Oct 27, 2007 - fb+1(n) = A(X, b + 1,n). = A(X, b, A(X, b + 1 ...... Python. 3. Python i = 9 ** 9 ** 9 b = 2. B=lambda y,
3
75 GDP
500
100
=10
10
79
14
10
68
1 6023 2
10
256
136
37
10 80000000000000000(8 10
) 63
4
20
Large Numbers 21
BEAF BEAF BEAF
2002
5
2013 2016
2007 2013 PDF 2
PDF
1
1
2 3 3 4
6
1 1 1
Kindle 2
Amazon 2
PDF -
4.0
3
15 PDF Limit of Empty Wiki
2017
1
6
MUPI
29
https://www.amazon.co.jp/dp/B01N4KCIJQ http://gyafun.jp/ln/ 3 CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0/deed.ja 2
Kindle
noan
7
3 1 1.1
. . . . . . . . . . . . . . . . . . . . . . .
11 11
1.1.1
. . . . . . . . . . . . . . . . . . . .
11
1.1.2
. . . . . . . . . . . . . . . . . . . . . . . .
13
1.1.3
. . . . . . . . . . . . . . . . . . . . . . . .
17
1.1.4
. . . . . . . . . . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . . . . . . . . . .
25
. . . . . . . . . . . . . . . . . . . . . . .
27
1.2.1
. . . . . . . . . . .
28
1.2.2
. . . . . . . . . . . . . . . . . . . .
31
2
1.1.5 1.2
3
1.2.3
. . . . . . . . . . . . . . . . .
32
. . . . . . . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . .
34
. . . . . . . . . . . . . . . . . . . . . . . .
36
. . . . . . . . . . . . . . . . . . . . . . .
37
1.3.1
. . . . . . . . . . . . . . .
38
1.3.2
. . . . . . . . . . . . . . . . . . . . .
39
1.2.4
E
1.2.5 1.2.6 1.3
4
1.4
1.5
. . . . . . . . . . . . . . . . . . . .
40
1.4.1
5
. . . . . . . . . . . . . .
40
1.4.2
. . . . . . . . . . . . . . . . . . . . .
41
. . . . . . . . . . . . . . . . . . .
44
8 2
45
2.1
. . . . . . . . . . . . . . . . . . . . .
45
2.1.1
. . . . . . . . . . . . . . . . . . . . . . . .
45
2.1.2
. . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.1.3
. . . . . . . . . . . . . . . . . . . .
49
2.1.4 2.1.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49 54
2.2
. . . . . . . . . . . . . . . . . . . . .
57
2.3
. . . . . . . . . . . . . . . . . . . . . . . . .
61
3
2
3.1
65 . . . . . . . . . . . . . . . . . . . . . . .
65
. . . . . . . . . . . . . . . . . . . . . . . . .
68
3.3
. . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.4
. . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.5
. . . . . . . . . . . . . . . . . .
76
3.2
2
4
81
4.1
. . . . . . . . . . . . . . . . . . .
81
4.2
. . . . . . . . . . . . . .
84
4.3
. . . . . . . . . . . . . . . . . . . . . . .
86
4.4
. . . . . . . . . . . . . . . . . . . . . . . . .
87
4.4.1
1 . . . . . . . . . . . . . .
88
4.4.2
2 . . . . . . . . . . . . . .
100
4.4.3
3 . . . . . . . . . . . . . .
103
. . . . . . . . . . . . . . . . . . .
106
4.5 5
115
5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
5.1.1
. . . . . . . . . . . . . . . . . . .
116
5.1.2
. . . .
120
5.1.3
. . . . . . . . . . . . . . . . .
122
5.1.4
. . . . . . . . . . . . . . . . .
124
9 5.2
. . . . . . . . . . . . . . . . . . . . . . . . . .
125
5.2.1
. . . . . . . . . . . . . . . . . . . . .
126
5.2.2
. . . . . . . . . . . . . . . . . . . . . .
130
5.2.3
. . . . . . . . . . . . . . . . . . . . . .
138
5.3
. . . . . . . . . . . . . . . . . . . . . . . . . .
139
5.3.1 5.3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140 140
5.3.3
. . . . . . . . . . . . . . . . . . . .
143
5.3.4
. . . . . . . . . . . . . . . . . . . . . .
145
6 6.1
147 . . . . . . . . . . . . . . . . . . . . . . . .
147
6.2
. . . . . . . . . . . . . . . . . . . .
149
6.3
5 . . . . . . . . . . . . . . . . . .
151
6.4
. . . . . . . . . . . . . . . . . . . . . . . . .
154
6.5
. . . . . . . . . . . . . . . . . . . . . . . . . .
160
6.6
. . . . . . . . . . . . . . . . .
167
6.7
. . . . . . . . . . . . . . . . . . .
169
. . . . . . . . . . . . . . . .
170
. . . . . . . . . . . . . . . . . . . .
175
6.8
F [ϵ0 ](n)
BEAF
6.9 7
177
7.1
. . . . . . . . . . . . . . . . . . . . . . . .
177
7.2
. . . . . . . . . . . . . . . . . . . . . . . .
181
. . . . . . . . . . . . . . . .
181
7.2.1 7.2.2
θ
7.2.3
ψ
7.2.4
ψ
7.2.5
ϑ
7.2.6 7.3
2 7.3.1 7.3.2
2
. . . . . . . . . . . . . . .
183
. . . . . . . . . . . . . . . .
187
. . . . . . . . . . . . . . . . . .
187
. . . . . . . . . . . . . . . .
192
. . . . . . . . . . . . . . . .
193
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
194
. . . . . . . . . . . . . . . . . . . . . . . . .
194
. . . . . . . . . . . . . . . . . . . . . . . .
194
10 7.3.3
. . . . . . . . . . . . . . . . . . . . .
196
. . . . . . . .
197
. . . . . . . . . . . . . . .
200
. . . . . . . . . . . . . . . . . . . . .
200
7.4.1
6 . . . . . . . . . . . . . .
201
7.4.2 7.4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205 215
7.4.4
. . . . . . . . . . . . . . . .
216
7.4.5
BAN . . . . . . . . . . . . . . . . . . . . . . . . . .
219
7.4.6
BEAF . . . . . . . . . . . . . . . . . . . . . . . . .
220
7.4.7
. . . . . . . . . . . . . . . . .
222
7.4.8
. . . . . . . . . . . . . . . . . . . . . .
224
7.4.9
7.3.4 7.3.5 7.4
2 C
. . . . . . . . . . . . . . . . . . . . . .
226
7.4.10
. . . . . . . . . . .
229
7.4.11
. . . . . . . . . . . . . . . . . . . .
232
7.4.12
. . . . . . . . . . . . . . . . . . .
234
8
239
8.1
. . . . . . . . . . . . . . . . . . . . .
239
8.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
243
8.3
. . . . . . . . . . . . . . . . . . . . . . . . . .
244
8.4
4 . . . . . . . . . . . . . . . . . .
245
8.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
248
8.6 8.7
7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
251 255
8.8
. . . . . . . . . . . . . . . . . . . . . . . . .
257 259 263 267
11
1
1.1
2
1.1.1 (Robert Munafo)
1996
1
0
1
3
2 2 2
2 0
(0–6): 2
1 Large
0
subitizing 1.1
Numbers http://www.mrob.com/pub/math/largenum.html E. L. et al. (1949). The discrimination of visual number. American Journal of Psychology 62 (4): 498–525. doi:10.2307/1418556 2 Kaufman,
12
1
1.1: 2 5 5 5
9
6
0
6
0 (6–106 ):
1
1 100
100
1
1
1 85
70 100
1 100 6
(106 –1010 ):
2 0
6
1 2
6
2
10
6
10 = 1000000
1010 = 101000000
2
1.1.
