
0
This change is about how to read omitted coordinates.
(See also previous change).
Step 1. Fill coordinates right to known coordinates by decreasing 0th coordinate by one until limit or zero value of 0th coordinate, or next right element has already known coordinates, or end of array.
Step 2. If coordinates of last element of array is unknown, set it (coordinates) to 0.
Step 3. Fill coordinates left to known coordinates by increasing 0th coordinate by one until next left element has already known coordinates, or beginning of array.
Note: old designation was similar, but without Step 1.
1, 5
In old designation: 1, 5 = 1, 0, 0, 5
In new designation: 1, 5 = 1, 5, 0, 0
<Ï‰ + 1>1, 3, 5
In old designation: <Ï‰ + 1>1, 3, 5
In new designation: <Ï‰ + 1>1, <Ï‰>3, 5 â€¦
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(Or just MANE).
This is based on and inspired by Hyp cos' Mixed Arrow Notation.
Let a âŠ• b is any binary operation, for example, a + b, a Â· b, a â†‘â†‘ b, a â†“â†“ b, a â†‘â†‘â†“â†‘â†“â†‘â†“â†“â†‘ b.
1. a  b = a â†“ b = a â†‘ b = a â†• b = a^{b}
2. a âŠ• 1 = a âŠ•â†“ 1 = a âŠ•â†‘ 1 = a âŠ•â†• 1 = a
3a. a âŠ• (b + 1) = a âŠ• a, b > 0
b. a âŠ•â†“ (b + 1) = (a âŠ•â†“ b) âŠ• a
c. a âŠ•â†‘ (b + 1) = a âŠ• (a âŠ•â†‘ b)
d. a âŠ•â†• (b + 1) = (a âŠ•â†• b) âŠ• (a âŠ•â†• b)
3a. a âŠ• b = a âŠ• a, b > 0
b. a âŠ•â†“ b = sup((a âŠ•â†“ c) âŠ• a), c < b
c. a âŠ•â†‘ b = sup(a âŠ• (a âŠ•â†‘ c)), c < b
d. a âŠ•â†• b = sup((a âŠ•â†• c) âŠ• (a âŠ•â†• d)), c, d < b
(At least for a âŠ• b such as if c â‰¥ a, d â‰¥ b then c âŠ• d â‰¥ a âŠ• b).
Other numbers instead of 1. For example,
a âŠ• 0 = a âŠ•â†“ 0 = a âŠ•â†‘ 0 = a âŠ•â†• 0 = a
a âŠ• (1) = a âŠ•â†“ (1) = a âŠ•â†‘ (1) = a âŠ•â†• (1) = a
a âŠ• 1 = a
a âŠ• 2 = a âŠ• a
a âŠ• â€¦
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Here I want to share a thought that came to me long time ago, but I still have not seen anything like it anywhere else.
Square root of four:
4^{1/2} is Â± 2.
That is, we take a number (4), apply to it an operation (square root), and get two numbers (2 and 2).
So, from one number (4) we get two numbers (2 and 2).
But, if we get two different numbers, maybe, initially we also had two different numbers, but just did not distinct them?
Maybe, 4 is actually two different numbers (let's denote them 4_{+} and 4_{}):
2^{2} = 4_{+}
(2)^{2} = 4_{}
So,
4_{+}^{1/2} = 2
4_{}^{1/2} = 2
Then, 2 itself should be two different numbers 2_{+} and 2_{}, so, 4_{+} should be two different numbers (2_{+}^{2} and 2_{}^{2}), as well as 4_{} (now we distinct four different numbers in 4, we may denote them 4_{++}, 4_{+}, 4_{+}, 4_{}), and so â€¦
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Here I tried to make up names of numbers. I named this "Gnomen numbers".
1 one
2 two
3 three
4 four
5 five
6 six
7 seven
8 eight
9 nine
10 ty
11 tyone
12 tytwo
13 tythree
14 tyfour
15 tyfive
16 tysix
17 tyseven
18 tyeight
19 tynine
20 twenty
21 twentyone
22 twentytwo
23 twentythree
30 thirty
40 forty
50 fifty
60 sixty
70 seventy
80 eighty
90 ninety
99 ninetynine
100 cy
101 cyone
102 cytwo
103 cythree
110 cyty
111 cytyone
112 cytytwo
113 cytythree
120 cytwenty
121 cytwentyone
130 cythirty
140 cyforty
150 cyfifty
200 twency
201 twencyone
202 twencytwo
203 twencythree
210 twencyty
211 twencytyone
212 twencytytwo
213 twencytythree
220 twencytwenty
221 twencytwentyone
230 twencythirty
300 thircy
400 forcy
500 fifcy
600 sixcy
700 sevency
800 eighcy
900 ninecy
999 nâ€¦
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Here I tried to count Madore's Î¨ function and express it through my generalized Veblen function.
Here I use my new array notation as Veblen function array of agruments.
Briefly, this notation is list of pairs "coordinates inside , then ordinal at this coordinates", separated with ",". For example,
Î±, 0, 0, 0, Î², 0, 0, Î³ = Î±, Î², Î³ = Î±, Î², Î³
( can be omitted, as well as adjacent coordinates left to known coordinates).
And, if an array is multidimensional, then coordinates also can be an array, written same way, with any number of nesting levels.
So, list of ordinals:
Î¨(0) = Îµ_{0} = Ï†(1, 0) = Ï†(1)
Î¨(1) = Îµ_{1} = Ï†(1, 1) = Ï†(1, 1)
Î¨(2) = Îµ_{2} = Ï†(1, 2)
Î¨(3) = Îµ_{3} = Ï†(1, 3)
Î¨(4) = Îµ_{4} = Ï†(1, 4)
Î¨(5) = Îµ_{5} = Ï†(1, 5)
Î¨(Ï‰) = Îµ_{Ï‰} = Ï†(1, Ï‰)
Î¨(Î¨(0)) = Îµ_{Îµ0} = Ï†(1, Îµ_{0})
Î¨(Î¨(Î¨(0))) = Îµ_{ÎµÎµ0â€¦}
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