\)<\(V_0\)>)

φ(φ(\(V_0\)<\(V_0\)>)<\(V_0\)>)

φ(φ(φ(\(V_0\)<\(V_0\)>)<\(V_0\)>)<\(V_0\)>)

...

φ(1<\(V_0\) + 1>)

φ(1<\(V_0\) + 2>)

φ(1<\(V_0\) + 3>)

...

φ(1<φ(1<\(V_0\) + 1>)>)

φ(1<φ(1<φ(1<\(V_0\) + 1>)>)>)

...

\(V_1\) = φ(1<1,>1) - second V number.

φ(1<φ(1<\(V_1\) + 1>)>)

φ(1<φ(1<φ(1<\(V_1\) + 1>)>)>)

...

\(V_2\) = φ(1<1,>2)

\(V_3\) = φ(1<1,>3)

...

\(V_{V_0}\)

\(V_{V_{V_0}}\)

...

φ(1<1,>1,)

φ(1<1,>1,1)

φ(1<1,>1,2)

...

φ(1<1,>2,)

φ(1<1,>3,)

...

φ(1<1,>1,,) = φ(1<1,>1<2>)

φ(1<1,>1,,,) = φ(1<1,>1<3>)

...

φ(1<1,>1<ω>)

...

φ(1<1,>1<\(V_1\)>)

φ(1<1,>1<φ(1<1,>1<\(V_1\)>)>)

φ(1<1,>1<φ(1<1,>1<φ(1<1,>1<\(V_1\)>)>)>)

...

φ(2<1,>) - second after \(V_0\) ordinal, changing second row of variables.

φ(3<1,>)

...

φ(\(V_0\)<1,>)

φ(φ(\(V_0\)<1,>)<1,>)

φ(φ(φ(\(V_0\)<1,>)<1,>)<1,>)

...

φ(1,0<1,>) = φ(1<1,1>) - least ordinal using second variable in second row

φ(1<1,2>)

φ(1<1,3>)

...

φ(1<1,\(V_0\)>)

φ(1<1,φ(1<\(V_0\)>)>)

φ(1<1,φ(1<φ(1<\(V_0\)>)>)>)

...

φ(1<2,>) - least ordinal using third row of variables

φ(1<3,>)

φ(1<4,>)

...

φ(1<\(V_0\),>)

φ(1<φ(1<\(V_0\),>),>)

φ(1<φ(1<φ(1<\(V_0\),>),>),>)

...

φ(1<1,,>) = φ(1<1<2>>) - least ordinal using second plane of variables

φ(1<φ(1<1<2>>)<2>>)

φ(1<φ(1<φ(1<1<2>>)<2>>)<2>>)

...

φ(1<1,,,>) = φ(1<1<3>>) - least ordinal using second 3D space of variables

φ(1<1<4>>)

φ(1<1<5>>)

...

φ(1<1<ω>>) - least ordinal using infinite-dimensional space of variables

...

φ(1<1<1,>>) = \(\psi(\Omega\uparrow\uparrow 4)\) (ѱ is Madore function)

φ(1<1<1<1,>>>) = \(\psi(\Omega\uparrow\uparrow 5)\)

φ(1<1<1<1<1,>>>>) = \(\psi(\Omega\uparrow\uparrow 6)\)

...

φ(1<1|1>) - Bachmann-Howard ordinal, least ordinal using non-zero type of separators.

## Examples of [X]ω

[0]ω = ω + 1

[1]ω = ω + 2

[2]ω = ω + 3

[1,0]ω = ω2

[1,1]ω = ω4

[1,2]ω = ω8

[2,0]ω = \(\omega^2\)

[2,1]ω = \(\omega^\omega\)

[2,2]ω = \(\omega^{\omega^\omega}\)

[3,0]ω = \(\epsilon_0\)

[2,0][3,0]ω = \(\epsilon_0^2\)

[2,1][3,0]ω = \(\epsilon_0^{\epsilon_0}\)

[2,2][3,0]ω = \(\epsilon_0^{\epsilon_0^{\epsilon_0}}\)

[2,ω][3,0]ω = \(\epsilon_1\)

[2,ω2][3,0]ω = \(\epsilon_2\)

[2,ω3][3,0]ω = \(\epsilon_3\)

[3,1]ω = \(\epsilon_{\epsilon_0}\)