2
13
6
1010 = 10(10 10 6
(10 ) = 10
10×6
= 10
6
)
60
2
999999
111111 100 100
10
100
1.1.2
2
10 = 10
8
= 10
12
= 1024 = 1040
= 1056 = 1064
Wiki -
1 16
= 10
= 1032
= 1044
= 1052
3
= 10 = 1028
3
100
20
= 1036
= 1048 = 1060 = 1068
http://ja.googology.wikia.com/wiki/
14
1
1088
108 1068
1627 8
1088
11
✓
10
✒
1068
(1631) 1068
(1634)
68
✏ ✑
= 1068
4
10 4
=10
=10
5
6
=10
=10
1
14
1 1012
1012
1016 1044
1080
=108 = 102
2 =1012 =1028
=1031
=1056 2
7
=1015 =1060
128
2
8
4
=102 = 1016 5
=102 = 1032
=1024
=1040
=1048
6
=102 = 1064 256
2
10 = 10 =10 = 10 =10 11 12 =102 = 102048 =102 = 104096
4
=1020
9
= 10
=10127 512
=10
2
10
= 10
3
=
1024
=108191
http://www.sf.airnet.ne.jp/˜ts/language/largenumber.html
1.1.
2
15 100
(1299
2
)
1052 1052
2500 km 10000 km
km
4 km
2
10 km3
1m
0.05–2 mm
0.1 mm 50 % 100 %
2 100 %
10 km3 = 1010 m3
0.001 mm3 = 10−12 m3
16
1 1022
100
5
2
1 2
6
0.2nm
1
1
=500m
1 4 6
1
= 4 × 10 7
1
4000
=1
24
2
=500m
=1
0.5mm
= 6.4 × 1024 8
138
9
5 http://www.otani.ac.jp/yomu
1022 × 1024 = 1046
100
=100
page/b yougo/nab3mq0000000r5r.html http://d.hatena.ne.jp/inyoko/20111015/How long is kalpa2 7 http://d.hatena.ne.jp/inyoko/20111008/How long is kalpa 8 Universe as an Infant: Fatter Than Expected and Kind of Lumpy (New York Times) http://nyti.ms/YrtRRV 9 WikiArc: : http://bit.ly/1b4Myz9 (labo.wikidharma.org) 6
1.1.
2
17
1.1.3
(k, kilo-, 103 )
(M, Mega-, 106 )
(G, Giga-109 )
(T,
12
Tera-10 ) Mbps
1
1000
210 = 1024
2 1000 (IEC)
1024 2
2
10
2 (kibi)
Ki 20
2 12
10
(Mi) 2
(kilobinary) 1000
30
KB (Gi) 2
40
(Ti) 40
= 1, 000, 000, 000, 000
2
10 %
= 1, 099, 511, 627, 776 2
10 2
SI 18
(E, Exa-10 )
21
(Z, Zetta-10 )
(P, Peta-1015 ) (Y, Yotta-1024 ) =1024
18
1
(Sir Arthur Stanley Eddington, 1882–1944)
1938
(Tarner lecture) 136 × 2256
10
(Eddington number)
10
79
1.574 × 10
1080 –1085 10
million (100
)
billion (10
100
)
(Nicolas Chuquet, 1450 )
–1500
3 (Le triparty en la science des nombres) 1520 (Estienne de La Roche, 1470–1530) (l’Arismetique) 1870
(Aristide Marre, 1823–1918) 1880 6 million, byllion, tryllion, quadrillion, quyillion,
sixlion, septyllion, ottyllion, nonyllion nonyllion million
1054 6
3 million
100
10 http://ja.googology.wikia.com/wiki/
billion
10
1.1.
2
19 1
billion
billion
10
1
billion
1
trillion
million
billion, trillion, quadrillion,
100
million = 106 , billion = 109 , trillion = 1012 , quadrillion = 1015 , quintillion = 1018 , sextillion = 1021 , septillion = 1024 , octillion = 1027 , nonillion = 1030 , decillion = 1033 , undecillion = 1036 , duodecillion = 1039 , tredecillion = 1042 , quattuordecillion = 1045 , quindecillion = 1048 , sexdecillion (sedecillion) = 1051 , septendecillion = 1054 , octodecillion = 1057 , novemdecillion (novendecillion) = 1060 , (vigintillion) = 10
63
(
10 12
120
11
)
(centillion) = 10303 (
10600 ) 13
(googolplex) = 10 ✓ ✒
(googol) = 10100
10100
✏ ✑
= 10100
(Edward Kasner, 1878–1955) 11 http://ja.googology.wikia.com/wiki/ 12 http://ja.googology.wikia.com/wiki/ 13 http://ja.googology.wikia.com/wiki/
(Mathematics and the
20
1
Imagination)
14
9 1
(Milton Sirotta) 1911
0
100
9
1920 3 IT
Google
Google
googol Google
(Lawrence Edward “Larry” Page) (Sergey Mikhailovich Brin) 1997
BackRub
9 (Sean Anderson)
2004 (David Koller)
15
14 Kasner, E. and Newman, J. R. (1940) Mathematics and the Imagination. Simon & Schuster, New York. 15 http://graphics.stanford.edu/˜dk/google name origin.html
1.1.
2
21
googol
google.com google.com
googol
google.com
Google
1.1.4 16
(Shannon number)
10120
1950 17
(Claude Elwood Shannon, 1916–2001)
1949
EDSAC
30
16
Wiki http://ja.googology.wikia.com/wiki/ C. (1950) Programming a computer for playing chess. Philosophical Magazine 41 (314) 17 Shannon,
22
1
1
1
1
2
10
3
40
10120 1
1 1090
10120 ✓ ✒
10120 ✏ ✑
= 10120
64! 32!(8!)2 (2!)6
32 2
64 8
2
6 1043
40 (Godfrey Harold
1.1.
2
23 1080
Hardy, 1877–1947)
1010 18
50
3 10
3 1043 ×2
(10 )
43
= 10
3 6×1043
50
75
75 8848 26
8848
≈ 1012500 1043
1997 6
5
IBM 2
1
3
1997
1062 –1070 19
115 115
80
80 220
≈ 10
18 Hardy, G. H. (1940) Ramanujan: twelve lectures on subjects suggested by his life and work. Cambridge University Press. 19 (2008) , IPSJ Symposium Series 2008(11) 116–119.
24
1
20
10
10224
224 21
(2016
6
25
)
70
10
20 2010 2012 2013 5
5 1
3
1
2016 3
(AlphaGo) 4
1
10360 2
2016
319 ≈ 1.74 × 10172 1
2.081681994 × 10170
22
20
2006
2 20
http://ja.googology.wikia.com/wiki/ https://www.ipsj.or.jp/event/shogi.html 22 Number of legal Go positions https://tromp.github.io/go/legal.html 21
Wiki -
1.1.
2
25
6 )
million (100
7
billion (10
( )
)
trillion
8 (1063 )
1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 23
103003
106000
(millinillion) = 103003
24
2
1.1.5 1.1
2
2
10 3.23 × 1021
10
10
23 Conway,
J. H. and Guy, R. K. (1996) The book of numbers. Copernicus. . 24 http://ja.googology.wikia.com/wiki/
26
1
1.1:
2 1046 1068 ≈ 1.57 × 1079 10100 10120 10303 103003 104096 101000000 = 1010
2
6
25
50
50 1.5747724136275002577605653961181555468044717914527 × 1079 80 25
(CASIO) http://keisan.casio.jp/
30
1.2.
