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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 Veblenlike functions Ï†_{1}(X_{2}), Ï†_{2}(X_{3}), Ï†_{3}(X_{4}), ..., and there are ordinals expressed as
Ï†(X)
Ï†(Ï†_{1}(X_{2}))
Ï†(Ï†_{1}(Ï†_{2}(X_{3})))
Ï†(Ï†_{1}(Ï†_{2}(Ï†_{3}(X_{4}))))
...
But what if we use Ï†_{Ï‰}(X_{Ï‰ + 1}), Ï†_{Ï‰ + 1}(X_{Ï‰ + 2}), ...? We should write transfinitely many Veblenlike functions?
In comments I wrote that instead of Ï†(Ï†_{1}(X_{2})), maybe, it will be something like Ï†(X_{2}), 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 withâ€¦
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Argument of generalized Veblen function Ï†(X) is array of ordinals X, and its value is ordinal. X may be a natural number, then an ordinal, then, beginning from ^{1}1, a multiordinal array, then, beginning from ^{11}1, a multidimensional array of ordinals, then, beginning from ^{111}1, a multitrimensional array of ordinals, etc.
To go further, we need larger arrays.
Consider least array X such as X = ^{X}1. (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 c_{0}. So,
c_{0} = ^{c0}1
Least array larger than c_{0} is
^{c0}1, 1.
Then
^{c0}1, 2
^{c0}1, Ï‰
^{c0}1, Î©
^{c0}1, ^{1}1
^{c0}1, ^{11}1
^{c0}1, ^{111}1
^{c0}2
^{c0}3
^{c0}Ï‰
^{c01, 1}1
^{c01, 11}1
^{c01, 111}1
c_{1}  secondâ€¦
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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/xmsdownload", 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 toâ€¦
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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 Ïˆ(Î±) instead of generalized Veblen function Ï†(X).
FS #1
0
1
2
3
4
5
6
7
8
9
FS #2
Ï‰
Ï‰ + 1
Ï‰ + 2
Ï‰ + 3
Ï‰ + 4
Ï‰ + 5
Ï‰ + 6
FS #3
Ï‰2
Ï‰2 + 1
Ï‰2 + 2
Ï‰2 + 3
Ï‰2 + 4
FS #4
Ï‰3
Ï‰3 + 1
Ï‰3 + 2
Ï‰3 + 3
Ï‰3 + 4
FS #5
Ï‰4
Ï‰5
Ï‰6
Ï‰7
FS #6
Ï‰^{2}
Ï‰^{2} + 1
Ï‰^{2} + 2
Ï‰^{2} + 3
Ï‰^{2} + 4
FS #7
Ï‰^{2} + Ï‰
Ï‰^{2} + Ï‰2
Ï‰^{2} + Ï‰3
Ï‰^{2} + Ï‰4
FS #8
Ï‰^{2}2
Ï‰^{2}3
Ï‰^{2}4
Ï‰^{2}5
FS #9
Ï‰^{3}
Ï‰^{3} + 1
Ï‰^{3} + 2
Ï‰^{3} + 3
Ï‰^{3} + 4
FS #10
Ï‰^{3} + Ï‰
Ï‰^{3} + Ï‰2
Ï‰^{3} + Ï‰3
Ï‰^{3} + Ï‰4
FS #11
Ï‰^{3} + Ï‰^{2}
Ï‰^{3} + Ï‰^{2}2
Ï‰^{3} + Ï‰^{2}3
Ï‰^{3} + Ï‰^{2}4
FS #12
Ï‰^{3}2
Ï‰^{3}3
Ï‰^{3}4
Ï‰^{3}5
FS #13
Ï‰^{4}
Ï‰^{5}
Ï‰^{6}
Ï‰^{7}
FS #14
Ï‰^{Ï‰}
Ï‰^{Ï‰} + 1
Ï‰^{Ï‰} + 2
Ï‰^{Ï‰} + 3
Ï‰^{Ï‰} + 4
FS #15
Ï‰^{Ï‰} + Ï‰
Ï‰^{Ï‰} + Ï‰^{2}
Ï‰^{Ï‰} + Ï‰^{3}
Ï‰^{Ï‰} + Ï‰^{4}
FS #16
Ï‰^{Ï‰}2
Ï‰^{Ï‰â€¦}
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Definitions of eo, cofbeo, leo, lest, X^{0} see here.
asttr means "array shift to the right"
asttr(X) is array such as
eo(asttr(X); X_{1}) = eo(X; 1 + X_{1}) if cofbeo(X_{1}) = 0 (that is X_{1} is ordinal)
eo(asttr(X); X_{1}) = eo(X; X_{1}) if cofbeo(X_{1}) â‰ 0 (that is X_{1} is not ordinal)
ate means "array to exponent"
ate(0) = 0
ate(âŸ¨X_{1}âŸ©Î±_{1}, âŸ¨X_{2}âŸ©Î±_{2}, âŸ¨X_{3}âŸ©Î±_{3}, ...) = Î©^{ate(X1)}Î±_{1} + Î©^{ate(X2)}Î±_{2} + Î©^{ate(X3)}Î±_{3}
ata means "array to argument"
ata(1, Î±) = Î±
ata(1 + Î², Î±) = Î©^{Î²}(1 + Î±) if Î² > 0
ata(X) = Î©^{ate(asttr(X))}(1 + Î±) if cofbeo(X) > 1
vtm means "Veblen to Madore"
Let Î±  standard form of ordinal expressed using generalized Veblen function Ï†(X). Then vtm(Î±) is Î± expressed using Madore function Ïˆ(Î±).
vtm(0) = 0
vtm(1) = 1
vtm(Î± + Î²) = vtm(Î±) + vtm(Î²)
vtm(Ï‰^{Î±}) = Ï‰^{vtm(Î±)}
If cofbeo(X_{0}) â‰ 0 and
 leo(X_{0}) = Ï†(Xâ€¦