10,098 Pages

# Rgetar

My favorite wikis
• ## Another change of my array notation

December 11, 2017 by Rgetar

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€¦

• ## Mixed Arrow Notation Extension

December 8, 2017 by Rgetar

(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 = ab

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€¦

• ## 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 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-):

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 â€¦

• ## 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 nâ€¦

• ## Madore's Î¨ function and generalized Veblen function up to BHO

December 6, 2017 by Rgetar

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

So, list of ordinals:

Î¨(0) = Îµ0 = Ï†(1, 0) = Ï†(1)

Î¨(1) = Îµ1 = Ï†(1, 1) = Ï†(1, 1)

Î¨(2) = Îµ2 = Ï†(1, 2)

Î¨(3) = Îµ3 = Ï†(1, 3)

Î¨(4) = Îµ4 = Ï†(1, 4)

Î¨(5) = Îµ5 = Ï†(1, 5)

Î¨(Ï‰) = ÎµÏ‰ = Ï†(1, Ï‰)

Î¨(Î¨(0)) = ÎµÎµ0 = Ï†(1, Îµ0)

Î¨(Î¨(Î¨(0))) = ÎµÎµÎµ0â€¦