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  • Rgetar

    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

    <ω + 1>1, 3, 5

    In old designation: <ω + 1>1, 3, 5

    In new designation: <ω + 1>1, <ω>3, 5 …



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  • Rgetar

    (Or just MANE).

    This is based on and inspired by Hyp cos' Mixed Arrow Notation.

    Let a ⊕ b is any binary operation, for example, a + b, a · b, a ↑↑ b, a ↓↓ b, a ↑↑↓↑↓↑↓↓↑ b.


    1. a | b = a ↓ b = a ↑ b = a ↕ b = ab

    2. a ⊕| 1 = a ⊕↓ 1 = a ⊕↑ 1 = a ⊕↕ 1 = a

    3a. a ⊕| (b + 1) = a ⊕ a, b > 0

      b. a ⊕↓ (b + 1) = (a ⊕↓ b) ⊕ a

      c. a ⊕↑ (b + 1) = a ⊕ (a ⊕↑ b)

      d. a ⊕↕ (b + 1) = (a ⊕↕ b) ⊕ (a ⊕↕ b)


    3a. a ⊕| b = a ⊕ a, b > 0

      b. a ⊕↓ b = sup((a ⊕↓ c) ⊕ a), c < b

      c. a ⊕↑ b = sup(a ⊕ (a ⊕↑ c)), c < b

      d. a ⊕↕ b = sup((a ⊕↕ c) ⊕ (a ⊕↕ d)), c, d < b

    (At least for a ⊕ b such as if c ≥ a, d ≥ b then c ⊕ d ≥ a ⊕ b).


    Other numbers instead of 1. For example,

    a ⊕| 0 = a ⊕↓ 0 = a ⊕↑ 0 = a ⊕↕ 0 = a

    a ⊕| (-1) = a ⊕↓ (-1) = a ⊕↑ (-1) = a ⊕↕ (-1) = a


    a ⊕| 1 = a

    a ⊕| 2 = a ⊕ a

    a ⊕| …





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  • Rgetar

    Number distinction

    December 7, 2017 by Rgetar

    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:

    41/2 is ± 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-):

    22 = 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 …

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  • Rgetar

    Gnomen numbers

    December 6, 2017 by Rgetar

    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 ty-one

    12 ty-two

    13 ty-three

    14 ty-four

    15 ty-five

    16 ty-six

    17 ty-seven

    18 ty-eight

    19 ty-nine

    20 twenty

    21 twenty-one

    22 twenty-two

    23 twenty-three

    30 thirty

    40 forty

    50 fifty

    60 sixty

    70 seventy

    80 eighty

    90 ninety

    99 ninety-nine

    100 cy

    101 cy-one

    102 cy-two

    103 cy-three

    110 cy-ty

    111 cy-ty-one

    112 cy-ty-two

    113 cy-ty-three

    120 cy-twenty

    121 cy-twenty-one

    130 cy-thirty

    140 cy-forty

    150 cy-fifty

    200 twency

    201 twency-one

    202 twency-two

    203 twency-three

    210 twency-ty

    211 twency-ty-one

    212 twency-ty-two

    213 twency-ty-three

    220 twency-twenty

    221 twency-twenty-one

    230 twency-thirty

    300 thircy

    400 forcy

    500 fifcy

    600 sixcy

    700 sevency

    800 eighcy

    900 ninecy

    999 n…

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  • Rgetar

    Here I tried to count Madore's Ψ 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,

    α, 0, 0, 0, β, 0, 0, γ = α, β, γ = α, β, γ

    ( can be omitted, as well as adjacent coordinates left to known coordinates).

    And, if an array is multi-dimensional, then coordinates also can be an array, written same way, with any number of nesting levels.

    So, list of ordinals:

    Ψ(0) = ε0 = φ(1, 0) = φ(1)

    Ψ(1) = ε1 = φ(1, 1) = φ(1, 1)

    Ψ(2) = ε2 = φ(1, 2)

    Ψ(3) = ε3 = φ(1, 3)

    Ψ(4) = ε4 = φ(1, 4)

    Ψ(5) = ε5 = φ(1, 5)

    Ψ(ω) = εω = φ(1, ω)

    Ψ(Ψ(0)) = εε0 = φ(1, ε0)

    Ψ(Ψ(Ψ(0))) = εεε0…

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