User blog:Rgetar/Less-than greater-than separators in decimal and ω-ry number systems

First I made up <> separators for my [X]a function, then I realized that it may be used for any functions of multiple variables.


 means n separators (commas) between variables with zeros between them (zeros may be omitted). So,  is n "empty" variables (zeros):

<1> = ,

<2> = ,0,

<3> = ,0,0,

For example, let f - function of 15 variables.

f(1,3,0,0,0,0,0,0,0,0,0,0,0,0,1) may be written as f(1,3<12>1)

Particularly, I used <> for Veblen function:

Γ0 = φ(1,0,0) = φ(1<2>)

φ(1,0,0,0) = φ(1<3>)

φ(1<ω>) - small Veblen ordinal

Multi-dimensional arrays of variables
Also <> may be used to describe multi-dimensional arrays of variables:

 is n "empty" variables

 is n "empty" rows

 is n "empty" planes

For example,

<1,2,3> is three "empty" variables, two "empty" rows and one "empty" plane

I used it for multi-dimensional extension Veblen function:

φ(1<1,0>) - large Veblen ordinal

This extension of Veblen function may be used up to Bachmann-Howard ordinal.

<> in positional number systems
Then I realized <> may be used in positional number systems, but with "separators" between digits instead of commas:

<1> = 0

<2> = 00

<3> = 000

etc.

Decimal number system
In decimal number system:

1<2> = 100

1<6> = 1000000

1.<5>1 = 0.000001

-1<6>.<5>1 = -1000000.000001

12345<25>3<7>.<9>1<5>25 = 12345000000000000000000000000030000000.00000000010000025

1 - googol

1<1 > - googolplex

1<1 >1 - ten googolplexes one

ω-ry number system
For infinitary number system we need infinite number of digits.

For example,

0

1

2

3

4

5

6

7

8

9

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

...

In ω-ry number system all natural numbers are one-digit numbers.

The least two-digit number is

10 = ω

More examples:

11 = ω+1

12 = ω+2

13 = ω+3

100 = ω2

101 = ω2+1

110 = ω2+ω

111 = ω2+ω+1

1000 = ω3

12300000101 = ω10+ω9·2+ω8·3+ω2+1

1<10> = ωω

1<11> = ωω+1

1 = ωω 2

1<1<10>> = ωω ω

1<1<1<10>>> = ωω ω ω

1<1<1<1<10>>>> = ωω ω ω ω

This may be used up to ε0.

Examples with "integer ordinals":

11 = ω+1

10-1 = ω-1

-11 = -1-ω

1-10 = 1-ω