FANDOM


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.


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<sup>2</sup>]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 + (a - b) = a

We'll use the second one.

a-1 is as for natural numbers, if a is a finite ordinal, and a-1 = a , if a is a countable ordinal.

Examples:

5 - 1 = 4

but

(ω + 5) - 1 = 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 + a)*2\(^{1+b}\) - 1 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 + a)*2\(^{1+a}\) - 1 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)

2,0][1,1,0]ω = φ(ω2,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

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