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Yesterday I was thinking about my Veblenlike functions, and I realized that its weak point is that maximal number of iterations is ω, but ω is too small ordinal, and it makes rules more complex.
Then I created OCF based on my ordinal array functions, since these functions do not have fixed points, and there should not be limited iterations. Also, today I slightly modified rules taken from ordinal array functions in order to make rules simpler, but, possibly, it makes calculations more difficult.
Here I publish this OCF (currently I limited it by Ω_{ω}).
Apparently, there are different definitions of fundamental sequences and cofinality. I use the following definitions:
 Fundamental sequence of ordinal α is increasing sequence of ordinals such as …
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Initially I was not going to publish it here, but since some users here are concerned about problem of the online community, I decided to share with my thoughts on the problems of the Internet, their causes and possible ways to eliminate them.
Internet is great invention, its advantages:
 1. People with similar interests easily can combine theirs efforts.
 2. It is easy to publish content.
 3. Safety and peace, since your enemies can't get to you. And it is pointless to start a war, since you can't get to your enemies, and if you do not like a web page or site, you just close it and search for another web page or site which you like.
These advantages should lead to production of large amount of content.
But something went wrong.
Internet suffers fro…
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Possibly, I'll use computer format for ordinals using five symbols: "(", ")", ".", "", "1":
empty string for 0
1 for 1
(a.b.cd) for Veblenlike function φ^{a}_{b}(c)d
d is just multiplier.
(b.cd) = (.b.cd)
(cd) = (.1.cd)
(a.b.c) = (a.b.c1)
(b.c) = (.b.c1)
(c) = (.1.c1)
(a..cd) = (a.a + 1.cd)
(a..c) = (a.a + 1.c)
Previously I used computer format with eight symbols: "(", ")", "", "+", ",", "0", "1".
Since now I use uncountable ordinals instead of arrays, may be used only one type of brackets, and "," may be replaced with "+".
But "+" may be just omitted. I remembered that an anonymous user wrote in my talk page that in my old computer format 11 can be written two ways: "1+1+1+1+1+1+1+1+1+1+1" and "11". But now it can be written only one way: "111111…
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There is a rule in Veblen function:
φ(α) = ω^{α}
where α is ordinal.
I was wondering, what if replace this rule with some another rule, for example,
φ(α) = ωα
or
φ(α) = ω + α
or
φ(α) = 1 + α
(Note: lately I usually write arrays of ordinals as single "larger" ordinals, but here I will write arrays as arrays).
Twovariable Veblen function for rule φ(α) = ωα is
φ(a, b) = ω^{ωa}b
Note:
φ(a, 0) = 0
(unlike Veblen function for rule φ(α) = ωα, where φ(a, 0) > 0)
α = φ(1, 0, a) must satisfy the equation
α = φ(α, 0) = 0
α = 0
So, there is only one fixed point of φ(α, 0). It is 0.
So, φ(1, 0, 0) = 0, and φ(1, 0, a) for a > 0 does not exist.
α = φ(1, 1, a) must satisfy the equation
α = φ(1, 0, α)
α = φ(1, 0, 0)
α = 0
So, φ(1, 1, 0) = 0, and φ(1, 1, a) for a > 0 does not exist.
App…
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Recently I was trying to extend set X{·}a (see Definitions update blog) beyond Ω^{ΩΩΩ...}. (Now I like to represent arrays of ordinals as "larger" ordinals).
I did not want to give up X{·}a, since it was used in short and independent of fundamental sequence systems definitions of ordinal array functions [X]a and generalized Veblen function φ(X). Also, fundamental sequences was not used in these definitions.
When I failed to extend X{·}a, I started to formulate equal values and comparison of generalized Veblen function in terms of "larger" ordinals instead of arrays of ordinals. I noticed that apparently it does work beyond Ω^{ΩΩΩ...}.
Then I suddenly realized, that sets used there can be used instead of X{·}a. (Later I found out that I already used…
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