3
27 26 27
1574772413 6275002577 6056539611 8155546804 4717914527 1167093662 3142507618 5631031296 2
(log)
10 e ≈ 2.71828
a
10a
exp(a) = ea 136 × 2256 log(136 × 2256 )
1079.197217798
≈
log(136) + 256 log(2)
≈
79.197217798
=
100.197217798+79
=
100.197217798 × 1079
≈
1.57477241 × 1079
100
2
100
1.2
3 3
6
(1010 –1010
106
):
2
10 2
26 Arbitrary
Precision Calculator http://apfloat.appspot.com/
27 http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm
28
1 1010
2 ✓
6
1010
10
106
3 ✏
c(0) = 6, c(n + 1) = 10c(n)
c(n) n
(i) n = 0
c(0)
(ii) n > 0 ✒
c(n − 1)
✑
c(n)
1.2.1 (Archimedes; 287
? –
212
Reckoner;
) )
(The Sand
28
1 10
myriad (
)
8
108
108
1 2 10
2 10
16
10
8
108
2
8
1016
3 108 (108 )(10
8
)
= 108·10
108
8
1 2
1 2 28 http://ja.googology.wikia.com/wiki/
108·10
1 8
10
8
2 2
2
10
8
108
1.2.
3
29 3
108 !
1.
108
(108 )(10
40
8
)
"(108 )
108 = 108×10
1 403 = 64000
1
1 64000
6
1
1
4000
2
1 2 3.
16
3
1
2.
2
8
(108·10 ) = 1016·10
1
6
4000 (109 )
10 100
2
4. 1
107
1015
1 180 m
5.
180 m 1
40
10000
1/3
1 1
0.45 mm 0.02 mm
1
8
30
1
0.1 mm 3
5 = 125
6.
0.02 mm
2
1
100 107
2 3
100 1021
10
7. 300 100 1 10 2 8.
3
10
18
km
km
1010 1000
7 10
51
9.
(Aristarchus, 310
–
230
)
1
2 8
Harrison (2000)
29
1063
1000
1063
1080
29 Harrison, E. R. (2000) Cosmology - The Science of the Universe. Cambridge University Press. pp. 481–482.
1.2.
3
31
1.2.2
45 100
=10
1
1
1 1 1 1
1
=
7
10
2 =107
=107×2 = 1014 =107 2
× 3
2
= 1056
=107 2
✒
10
2
= 1028
2
1
107
n 30
✓
× 2
× n
2
122
7×2122
= 107×2
✏
122
31
✑
60
80 = 105×2 (Thomas Cleary)
120
32 33
30 http://ja.googology.wikia.com/wiki/ 31
(2016) (1957) 33 Cleary, T. (1993) The flower ornament scripture: a translation of the Avatamsaka Sutra. Shambhala, Colorado, USA. 32
32
1 unspeakable = 1010×2
120
, square untold = 1010×2
123
34
1.2.3 35
1010
✓
(googolplex)
(10googol )
10
100
✒
= 1010
✏
100
✑
9
1
1
1
0
0
1
0 1080 1
1085
0 10 10 3
4
3 7 × 2122
34 http://iteror.org/big/book/ch1/ch1 35 http://ja.googology.wikia.com/wiki/
7.html
1.2.
3
33
10x 7 × 2122
10100
7 × 2122 = 37218383881977644441306597687849648128 ≈ 3.72 × 1037 10100
1.2.4
E (Sbiis Saibian)
E
37
36
(hyper-E notation)
E
1
an
#
E(b)a1 #a2 #...#an
b
10
✓
✏
E
E(b)x
=
bx
E(b)a1 #a2 · · · #an #1
=
E(b)a1 #a2 · · · #an
=
E(b)a1 · · · #an−2 #(E(b)a1 · · · #an − 1)
✒
E(b)a1 #a2 · · · #an
bx
1. 1
2. 2
36 http://ja.googology.wikia.com/wiki/Sbiis 37 http://ja.googology.wikia.com/wiki/
1
Saibian E
✑
34
1 3. 3
1 z
2 (z)
Ea
=
E100 = E100#2 = E100#3 = Ea#b
Ea#1 = 10a E100#1 = 10100 = E(E100) = E10100 = 1010 E(E100#2) = 10 10
1010
a
E100#100#100#100 E
1
=
100
b
E
the Finite)
100
–
E
E
Web (One to Infinity: A Guide to
38
1.2.5 39
maxima)
N
N
N! 38 One
to Infinity https://sites.google.com/site/largenumbers/
39 http://ja.googology.wikia.com/wiki/
(pro-
1.2.
3
35
10−35 m
1026 m (1026 /10−35 )3 = 10183 N
1010
185
N ! = 10183 ! = E185#2 10−43 N! (N !)t t
1 t
5 × 1017
(N !)t = 1010
245
= E245#2
(probability maximum)
1010 1010
245
1035
1081
1035 106.5446×10
245
343
36
1
1.2.6 40
(abundant number) 36
2 1, 2, 3, 4, 6, 9, 12, 18
55
36 12, 18, 20, 24, 30, 36, 40, 42, 48, ... 945 2
3 5391411025 = 52 × 7 × 11 × 13 × 17 × 19 × 23 × 29 11 2
13 × 172 × 19 × 23 × 29 × 31 × 37 × 41 × 43 × 47 × 53 × 59 × 61 × 67 × 71 × 73 × 79 × 83 × 89 × 97 × 101 × 103 × 107 × 109 × 113 × 127 × 131 × 137 × 139 × 149 × 151 × 157 × 163 × 167 × 173 × 179 × 181 × 191 × 193 × 197 × 199 × 211 × 223 × 227 ≈ 7.970466327 × 1087
2
3 n
σ(n)
σ(n) > 2n
σ(n) > 1000n
1000 N = ceil(e1000 )! ceil(x)
ceil
x
!
N N
N k
k
ceil(e1000 )
σ(N ) > N
#
k=1
(log
1 ≤ k ≤ ceil(e1000 )
1 > N log(ceil(e1000 )) > N log(e1000 ) = 1000N k
) N
1000
E446.94#2 40
Wiki -
ceil(e1000 )
http://ja.googology.wikia.com/wiki/
N < 1010
446.94
=
1.3.
4
37
σ(n) > 1000n 1984
Guy Robin
n > 5040 σ(n) < eγ n log log n 41
γ
0.5772156649
σ(n) > 1000n 1000n < 1000 < eγ n >
σ(n) < eγ n log log n log log n ee
1000/eγ
> 1010
243.47
= E243.47#2
σ(n) > 1000n E243.47#2 < n < E446.94#2 3 m 1000n 1, 2, 4, 6, 12, 24,
36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600, ...
1.3
42
4 4
E6#4 = 1010 2 3
(E6#3–E6#4): 1010
4
6
41 Robin, G. (1984) Grandes valeurs de la fonction somme des diviseurs et hypothse de Riemann. Journal de Mathmatiques Pures et Appliques 63, 187–213. 42 https://oeis.org/A004394
38
1 10
10
1.3.1 10
100
10
10 10
N
N
(googolplexplex) (googolplusplex)
(gargoogolplex)
(googolplexian) (googolduplex) du 3
2 10
tri
E100
=
E100#2 = E100#3 = E100#4 = E100#5 = E100#6 = E100#7 =
(googoltriplex)
10100 = 1010 10
100
1010
(googol)
=
(googolplex)
100
=
(googolduplex) (googoltriplex) (googolquadriplex) (googolquinplex) (googolsextiplex)
E100#8 =
(googolseptiplex)
E100#9 =
(googoloctiplex)
1.3.
