User blog:Rgetar/Thoughts on further generalization of Veblen function

Argument of generalized Veblen function φ(X) is array of ordinals X, and its value is ordinal. X may be a natural number, then an ordinal, then, beginning from 11, a multi-ordinal array, then, beginning from 11 1, a multi-dimensional array of ordinals, then, beginning from 11 1 1, a multi-trimensional array of ordinals, etc.

To go further, we need larger arrays.

Larger arrays
Consider least array X such as X = X1. (Let me remind you how arrays are compared: first compared theirs elements with larger equal coordinates, then, if this elements are equal, elements with lesser equal coordinates, and so on).

Let's denote this array as c0. So,

c0 = c01

Least array larger than c0 is

c01, 1.

Then

c01, 2

c01, ω

c01, Ω

c01, 11

c01, 11 1

c01, 11 1 1

c02

c03

c0ω

c01, 1 1

c01, 1 1 1

c01, 1 1 1 1

c1 - second array X such as X = X1

c11, 1

c11, 11

c11, 11 1

c11, 11 1 1

c11, c01

c11, c02

c11, c01, 1 1

c11, c01, 1 1 1

c11, c01, 1 1 1 1

c12

c11, 1 1

c11, 1 1 1

c11, 1 1 1 1

c2

c22

c21, 1 1

c21, 1 1 1

c21, 1 1 1 1

c3

c4

c5

cω

...

I think, that φ(c0) = BHO. And, I think, least ordinal such as α = φ(cα) we can designate as φ(c1, 0). So, subscript of c is array of ordinals, not just an ordinal. So,

c1, 0 = c11

c11 1

c11 1 1

cc 0

cc c 0

cc c c 0

cc c c c 0

d0

Arrays of arrays, Veblen-like functions
I think, that we may designate these large arrays with Veblen-like function φ1(X1). Argument of φ1 is array of arrays of ordinals X1, and its value is array of ordinals (whereas argument of Veblen function φ is arrays of ordinals, and its value is ordinal).

φ1(X1) is defined same way as φ(X), but if X1 is array of ordinals X, then φ1(X) = X1 (instead of φ(α) = ωα).

To designate large arrays of arrays of ordinals we may similarly define another Veblen-like function φ2(X2), which argument is array of arrays of arrays of ordinals, and value is array of arrays of ordinals. And if X2 is array of arrays of ordinals X1, then

φ2(X1) = <1|X1>1

Then, to designate large arrays of arrays of arrays of ordinals, we may introduce another Veblen-like function φ3(X3), then φ4(X4), φ5(X5), and so on.

To designate arrays of arrays we need new separators. I think, that we may use something like html tags:

⟨order|coordinates⟩element⟨/order⟩

where, say, for arrays of ordinals order = 0, for arrays of arrays of ordinals order = 1, for arrays of arrays of arrays of ordinals order = 2, and so on.

For 0-th order:

⟨0|c⟩ = ⟨c⟩

⟨/0⟩ = ,

So,

⟨0|c⟩e⟨/0⟩ = ⟨c⟩e, = ce,

where e is ordinal, and c is array of ordinals.

For 1-st order let's introduce some separator for ⟨/1⟩, for example, ;:

⟨/1⟩ = ;

So,

⟨1|c⟩e⟨/1⟩ = ⟨1|c⟩e;

where e is array of ordinal, and c is array of arrays of ordinals.

2-nd order:

⟨2|c⟩e⟨/2⟩

where e is array of arrays of ordinal, and c is array of arrays of arras of ordinals.

Etc.

A seperator (that is "closing tag") may be omitted, if this is last c-e pair in the array.

So,

φ1(0) = 01 = 1

φ1(1) = 11

φ1(1, 0) = φ1(11) = 11 1

φ1(11 1) = 11 1 1

φ1(11 1 1) = 11 1 1 1

φ1(1; 0) = c0

φ1(c0) = c01 = c0

φ1(c01, 1) = c01, 1 1

φ1(c01, 1 1) = c01, 1 1 1

φ1(c01, 1 1 1) = c01, 1 1 1 1

φ1(1; 1) = c1

φ1(1; 2) = c2

φ1(1; 3) = c3

φ1(1; ω) = cω

φ1(1; 1, 0) = c1, 0

φ1(1; 11 1) = c11 1

φ1(2; 0) = d0

...