[3,2]ω = \(\epsilon_{\epsilon_{\epsilon_0}}\)

[4,0]ω = \(\zeta_0\) - Cantor's ordinal

[2,ω][4,0]ω = \(\epsilon_{\zeta_0 + 1}\)

[2,ω2][4,0]ω = \(\epsilon_{\zeta_0 + 2}\)

[2,ω3][4,0]ω = \(\epsilon_{\zeta_0 + 3}\)

[3,0][4,0]ω = \(\epsilon_{\zeta_0 2}\)

[3,1][4,0]ω = \(\epsilon_{\epsilon_{\zeta_0 2}}\)

[3,2][4,0]ω = \(\epsilon_{\epsilon_{\epsilon_{\zeta_0 2}}}\)

[3,ω][4,0]ω = \(\zeta_1\)

[3,ω2][4,0]ω = \(\zeta_2\)

[3,ω3][4,0]ω = \(\zeta_3\)

[4,1]ω = \(\zeta_{\zeta_0}\)

[4,2]ω = \(\zeta_{\zeta_{\zeta_0}}\)

[5,0]ω = φ(3,0)

[5,1]ω = φ(3,φ(3,0))

[5,2]ω = φ(3,φ(3,φ(3,0)))

[6,0]ω = φ(4,0)

[7,0]ω = φ(5,0)

[1,0,0]ω = φ(ω,0)

[ω,0][1,0,0]ω = φ(ω,1)

[ω,1][1,0,0]ω = φ(ω,2)

[ω,2][1,0,0]ω = φ(ω,3)

[ω,ω][1,0,0]ω = φ(ω,ω)

[ω+1,0][1,0,0]ω = φ(ω,φ(ω,0))

[ω+1,1][1,0,0]ω = φ(ω,φ(ω,φ(ω,0)))

[ω+1,2][1,0,0]ω = φ(ω,φ(ω,φ(ω,φ(ω,0))))

[ω+1,ω][1,0,0]ω = φ(ω+1,0)

[ω+1,ω2][1,0,0]ω = φ(ω+1,1)

[ω+1,ω3][1,0,0]ω = φ(ω+1,2)

[ω+2,0][1,0,0]ω = φ(ω+1,φ(ω+1,0))

[ω+2,1][1,0,0]ω = φ(ω+1,φ(ω+1,φ(ω+1,0)))

[ω+2,2][1,0,0]ω = φ(ω+1,φ(ω+1,φ(ω+1,φ(ω+1,0))))

[ω+2,ω][1,0,0]ω = φ(ω+2,0)

[ω+3,0][1,0,0]ω = φ(ω+2,φ(ω+2,0))

[ω+3,ω][1,0,0]ω = φ(ω+3,0)

[1,0,1]ω = φ(φ(ω,0),0)

[1,0,2]ω = φ(φ(φ(ω,0),0),0)

[1,0,3]ω = φ(φ(φ(φ(ω,0),0),0),0)

[1,1,0]ω = φ(1,0,0) = Γ₀ - Feferman–Schütte ordinal

[0][1,1,0]ω = Γ₀+1

[1,0][1,1,0]ω = Γ₀·2

[2,0][1,1,0]ω = Γ₀↑2

[2,1][1,1,0]ω = Γ₀↑↑2

[2,2][1,1,0]ω = Γ₀↑↑3

[2,ω][1,1,0]ω = \(\epsilon_{Γ₀+1}\)

[3,0][1,1,0]ω = \(\epsilon_{\epsilon_{Γ₀+1}}\)

[3,ω][1,1,0]ω = \(\zeta_{Γ₀+1}\)

[4,0][1,1,0]ω = \(\zeta_{\zeta_{Γ₀+1}}\)

[4,ω][1,1,0]ω = φ(3,Γ₀+1)

[5,ω][1,1,0]ω = φ(4,Γ₀+1)

[ω,0][1,1,0]ω = φ(ω,Γ₀+1)

[ω+1,0][1,1,0]ω = φ(ω,Γ₀2)

[ω+2,0][1,1,0]ω = φ(ω+1,Γ₀2)

[ω2,0][1,1,0]ω = φ(ω2,Γ₀+1)

[ω^{2},0][1,1,0]ω = φ(ω^{2},Γ₀+1)