4
39
E100#10
=
(googolnoniplex)
E100#11
=
(googoldeciplex) 5
1.3.2 (prime counting function) π(x)
x x ̸= 1
li(x) li(x) =
$
x 0
dt log(t)
log
π(x) π(x) < li(x)
li(x) π(x) > li(x) π(x)
x
li(x) π(x) > li(x)
43 44
(Skewes number)
x
2 Sk1 = ee
1933 E(e)79#3 45
Sk2 = e
ee
e7.705
1 2
= E(e)7.705#4 46
43
=
π(x) > li(x) Sk1
1995
e79
π(x) > li(x)
x
Bays and Hudson (2000)
47
Wiki http://ja.googology.wikia.com/wiki/ - Skewes Number http://mathworld.wolfram.com/SkewesNumber.html 45 Skewes, S. (1933) On the difference pi(x)-li(x). Journal of the London Mathemetical Society s1-8(4): 277–283. doi:10.1112/jlms/s1-8.4.277 46 Skewes, S. (1955) On the difference pi(x)-li(x). II. Proceedings of the London Mathematical Society s3-5 (1): 48–70. doi:10.1112/plms/s3-5.1.48 47 Bays, C. and Hudson, R. H. (2000), A new bound for the smallest x with π(x) > li(x). Mathematics of Computation 69 (231): 1285–1296. doi:10.1090/S0025-5718-9901104-7 44 MathWorld
40
1 1.3983 × 10316
1010
10 ✓ ✒
103
Sk2
4
E963#3 = 1010
Sk1
10963
E34#3 = 1010
1034
E3#4 = ✏
2 = ee
2
7.705 ee
≈ 1010
1010
3
✑
= E3#4
48
Hypercalc
(Hypercalc)
2
: e^e^e^e^7.705 : 10 ^ [ 10 ^ ( 3.299943221955 x 10 ^ 963 ) ] 2
1.4
5
1.4.1 ✓
10^10^10^10^10^1.1 = E1.1#5 ✒ 49 50
Page)
✏ (Don Nelson
✑
51
48 Hypercalc
http://www.mrob.com/pub/perl/hypercalc.html
49 http://ja.googology.wikia.com/wiki/ 50 Longest
Possible Time http://www.numberphile.com/videos/longest time.html D. N. (1994) Information loss in black holes and/or conscious beings? http://arxiv.org/pdf/hep-th/9411193 51 Page,
1.4.
5
41 1 52
)
1000
(5.391 × 10−44
Page (1994)
5
4 1.1
1.4.2 (Jonathan Bow53
ers)
(Forever endeavor) 54
Wiki
10
100
1 10
52 https://twitter.com/astrophys
tan/status/391928927012134912 Wiki http://ja.googology.wikia.com/wiki/ 54 http://www.polytope.net/hedrondude/foreverendeavor.htm 53
42
1
10 10
0 9 1
1
2 1
V
1 0
1
10 0
2 2
0000000000 3 1 0000000001, 0000000002, 0000000003, ...
1
1
18
500
1.4.
5
43
550
100
3 10 3
000 . . 000( % .&' 100
4
10
3 4
100 1 4 5
1010
10000000000
10
10 55
✓ 1 + 10 +
✒
55
8 #
E10#i = 1 + 10 + 10
10
(Bentley’s Number)
+ 10
1010
+ ... + 10
1010
✏ 1010
i=1
9
Wiki -
http://ja.googology.wikia.com/wiki/
1010
1010
✑
44
1
1.5
56 57 58 59
pixiv
2017
8
10 60
20
Googology Wiki
Googology Wiki
Wiki
62
Wiki Googology Wiki
56 pixiv
http://comic.pixiv.net/works/1505
57 http://ja.googology.wikia.com/wiki/ 58
http://www.geocities.co.jp/Technopolis/9946/
59 http://www.slideshare.net/DoomKobayashi/presentations 60
(2016) Wiki http://googology.wikia.com/ Wiki http://ja.googology.wikia.com/
61 Googology 62
61
45
2
2.1 2.1.1 TEX
(Donald Knuth) 1 2 3
TEX
1
(arrow notation) LATEX
Wiki http://ja.googology.wikia.com/wiki/ - Arrow Notation http://mathworld.wolfram.com/ArrowNotation.html 3 Knuth, D. E. (1976) Mathematics and Computer Science: Coping with Finiteness. Science 194, 1235–1242. doi:10.1126/science.194.4271.1235 2 MathWorld
46
2
✓ x
y
x x ✒
n
x
xy
=
2 =
x
2 =
x
y
✏
=
x, x
y=x
x, x
x(n
(y − 1))
(x
y=x )y = x
ab
n−1
(y − 1))
(x
(x
n
✑
(y − 1))
b a
b ˆ
a
b
a ˆ
a
b
b
a ˆˆ b
2.1.2 1 Hypercalc
3
3
3 =
33 = 27
3
3
3 =
33 = 327 = 7625597484987
3
3
3
3 =
33
3
3
3
3
3 =
3
3
3
3
3 =
Hypercalc
10
3
33
= 1.35 × 10
3638334640024
10
(6.46 × 10
3638334640023)
10
10
10
4 PT ( 6.46
10
10
(6.46 × 10 (6.46
10
3638334640023) 3638334640023)
10 ˆ 3638334640023 )
2.1.
47 10
6.46 × 10
4
4 PT
3638334640023 = 10
1
3
10
12.56
3
10
3
3 =
27
3
3
3 =
10
12.88
3
3
3
3 =
10
10
12.56
3
3
3
3
3 =
10
10
10
12.56
3
3
3
3
3
3 =
10
10
10
10
12.56
3
3
3
3
3
3 =
10
10
10
10
10
3
12.56
10
27,12.88,12.56 3
10
10
3
2
2 2=2
2
2=2
2=4 3
3 3 1
3
3
3 3
3
3
3
3
3
3
2=2
2 4 3
3
3
3
3
1 3
5
2
3
48
2 3
10 3x
x
10x
10x = (3log3 (10) )x = 3x log3 (10) ≈ 32.1x x
1010
2.1
x
log3 (1010 ) 1010 x
2.1 × 10x ≈ 2.1 × 32.1x
≈
x
32.1×3
≈
2.1
2.1x
= 33
2.1x+log3 (2.1)
2.1x+0.68
2.1
0.68
3
2
3
3
3
3 = 10
10
x = 12.56 3
x
3
1
3
≈ 33
33
0.68
10
3
x
10
10
12.56
2.1x + 0.68 = 27.056
3 = 27
2.1x + 0.68
Hypercalc
x = 10 10
10
10
10
10
10
2x
10
xx 2x ≈ x x
10 ≈
0 ≈
2x
xx
4
4
log(log(log(log(2x )))) ≈ log(log(log(log(xx )))) ≈ 10 2x ≈ x x
2.1.
49
2.1.3 4
2
x
y
(y − 1))
=
x
(x
=
x
(x
(x
=
x
(x
(x
=
x %
x y
5
(power tower)
&'
x
(y − 2))) (y − 3))))
(x x (
(tetration) 6
4
tetra9 # i=0
10 ↑↑ i
2.1.4 4 4
(hyper-4)
n
Hn (a, b)
7
4 MathWorld
- Power Tower http://mathworld.wolfram.com/PowerTower.html
5 http://ja.googology.wikia.com/wiki/ 6 Goodstein, R. L. (1947). Transfinite Ordinals in Recursive Number Theory. Journal of Symbolic Logic 12(4): 123–129. doi:10.2307/2266486 7 http://ja.googology.wikia.com/wiki/
50
2 n
Hn (a, b) =
Hn (a, b)
⎧ ⎪ b+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ a
(n = 0) (n = 1, b = 0)
0 1
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(n = 2, b = 0) (n ≥ 3, b = 0)
Hn−1 (a, Hn (a, b − 1)) (
)
H0 (a, b) = b + 1
H1 (a, b) = a + b H2 (a, b) = a × b H3 (a, b) = ab H4 (a, b) = a
b
H5 (a, b) = a
b
H6 (a, b) = a
4
Hn (a, b) = a
n−2
3
b b(n ≥ 3)
3
3 =
3
3
3 =
3
=
3 %
3
3 = 7625597484987 7625597484987
3
3
&'
7625597484987
3
7625597484986
Hypercalc
3ˆ
3 (
7 7625597484983 PT ( 6.46
10 ˆ 3638334640023 ) 3
3 3
3
3 3
3
3 3
3=3 % 3
3 3
3
&'
3 3
3
3 (
Hypercalc
2.1.
