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  • Rgetar

    Designations of arrays

    January 29, 2018 by Rgetar

    Recently I was thinking about further generalization of Veblen function, and there is a problem to avoid transfinitely long expressions (Alemagno12 wrote about this in comments).

    Specifically, I introduced Veblen-like functions φ1(X2), φ2(X3), φ3(X4), ..., and there are ordinals expressed as

    φ(X)

    φ(φ1(X2))

    φ(φ12(X3)))

    φ(φ123(X4))))

    ...

    But what if we use φω(Xω + 1), φω + 1(Xω + 2), ...? We should write transfinitely many Veblen-like functions?

    In comments I wrote that instead of φ(φ1(X2)), maybe, it will be something like φ(X2), that is argument of Veblen function will be not only array of ordinals, but also array of arrays of ordinals, array of arrays of arrays of ordinals etc. (there is something like this in OCFs). But then I came up with…

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  • Rgetar

    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 111, a multi-dimensional array of ordinals, then, beginning from 1111, a multi-trimensional array of ordinals, etc.

    To go further, we need 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 these 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, 111

    c01, 1111

    c02

    c03

    c0ω

    c01, 11

    c01, 111

    c01, 1111

    c1 - second…


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  • Rgetar

    Program

    January 5, 2018 by Rgetar

    I wrote program I was going to write. I tried to upload it to Googology Wiki, but I did not succeed, since exe is not permitted file type here. Then I tried to change its extension to jpg and upload, but it also did not work, and a message appeared: "Cannot upload this file because Internet Explorer would detect it as "application/x-msdownload", which is a disallowed and potentially dangerous file type". Then I googled "free file hosting", and first link was filedropper.com. I successfully uploaded the program there, here is the link:

    https://ufile.io/jrvjq


    1. Download the program Project2 (link above).

    2. Run file Project2.exe.

    3. Click Start button.

    4. Open your web browser.

    5. Type localhost.

    6. Click ordinal to add its fundamental sequence to…


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  • Rgetar

    I added converting generalized Veblen function to Madore function to my program.

    So, here is list of ordinals, generated by this program, it is the same list as my previous list of 4368 ordinals from 0 to BHO in 1073 fundamental sequences, but the ordinals are expressed through Madore ordinal collapsing function ψ(α) instead of generalized Veblen function φ(X).

    FS #1

    0

    1

    2

    3

    4

    5

    6

    7

    8

    9

    FS #2

    ω

    ω + 1

    ω + 2

    ω + 3

    ω + 4

    ω + 5

    ω + 6

    FS #3

    ω2

    ω2 + 1

    ω2 + 2

    ω2 + 3

    ω2 + 4

    FS #4

    ω3

    ω3 + 1

    ω3 + 2

    ω3 + 3

    ω3 + 4

    FS #5

    ω4

    ω5

    ω6

    ω7

    FS #6

    ω2

    ω2 + 1

    ω2 + 2

    ω2 + 3

    ω2 + 4

    FS #7

    ω2 + ω

    ω2 + ω2

    ω2 + ω3

    ω2 + ω4

    FS #8

    ω22

    ω23

    ω24

    ω25

    FS #9

    ω3

    ω3 + 1

    ω3 + 2

    ω3 + 3

    ω3 + 4

    FS #10

    ω3 + ω

    ω3 + ω2

    ω3 + ω3

    ω3 + ω4

    FS #11

    ω3 + ω2

    ω3 + ω22

    ω3 + ω23

    ω3 + ω24

    FS #12

    ω32

    ω33

    ω34

    ω35

    FS #13

    ω4

    ω5

    ω6

    ω7

    FS #14

    ωω

    ωω + 1

    ωω + 2

    ωω + 3

    ωω + 4

    FS #15

    ωω + ω

    ωω + ω2

    ωω + ω3

    ωω + ω4

    FS #16

    ωω2

    ωω…

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  • Rgetar

    Definitions of eo, cofbeo, leo, lest, X0 see here.

    asttr means "array shift to the right"

    asttr(X) is array such as

    eo(asttr(X); X1) = eo(X; 1 + X1) if cofbeo(X1) = 0 (that is X1 is ordinal)

    eo(asttr(X); X1) = eo(X; X1) if cofbeo(X1) ≠ 0 (that is X1 is not ordinal)

    ate means "array to exponent"

    ate(0) = 0

    ate(⟨X1⟩α1, ⟨X2⟩α2, ⟨X3⟩α3, ...) = Ωate(X1)α1 + Ωate(X2)α2 + Ωate(X3)α3

    ata means "array to argument"

    ata(1, α) = α

    ata(1 + β, α) = Ωβ(1 + α) if β > 0

    ata(X) = Ωate(asttr(X))(1 + α) if cofbeo(X) > 1

    vtm means "Veblen to Madore"

    Let α - standard form of ordinal expressed using generalized Veblen function φ(X). Then vtm(α) is α expressed using Madore function ψ(α).

    vtm(0) = 0

    vtm(1) = 1

    vtm(α + β) = vtm(α) + vtm(β)

    vtm(ωα) = ωvtm(α)

    If cofbeo(X0) ≠ 0 and

    • leo(X0) = φ(X…
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