[ε₀,0][1,1,0]ω = φ(ε₀,Γ₀+1)

[1,0,0][1,1,0]ω = φ(Γ₀,1)

[1,0,1][1,1,0]ω = φ(φ(Γ₀,1),0)

[1,0,2][1,1,0]ω = φ(φ(φ(Γ₀,1),0),0)

[1,0,ω][1,1,0]ω = φ(1,0,1)

[1,0,ω·2][1,1,0]ω = φ(1,0,2)

[1,0,ω·3][1,1,0]ω = φ(1,0,3)

[1,1,1]ω = φ(1,0,Γ₀)

[1,1,2]ω = φ(1,0,φ(1,0,φ(1,0,Γ₀)))

[1,1,3]ω = φ(1,0,φ(1,0,φ(1,0,φ(1,0,Γ₀))))

[1,2,0]ω = φ(1,1,0)

[1,1,ω][1,2,0]ω = φ(1,1,1)

[1,1,ω·2][1,2,0]ω = φ(1,1,2)

[1,1,ω·3][1,2,0]ω = φ(1,1,3)

[1,2,1]ω = φ(1,1,φ(1,1,0))

[1,2,2]ω = φ(1,1,φ(1,1,φ(1,1,0)))

[1,3,0]ω = φ(1,2,0)

[2,0,0]ω = φ(1,ω,0)

[2,0,1]ω = φ(1,φ(1,ω,0),0)

[2,0,2]ω = φ(1,φ(1,φ(1,ω,0),0),0)

[2,1,0]ω = φ(2,0,0)

[2,0,0][2,1,0]ω = φ(1,φ(2,0,0)+1,0)

[2,0,1][2,1,0]ω = φ(1,φ(1,φ(2,0,0)+1,0)+1,0)

[2,0,ω][2,1,0]ω = φ(2,0,1)

[2,1,1]ω = φ(2,0,φ(2,0,0))

[2,1,2]ω = φ(2,0,φ(2,0,φ(2,0,0)))

[2,2,0]ω = φ(2,1,0)

[2,3,0]ω = φ(2,2,0)

[3,0,0]ω = φ(2,ω,0)

[3,1,0]ω = φ(3,0,0)

[1,0,0,0]ω = φ(ω,0,0)

[1,0,0,1]ω = φ(φ(ω,0,0),0,0)

[1,0,0,2]ω = φ(φ(φ(ω,0,0),0,0),0,0)

[1,0,1,0]ω = φ(1,0,0,0) - Ackermann ordinal

[1,0,0,ω][1,0,1,0]ω = φ(1,0,0,1)

[1,0,1,1]ω = φ(1,0,0,φ(1,0,0,0))

[1,0,1,2]ω = φ(1,0,0,φ(1,0,0,φ(1,0,0,0)))

[1,0,2,0]ω = φ(1,0,1,0)

[1,0,1,ω][1,0,2,0]ω = φ(1,0,1,1)

[1,0,2,1]ω = φ(1,0,1,φ(1,0,1,1))

[1,0,3,0]ω = φ(1,0,2,0)

[1,1,0,0]ω = φ(1,0,ω,0)

[1,2,0,0]ω = φ(1,1,ω,0)

[2,0,0,0]ω = φ(1,ω,0,0)

[3,0,0,0]ω = φ(2,ω,0,0)

[1,0,0,0,0]ω = φ(ω,0,0,0)

[1,0,0,0,1]ω = φ(φ(ω,0,0,0),0,0,0)

[1,0,0,0,2]ω = φ(φ(φ(ω,0,0,0),0,0,0),0,0,0)

[1,0,0,1,0]ω = φ(1,0,0,0,0)

[1,0,1,0,0]ω = φ(1,0,0,ω,0)

[1,1,0,0,0]ω = φ(1,0,ω,0,0)

[2,0,0,0,0]ω = φ(1,ω,0,0,0)

[1,0,0,0,0,0]ω = φ(ω,0,0,0,0)

[1,0,0,0,1,0]ω = φ(1,0,0,0,0,0)

[1<1,0>]ω = φ(1<ω>) - small Veblen ordinal

[1<1,0>1,0]ω = φ(1<1,0>) = φ(1<1<1>>) - large Veblen ordinal

[1<1|1>]ω = φ(1<1|1>) - Bachmann-Howard ordinal