51 Hypercalc (1010 )
10
10
(3
3
3)
Hypercalc
✓
hc(n)
✏
hc(0) = 6, hc(n + 1) = c(hc(n))
c(n)
(p.27
1.2
)
n hc(0)
(i) n = 0 (ii) n > 0 ✒
hc(n − 1)
✑
hc(n)
hc(n+ c(n + 1) = 10c(n)
1) = c(hc(n)) 10
x
c(x) 1
6
1
2
N
3
3
N 3
2 3
2 3
3
3
3
1 3 3
3 1
3
Hypercalc
3
5 8
(pentation)
8 http://ja.googology.wikia.com/wiki/
52
2 3
✓
9
3
=3
✒
3=3
3
✏
3
7625597484986 10
(tritri)
✑
2 10
100
100 4
6 11
3
(hexation) 3 =
3
3
3
= 3 = 3 %
3
&'
3
3 (
3
3 f (4)
Hypercalc 1
9 http://ja.googology.wikia.com/wiki/ 10 http://ja.googology.wikia.com/wiki/ 11 http://ja.googology.wikia.com/wiki/
2.1.
53
1.
2. 3.
100 100000
100000 100
3
3
100
3
100
100
1) % &' ( b
X →c 1 =
X
→c−1
3 http://peterhurford.com/ 4
Wiki -
http://ja.googology.wikia.com/wiki/
4.4.
87 X →c 1 →c a
=
X
X →c (a + 1) →c (b + 1)
=
X →c (X →c a →c (b + 1)) →c b
3 2
2
1
3 →2 x 3 →2 x →2
=
3→3→3
3 →2 4
=
3→3→3→3
3 →2 5
=
3→3→3→3→3
x 3 2
→2
2 3 →x x
3 →2 3
4
3 →n x
3 3 →3 x 3
1
x n−1
+2
→2
4.4 5
5 http://ja.googology.wikia.com/wiki/
(Fish number)
3 3
88
4
4.4.1
1 6
2ch
161
02/06/20 22:25 >> 156 3
3 3
1 3
2
3
1
3
63 63 1
6 http://ja.googology.wikia.com/wiki/
4.4.
89
7
(2ch
317-319
m
)
f (x)
n
m, f (x), n
m
f (x)
n
g(x) S : [m, f (x)]
f (x) =
[n, g(x)]
x
f (4) f (f (4)) 64
64 g(x) = f (x)
f (4) 64
64
S : [m, f (x)]
f (x)
[f m (m), f m (x)] f 64 (64)
m = 64, f (x)
S >> 161 S
10 S
Ackermann
S 7 2ch
http://www.geocities.co.jp/Technopolis/9946/log/ln023.html
90
4
S Ackermann B(0, n)
=
f (n)
B(m + 1, 0) =
B(m, 1)
B(m + 1, n + 1)
=
B(m, B(m + 1, n))
g(x)
=
B(x, x)
S : [m, f (x)]
[g(m), g(x)]
[3, f (x) = x + 1] S
10
>> 161
S S
f (m)
S2
m, f (x), S
S2
SS : [m, f (x), S]
[n, g(x), S2]
g(x) = S2[m, f (x)], n = g(m) SS [3, x + 1, S]
SS
1
S 1
4 S
4.4.
91
S 8
1 1.
S S
S(m, f (x)) = (g(m), g(x)) g(x) B(0, n)
=
B(m + 1, 0) =
f (n) B(m, 1)
B(m + 1, n + 1)
=
B(m, B(m + 1, n))
g(x)
=
B(x, x)
2.
SS SS
SS(m, f, S) = (S f (m) (m, f ), S f (m) ) (( (
,
3. 3
,
,
)
(m0 , f0 , S0 )
),
)
3 m0 = 3, f0 (x) = x + 1, S0 SS 63 (m0 , f0 , S0 )
1
1
2
1
8 http://ja.googology.wikia.com/wiki/
1
S
92
4
1 1
F1
1 N
N
F1 (x)
FN
FN (x)
63 3
3
0
63 63
S : [m, f (x)]
[g(m), g(x)]
SS : [m, f (x), S]
[n, g(x), S2]
S2 S2 : [m, f (x)]
=
S f (m) [n, g(x)]
9
329,331,377379 [3, f (x) = x + 1]
1 B(0, n)
=
B(m + 1, 0) =
n+1 B(m, 1)
B(m + 1, n + 1)
=
B(m, B(m + 1, n))
g(x)
=
B(x, x)
B(m, n)
g(x) =
A(x, x) S : [3, x + 1]
[A(3, 3), A(x, x)]
9 http://ja.googology.wikia.com/wiki/
:Mikadukim/SS
4.4.
93
10
A(3, 3) = 61
1
2 B(0, n)
=
A(n, n)
B(m + 1, 0) =
B(m, 1)
B(m + 1, n + 1)
=
B(m, B(m + 1, n))
g(x)
=
B(x, x)
g(x) g(1) =
B(1, 1) = B(0, B(1, 0))
=
B(0, B(0, 1)) = B(0, A(1, 1))
=
B(0, 3) = A(3, 3) = 61
g(2) =
B(2, 2) = B(1, B(2, 1))
=
B(1, B(1, B(2, 0))) = B(1, B(1, B(1, 1)))
=
B(1, B(1, 61))
=
B(1, B(0, B(1, 60)))
B(1, 1) =
61
B(1, 2) =
A(61, 61)
B(1, 3) =
A(A(61, 61), A(61, 61)) B(1, 61)
g(2) = B(1, B(1, 61))
10
2
g(2)
http://comic.pixiv.net/viewer/stories/6995
94
4 g(2)
g(x)
x
g(61) 2
g(61)
B(0, n)
=
B(m + 1, 0) = B(m + 1, n + 1)
=
gg(x) =
g(n) B(m, 1) B(m, B(m + 1, n)) B(x, x)
gg(x) gg(1)
=
B(1, 1) = B(0, B(1, 0))
=
B(0, B(0, 1)) = B(0, g(1))
=
B(0, 61) = g(61)
=
B(2, 2) = B(1, B(2, 1))
=
B(1, B(1, B(2, 0))) = B(1, B(1, B(1, 1)))
=
B(1, g(61))
B(1, 1)
=
g(61)
B(1, 2)
=
g(g(61))
B(1, 3)
=
g(g(g(61)))
gg(2)
gg(2)
61
g(x)
g(61)
gg(3), gg(4)...
SS
gg(g(61))
SS S
S
m, f (x), S
f (m)
S f (m)
S f (m)
4.4.
95 SS
S
m
S f (x) S
S S
380
02/07/02 19:47
386
02/07/02 20:25
388
02/07/02 20:32
10
88
96
4
11
Code Ass (@aycabta)
12
1
Ruby
13
Code Ass 14
15
16
F1
S
f (x)
g(x)
F1 2
2
2
2
2
1 2
2 F1
F1
f (x)
Ax (z, z) g(z) = Ax (z, z) 3
11 2ch
S
2
g(x)
S
f (z) =
3 2
2 [3, x + 1]
2
F1 F1
S
i
Si (x)
http://www.geocities.co.jp/Technopolis/9946/log/ln024.html @aycabta https://twitter.com/aycabta 13 https://github.com/aycabta/fish-number 14 https://github.com/aycabta/ackermann 15 https://github.com/aycabta/arrow-notation 16 https://github.com/aycabta/chained-arrow-notation 12 Twitter
S
4.4.
