Oology:

excefrom Transfini Mathematiques pur Surreel Amateurs de Rôles Muet by André Joyce

translated by Razilee Purdue, edited by Michael Joseph Halm

It was in 1915 that P. E. B. Jourdain translated and so popularized georg Cantor's work with transfinite cardinal numbers (such as aleph-null and aleph-one) and only in 1963 when B. S. Johnson defined the square root of aleph-one. In 1972 when John Horton Conway defined surreal numbers, using the me mathematics changed. Many undefined terms suddenly were definable. The number of kinds of infinities increased more than infinitely.

Making use of the roulette 00 (double zero), later appropriated by users of the internet for the lazy eight symbol, ∞, for the first inaccessible number named the branch of googology that goes to, as Buzz Lightyear says, "Infinity and beyond!" oogology. The representations of the surreal numbers can assigned oologisms very much like the finite googolisms.

Making use of the then rarely used backslash, \ , Joyce defined and explored new numbers like ciscendentals, enfinities, enrationals, arrationals (Mpossibilities 68:1, neologisms) that made surreal mathematics easier by an aleph-nullth.

The backslash is read as "one conquered by one" or "one under one" (the complimentary operation to "divided by" from the phrase "divide and conquer" or "one over one". The overlining indicates a repeated sequence of digits just as with repeating decimals in the opposite direction. Oogology is like traveling with Alice through the looking glass, seeing even familiar addition and multiplication topsy-turvy.

0.1 = 00 + 0.1 = 1\1

01 = 00 + 1 = 10\1

02 = 00 + 2 = 5\1

03 = 00 + 3 = 10\3

012486374987513625 = 19\10

0319371256596806287434 = 23\10

032967 = 13\10

0628743403193712565968 = 23\20

329670 = 13\1

31 = (g(2, 00, 10) - 1)/3 - 2 = 15\2

3 = (g(2, 00, 10) - 1)/3 = 3\1

35 = 3\1 + 2 = 15\8

37 = 3\1 + 4 = 15\11

967032 = 13\3

9680628743403193712565 = 23\13

987513625012486374 = 19\9

2\2 = 2(00) + 0.2

4\4 = (2\2)(2\2) = g(2, 2, (2(00) + .2))

5\5 = 5(00) + 0.5

7\7 = 7(00) + 0.7

8\8 = (2\2)(2\2)(2\2) = g(2, 3, (2(00) + .2))

9001 = g(2, 10^00, 10) - 999 = ooduplexminim

9501 = g(2, 00, 10) - 499 = ooplexminid

901 = g(2, 00, 10) - 99 = ooplexminic

951 = g(2, 00, 10) - 49 = ooplexminil

91 = g(2, 00, 10) - 9 = ooplexminix

96 = g(2, 00, 10) - 4 = ooplexminiv

98 = g(2, 00, 10) - 2 = ooplexminij

9 = g(2, 00, 10) - 1 = 3\3 = ooplexmini

(g2, 2, 00 + 1)/00 = garoopliperoo = infiniti [Steven John Robinson]

angelic number

(ending in an infinite number of zeroes, coined by Matthew Fox in The Physics of Angels)

100 = g(2, 00, 10) = ooplex

1000 = g(2, 10(00), 10) = xooplex

10000 = g(2, 100(00), 10) = cooplex

100000 = g(2, 1000(00), 10) = mooplex

g(2, g(2, 00, 10), 10) = ooduplex

g(2, g(2, g(2, 00, 10), 10), 10) = ootriplex

g(3, (3, 2, 00), g(3, 2, 00)) = ooplexpleooplex

Beyond these are what might be called the archangelic numbers:

g(2, 00, 00) = g(3, 2, 00) = oopleoo

g(3, 00, 00) = g(4, 2, 00) = aleph-one

g(2, 4, 2, 00) = aleph-two

g(00, 4, 2, 00) = aleph-aleph-null

Even when we have reached

g(00, 00, 00, 00, 00, 00)

We are alas negligibly closer to ultimate infinity, the Most High, Ω.