Numbers larger than any ordinal?
Array 1, 0 is larger than any array α, but α is ordinal, so, arrays of ordinal may be considered as "extension" of ordinals. For example, 1, 0 may be considered as least number larger than any ordinal.

So, the same way as ordinals may be considered as transfinite numbers, arrays of ordinals may be considered as "transordinal" numbers.

Then, 1; 0 is larger than any array of ordinals, ⟨2|1⟩1⟨/2⟩ 0 is larger than any array of arrays of ordinals, ⟨3|1⟩1⟨/3⟩ 0 is larger than any array of arrays of arrays of ordinals, etc.

But I suspect that this may be incorrect. Can object larger than any ordinal exist? Possibly, such an object also would be an ordinal, so, it cannot exist, but I'm not sure.

Ordinals as arrays
If arrays of ordinals are "transordinal" numbers, maybe, ordinals may be considered as arrays of natural numbers?

Apparently, yes. So, 1, 0 is least ordinal larger then any natural number (that is ω), 1, 0, 0 is ω2, etc.

In fact, this array is Cantor normal form, where + is separator, coefficients are elements, and exponents of ω are coordinates, that is

... ce, ... = ... ωce + ...

And rule for Veblen function φ(X), X = α is similar to rule for Veblen-like function φ1(X1), X1 = X.

Rule for Veblen-like function:

φ1(X) = X1

Rule for Veblen function:

φ(α) = α1 = ωα

Arrays as uncountable ordinals
If ordinals are arrays of natural numbers, then all elements of these arrays are less than ω. I'm still not sure, can numbers larger than any ordinal exist, and if not, then, maybe, arrays of ordinals are uncountable ordinals, not "transordinal" numbers?

If so, then arrays of ordinals contain only elements lesser than first uncountable ordinal Ω, and 1, 0 = Ω.

Then, array of arrays of ordinals is array of ordinals lesser than Ω2, and 1; 0 = Ω2.

Arrays as positional numbering systems
But why 1, 0 is Ω, not some other large ordinal, for example, Church-Kleene ordinal ω1CK, or Ωω? Maybe, this is free choice?

If so, maybe, we can choose any ordinal α as limitation of array elements, so as all elements should not be larger than α? I think, yes. Then

1, 0 = α.

For example, if α = 10 then

1, 0 = 10

1, 0, 0 = 100

etc.

So, we get decimal numbering system.

(Actually, I think, elements can be ≥ α, just 1, 0 = α. For example, in decimal system array 15, 11 is 15 * 10 + 11 * 1 = 161, but in this case different arrays can be equal: 15, 11 = 1, 6, 1).

In general, it appears that array with limitation of elements α is α-ry positional numbering system.

For example, for α = ω array is Cantor normal form, and, in fact, Cantor normal form is ω-ry numbering system.

Array-ordinal correspondence
Is it necessary to imply that an array of ordinals has "limitation of elements"?

I think, if no, that is if arrays without limitation can exist, then "transordinal" numbers exist, and array 1, 0 without limitation is least "transordinal" number.

And if yes, then we should imply that for given class of arrays 1, 0 is some ordinal. So, there is array-ordinal correspondence, and arrays of ordinals can be considered as ordinals.

For example, array of Veblen function variables X. If we imply that for this array 1, 0 = Ω, then we can consider X as single ordinal written in Ω-ry positional numbering system.

Example of Veblen function with X represented as single ordinal:

φ(Ω) = ε0

φ(Ω + 1) = ε1

φ(Ω + ω) = εω

φ(Ω2) = ζ0

φ(Ω3) = η0

φ(Ωω) = φ(ω, 0)

φ(Ω2) = Γ0

φ(Ω23 + Ω5 + 7) = φ(3, 5, 7)

φ(Ωω) = φ(ω1) = SVO

φ(ΩΩ) = φ(Ω1, 0) = φ(Ω1) = φ(1, 01) = LVO

...