97
Si+1 (0, n)
=
Si (n, n)
Si (m + 1, 0)
=
Si (m, 1)
Si (m + 1, n + 1)
=
Si (m, Si (m + 1, n))
3 F1
S
i
S
B(m, n) = A(i, m.n)
1
g(2)
A(1, 2, 2)
A(1, 1, 0)
=
A(1, 0, 1) = A(1, 1) = 3
A(1, 1, 1)
=
A(1, 0, 3) = A(3, 3) = 61
A(1, 1, 2)
=
A(1, 0, 61) = A(61, 61) > 3
A(1, 1, 3)
>
A(1, 0, f 2 (1) + 2) > 3
A(1, 1, x)
>
3
3
x
A(1, 1, 65)
>
3
3
65
3
3 3
2
2+2
2+2
2 2>
A(1, 1, 65) A(1, 2, 0)
=
A(1, 1, 1) = 61
A(1, 2, 1)
=
A(1, 1, 61) > 3
A(1, 2, 2)
> A(1, 1, 3 >
3
3
3 (3
3 61
3
61
2
2) 61
2)
2
A(1, 2, 2) F1
4
4
A(1, 0, 1, 0)
=
A(1, 0, 0, 1) = A(1, 0, 1) = A(1, 1) = 3
A(1, 0, 1, 1)
=
A(1, 0, 0, (A(1, 0, 1, 0))
98
4 = A(1, 0, 1, 1)
A(1, 0, 0, 3) = A(3, 0, 3) = A(2, 3, 3)
2
A(1, 0, 1, 2)
=
A(1, 0, 0, A(1, 0, 1, 1))
=
A(1, 0, 0, A(2, 3, 3))
=
(A(2, 3, 3), 0, (2, 3, 3))
A(1, 0, 1, 3) =
A(1, 0, 0, A(1, 0, 1, 2))
A(1, 0, 1, n + 1)
A(1, 0, 1, 2) SS
2
=
A(A(1, 0, 1, 2), 0, A(1, 0, 1, 2))
=
A(1, 0, 0, A(1, 0, 1, n))
=
A(A(1, 0, 1, n), 0, A(1, 0, 1, n))
A(2, 3, 3)
S
F1
2
(
2
2
2
2
3
SS
S A(1, 0, 1, n)
n
m m
1
SS
f (m)
f (m)
SS
2
1
S
4
A(1, 0, 1, 1) A(1, 0, 1, 2) F1
SS
2
2
S
SS SS
A(1, 0, 1, 63)
A(1, 0, 2, 0)
=
A(1, 0, 1, 1) = A(2, 3, 3)
A(1, 0, 2, 1)
=
A(1, 0, 1, (A(1, 0, 2, 0)))
2 63
4.4.
99 =
A(1, 0, 2, 1)
S
SS
F1 SSSS
A(1, 0, 1, (A(2, 3, 3))) A(2, 3, 3)
SSS
A(1, 0, 2, 1)
A(1, 0, 3, 1) A(1, 1, 1, 1) A(1, 1, 1, 1)
SSSS
=
A(1, 1, 0, A(1, 1, 1, 0))
=
A(1, 1, 0, A(1, 1, 0, 1))
=
A(1, 1, 0, A(1, 0, 1, 1))
=
A(1, 1, 0, A(2, 3, 3))
=
A(1, 0, A(2, 3, 3), A(2, 3, 3))
S
A(2, 3, 3)
F1
A(1, 0, 1, 63) < F1 < A(1, 0, 2, 1) < A(1, 1, 1, 1) F1 S
1 17
87
W7plq.175s 3 2
a
B(x, y)
17
log/ln032.html
a−1
bN 2
S
2
3 F2 (x)
3
SS 63
3
S
102
4
SS
a
S
b
ga,b (n) = Ba,b (0, n)
A(a, b, 0, n)
a = 0, b = 0
B0,0 (0, n) = n + 1 = A(0, 0, 0, n) Ba,b (m, n) = A(a, b, 0, n)
S
1
A(a, b + 1, 0, n) S Ba,b (0, n)
=
A(a, b, 0, n)
Ba,b (m + 1, 0) = Ba,b (m + 1, n + 1)
Ba,b (m, 1)
=
Ba,b (m, B(m + 1, n))
Ba,b+1 (0, n) =
Ba,b (n, n)
Ba,b (m, n) = A(a, b, m, n) A(a, b, 0, n)
=
A(a, b, m, 1)
A(a, b, m + 1, n + 1)
=
A(a, b, m, A(a, b, m + 1, n))
A(a, b, 0, n)
SS SS
A(a, b, 0, n)
A(a, b, m + 1, 0)
A(a, b + 1, 0, n) =
S
=
SS 1
A(a, b, n, n)
A(a, b + 1, 0, n)
a
Ba,0 (0, n)
Ba+1,0 (0, n)
S
n
Ba,n (0, n)
f (m) n
Ba+1,0 (0, n)
=
A(a + 1, 0, 0, n) =
Ba,n (0, n)
A(a, n, 0, n)
4.4.
103 ga,b (n) = Ba,b (0, n) ≈ A(a, b, 0, n)
SS
63
A(63, 0, 0, n)
A(1, 0, 0, 0, 63) = A(63, 0, 0, 63)
4.4.3
3 3(F3 )
21
3 1.
f (x)
s(1)f s(n)f 2.
3.
f (x)
g(x)
s(n) (n > 0)
g; g(x) = f x (x)[
:=
] x
g; g(x) = [s(n − 1) ]f (x)(n > 1)[n
:=
g(x)
]
ss(n) (n > 0)
ss(1)f
:=
g; g(x) = s(x)f (x)[
]
ss(n)f
:=
g; g(x) = [ss(n − 1)x ]f (x)(n > 1)[
]
F3 (x) F3 (x) := ss(2)63 f ; f (x) = x + 1
4.
F3 := F363 (3) 22
F3
21 http://ja.googology.wikia.com/wiki/ 22
http://ja.googology.wikia.com/wiki/ 3
3 :Mikadukim/
104
4
f∗
f f ∗ (x) = f x (x) T∗
T
(T ∗ f )(x) = (T x f )(x) s(n), ss(n), F3 s(1)f
=
f∗
s(n)
=
s(n − 1)∗
(ss(1)f )(x)
=
(s(x)f )(x)
ss(n)
=
ss(n − 1)∗
F3 F1
63
=
(n ≥ 2) (n ≥ 2)
63
(ss(2) f0 ) (3),
f0 (x) = x + 1
F2
F2
F3 (
F3 )
F3 F3
F3
F1 ,F2 ,
F3 ,
F1
F3
F3
4.1 F3 F3
F3
F3
F3
F3
2
1
SS S
S
S S
S
4.4.
105
4.1: F1
F1
F3
2
S
S F2
(SS
S
(2
)
)
SS
(3
)
(SS F3
)
F2
S
,SS
,...
s(n)
(
)
ss(1) F3
F3
s(n)
s(1)
2
SS
S
SS s(n) F3
2
S 2
2 2
3 F3
s(1)
s(2) 2
F3 s(1)f
:=
g; g(x) = f x (x)
s(2)f
:=
g; g(x) = [s(1)x ]f (x)
F1
F2
s(2)
S s(1)f := g; g(x) = f x+1 (1)
106 s(2)
4 F1
F3
F2
S
s(n)
f (x) = x + 1 [s(1)a1 ][s(2)a2 ]...[s(n)an ]f (x) ≈ A(an , ..., a2 , a1 , x) s(1)f := g; g(x) = f x+1 (1) F3 s(n)
s(1)
F3 ss(1)
s(n)
g(x) = s(x)f (x)
f (x)
x
4.5 (Chris Bird)
23
uglypc.ggh.org.uk 2002
n 2 3 4 23
n 2 x →x x
Wiki - Chris Bird http://ja.googology.wikia.com/wiki/Chris Bird
4.5.
107 4
4
2
4
A(1, 0, 1, 2, 2)
24
(Bird’s linear notation) BEAF
(Jonathan Bowers)
25 26
2002
(John Spencer) (Bowers Exploding Array Function)
24 Chris
BEAF 2007
Bird’s Super Huge Numbers http://mrob.com/users/chrisb/ Bowers 26 Hedrondude’s Home Page http://www.polytope.net/hedrondude/home.htm 27 Wiki - BEAF http://ja.googology.wikia.com/wiki/BEAF 25 http://ja.googology.wikia.com/wiki/Jonathan
27
108
4
2002
1 2016
2
352
2008 Web 2015
E 3
E
15610 (Lawrence Hollom) (Aarex Tiaokhiao)
Hyp cos
28
BEAF
(array notation) BEAF
28
Wiki -
http://ja.googology.wikia.com/wiki/
BEAF
4.5.
109 BEAF A = (a1 , a2 , . . . , an ) 1
v(A) = {a1 , a2 , . . . , an }
1. {a} = a, {a, b} = ab 2. {a, b, c, . . . , n, 1} = {a, b, c, . . . , n} 3. {a, 1, b, c, . . . , n} = a 4. 3
1 {a, b, 1, . . . , 1, c, d, . . . , n} % &' ( x
=
1
{a, a, . . . , a, {a, b − 1, 1, . . . , 1, c, d, . . . , n}, c − 1, d, . . . , n} % &' ( % &' ( x+1
•
a
x
1
1
•
2
• 5.
1
1
1 4 {a, b, c, d, . . . , n} = {a, {a, b − 1, c, d, . . . , n}, c − 1, d, . . . , n} 3 {3, 3, 3}
=
{3, {3, 2, 3}, 2}
=
{3, {3, {3, 1, 3}, 2}, 2}
=
{3, {3, 3, 2}, 2} = {3, {3, {3, 2, 2}, 1}, 2}
=
{3, {3, {3, 2, 2}}, 2}
1
110
4 {3, 3{3,2,2} , 2}
=
{3, 3{3,{3,1,2},1} , 2}
=
{3, 3{3,3} , 2} = {3, 33 , 2}
=
{3, 3
3, 2} = {3, {3, 3
= 3
{3, 3
= 3
3
{3, 3
= 3
3
3
=
{3, 3, 3}
3 − 2, 2}
{3, 3 3
3 − 3, 2} 3
3
3
3
3 3
{a, b, c} = a
b
c
=
{a, {a, b − 1, c}, c − 1}
=
a
c−1
=
a
c−1
=
... = a %
{a, b − 1, c} = a c−1
b
c−1
&'
...
a
c−1
b
{a, {a, b − 2, c}, c − 1}
{a, b − 2, c} c−1
a
b
{a, b} = a
c
c−1
c
c=a
c−1
a
3 BEAF
c=1
{a, b, c}
3 − 1, 2}}
3 − 1, 2}
... = 3
= 3 {3, 3, 3}
3
=
a=a (
c
b
4 {3, 65, 1, 2}
=
{3, 3, {3, 64, 1, 2}, 1} = {3, 3, {3, 64, 1, 2}}
✷
4.5.
111 =
{3, 3, {3, 3, {3, 63, 1, 2}}}
=
{3, 3, {3, 3, {3, 3, {3, 62, 1, 2}}}}
{65, 2, 2, 2} = {65, {65, 1, 2, 2}, 1, 2} = {65, 65, 1, 2}
1.
1
0
2.
3.
{a, b} = ab
A(a) = a + 1
2 1
{a, b, 1, . . . , 1, c, d, . . . , n} =
4.
{a, a, a, . . . , {a, b − 1, 1, . . . , 1, c, d, . . . , n}, c − 1, d, . . . , n} a
, a) = A(X, b, a,
A(X, b+1, 0,
, a)
1 4
A(1, 0, 0, 0, 0, 5)
1
=
A(1, 5, 0, 0, 0, 5)
=
A(1, 4, 5, 0, 0, 5)
=
A(1, 4, 4, 5, 0, 5)
=
A(1, 4, 4, 4, 5, 5)
112
4
f (m) = {n + 1, m, a0 , X}(X {n + 1, 2, a0 + 1, X} {n + 1, 3, a0 + 1, X}
{n + 1, 4, a0 + 1, X} {n + 1, m + 1, a0 + 1, X} 2
)
=
{n + 1, {n + 1, 1, a0 + 1, X}, a0 , X}
=
{n + 1, n + 1, a0 , X} = f (n + 1)
=
{n + 1, {n + 1, 2, a0 + 1, X}, a0 , X}
=
{n + 1, f (n + 1), a0 , X}
=
f 2 (n + 1)
=
{n + 1, {n + 1, 3, a0 + 1, X}, a0 , X}
=
f 3 (n + 1)
=
f m (n + 1)
m+1
f
{n + 1, 2, a0 + 2, X}
= =
3
0
{n+1, 2, a0 +1, X} = f (n+1) 1
m
{n + 1, n + 1, a0 + 1, X} f n (n + 1)
{n+1, 2, a0 +2, X} = f n (n+1) f n (n)
3
{n + 1, m + 1, 1, a1 + 1, X} {n + 1, n + 1, {n + 1, m, 1, a1 + 1, X}, a1 , X}
= 4
{n + 1, m, 1, a1 +
1
1, X}
2
4 1
2 5
3 1
4
2 n+2
3 n
4.5.
113
n>1 {n + 1, n + 1, n + 1, n + 1} = {n + 1, 2, 1, 1, 2} ≈ {n + 1, ..., n + 1} = {n + 1, 2, 1, 1, 1, 2} ≈ % &' (
A(1, 0, 0, n) A(1, 0, 0, 0, n)
5
{n + 1, 2, a0 + 1, a1 + 1, ..., ak + 1} ≈
A(ak , ..., a1 , a0 , n)
f (n) = A(ak , ..., a2 , a1 , a0 , n) {n + 1, m + 1, a0 + 1, a1 + 1, a2 + 1, ..., ak + 1} ≈ f m (n)
{1, a0 , a1 , ..., ak }
= 1
{2, a0 , a1 , ..., ak }
= 4 (k > 1, a0 > 1, ak > 1
)
4
{2, a0 , a1 , ..., ak }
=
{2, {2, a0 − 1, a1 , ..., ak }, a1 − 1, ..., ak }
=
... = {2, X, 1, ..., ak }(
=
{2, 2, Y, ..., ak }(
=
{2, {2, 1, Y, ..., ak }, Y − 1, ..., ak }
=
{2, 2, Y − 1, a2 , ..., ak }
=
{2, 2, Y − 2, a2 , ..., ak }
=
{2, 2, 1, a2 , ..., ak }
=
{2, 2, 1, 1, a3 , ..., ak }
=
{2, 2, 1, 1, ..., 1}
=
{2, 2} = 4
3 2
2
X
)
Y = {2, X − 1, 1, ..., ak })
1 2
4
4 3
114
4 3
5
1
4
1
4 3
Googology Wiki
29
1
30
4 (a, b, c, d) a
b
d > {a, b, c, d}
c
a = 2, b > 2, c > 0, d > 1
n−1
n n−2
5
4
2 3 5
31 32
2
(Bird’s proof)
3 4
=4
(p. 219)
29
http://ja.googology.wikia.com/wiki/ :1793 http://googology.wikia.com/wiki/User blog:Wythagoras/Results of the First International Googological Olympiad 31 Wiki http://ja.googology.wikia.com/wiki/ 32 5 http://www.mrob.com/users/chrisb/Proof.pdf 30
1
115
5
2
2 •
•
116
5
5.1 (Georg Cantor, 1845–1918)
(ordinal number)
(transfinite number)
5.1.1 ✓ ✒
1 MathWorld 2
Wikipedia
1 2
✏ ✑
- Ordinal Number http://mathworld.wolfram.com/OrdinalNumber.html MathWorld ZF Quine
5.1.
117
3
(ordered set) 2
(totally ordered set) (partially ordered set)
2 a, b ⊂
X X
2
X ≤
≤
≤
(X, ≤)
1. X
≤
3
a≤a
a
2. x ≤ y
y≤z
x≤z
3. x ≤ y
y≤x
x=y
4. X
3 Wikipedia
x, y
-
x≤y
http://ja.wikipedia.org/wiki/
y≤x
X (X, ≤)
118
5 4 5
(well-order)
(well-ordered set)
X ≤
✓
≤
A
(well-founded)
X a0
A A
a0 ≤ a
a
X
✒
✑ 0
0 a
a/2 an = 1/n
(axiom of choice)6
7
theory)
(naive set theory)
4
Wiki http://ja.math.wikia.com/wiki/ http://ja.wikipedia.org/wiki/ 6 Wikipedia http://ja.wikipedia.org/wiki/ 7 http://alg-d.com/math/ac/ 5 Wikipedia
✏
(axiomatic set
5.1.
119
(Ernst Friedrich Ferdinand Zermelo, 1871–1953) (Abraham Halevi Adolf Fraenkel, 1891–1965) (Zermelo-Fraenkel set theory) ZFC ZF (choice)
ZF Zermelo
Fraenkel
C
8 9 10
ZFC
ZFC
(order type) (A, ≤)
2
A
✓ a1 , a 2
B
(A, ≤)
(order isomorphic)
(B, ≤)
A
✏
a1 ≤ a2 ⇐⇒ f (a1 ) ≤ f (a2 ) ✒ 8
A
B
f
Wiki http://ja.math.wikia.com/wiki/ http://ja.wikipedia.org/wiki/ 10 MathWorld - Order Type http://mathworld.wolfram.com/OrderType.html 9 Wikipedia
✑
120
5 (bijective)
1.
: f (A) = B
2.
:
11
A
f :A
a1 , a2
B
2
f (a1 ) = f (a2 )
a1 = a2
5.1.2 (finite ordinal) A k
B
k 0, 1, 2, 3, ...
(
k
) (inifinite ordinal) N = {0, 1, 2, 3, . . .}
ω
(transfinite number)
0, 1, 2, 3,
ω
0, 1, 2, 3,
(fundamental sequence) 1
ω 0, 2, 4, 6, ...
ω
ω ω
ω + 1, ω + 2, ... 1+ω
11 Wikipedia
-
http://ja.wikipedia.org/wiki/
5.1.
121
1 + 1, 1 + 2, 1 + 3, ...
1+ω =ω
(A, ≤)
α
α
(John von Neumann, 1903 - 1957)
0
=
{} (
1
=
{0} = {{}} (1
2
=
{0, 1} = {{}, {{}}} (2
3
=
{0, 1, 2}
4
=
{0, 1, 2, 3}
ω
=
{0, 1, 2, ...} (
=0
) ) )
)
ω+1 =
{0, 1, 2, ..., ω}
ω+2 =
{0, 1, 2, ..., ω, ω + 1}
ω+3 =
{0, 1, 2, ..., ω, ω + 1, ω + 2}
ω+ω
{0, 1, 2, ..., ω, ω + 1, ω + 2, ...}
=
(cardinality) A
B
A
ω {0, 1, 2, ..., ω} B
A
B
A = {0, 1, 2, ...} B
ω+1
B=
f
f (0) = ω, f (k) = k − 1(k > 0) a1 ≤ a2 ⇐⇒ f (a1 ) ≤ f (a2 )
A
122
5 a1 = 0, a2 = 1 f
f (k) = ω
k
f (k + 1)
ω
B
ω
A
(cardinal number)
ω+1 A ω
ω 0 ℵ0 )
ω+1
ω0
ω
5.1.3 α
β
α
α = β+1
(successor ordinal)
0
(limit ordinal)
ω
ω + a(a = 1, 2, 3, ...)
0
✓ α
ξ ✒
ζ
✏
ζ · · · >
ω βi c i
i=1 β1
ω c 1 + ω β2 c 2 + · · · + ω βk c k
✒
✑
ϵ 0 = ω ϵ0
ϵ0
ϵ0 ϵ0
ω
ϵ0 ϵ0
α
f (α)
f (α)
• f (α) = 1 : 0, 1, 2, ..., 9, ω • f (α) = 2 : 10, 11, 12, ...., 99, ω2, ω3, ..., ω9, ω 2 , ω 3 , ..., ω 9 , ω ω
5.2 α
✏
N→N
(ordinal hierarchy)
126
5
5.2.1 1904
(Godfrey Harold 13 14
Hardy, 1877–1947)
(Hardy function)
H[α] : N → N
α ✓ H[0](n)
=
n
H[α + 1](n)
=
H[α](n + 1)
H[α](n)
=
H[α[n]](n)(α
✒
✏
✑
)
(
+1
)
x+1
α H[α] α>β H[α] > H[β]
H[ 1
(canonical
fundamental sequence) α[0], α[1], α[2], ... 1 (Wainer hierarchy)
]
H[α] α ≤ ϵ0
13 http://ja.googology.wikia.com/wiki/ 14 Hardy, G.H. (1904) A theorem concerning the infinite cardinal numbers. Quarterly Journal of Mathematics 35: 87–94.
5.2.
127
✓
α ≤ ϵ0
α
α[0], α[1], α[2], ... ω[n]
=
n
[n]
=
ωα n
ω α [n]
=
ω α[n] (α
(ω α1 + ω α2 + · · · + ω αk )[n]
=
ω α1 + ω α2 + · · · + ω αk [n]
ω
α+1
✏
(α1 ≥ α2 ≥ · · · ≥ αk
✒
ϵ0 [0]
=
1, ϵ0 [n + 1] = ω ϵ0 [n]
ϵ0 [0] = 0
✑
ϵ0 ω[n] = n
ω
0, 1, 2, 3, ...
H[3](n) =
H[2](n + 1) = H[1](n + 2) = H[0](n + 3) = n + 3
H[m](n) =
n+m
H[ω](n) =
H[n](n) = n + n = 2n
H[ω + 2](n) =
H[ω + 1](n + 1) = H[ω](n + 2) = 2(n + 2)
H[ω + k](n) =
2(n + k)
H[ω × 2](n) =
H[ω + n](n) = 2(n + n) = 4n
H[ω × 2 + 2](n) = H[ω × 3](n) = H[ω × k](n) = H[ω 2 ](n) =
H[ω 2 + 3](n) = H[ω 2 + ω](n) = H[ω 2 + ω + k](n) =
H[ω × 2 + 1](n + 1) = H[ω × 2](n + 2) = 4(n + 2) H[ω × 2 + ω](n) = 4(n + n) = 8n = 23 n
2k n
H[ω × n](n) = 2n n
H[ω 2 ](n + 3) = 2n+3 (n + 3) H[ω 2 + n](n) = 2n+n (n + n) = 22n (2n) H[ω 2 + ω](n + k) = 22(n+k) 2(n + k)
128
5
H[ω 2 + ω × 2](n) =
H[ω 2 + ω + n](n) = 22(n+n) 2(n + n) = 24n (4n)
H[ω 2 + ω × 3](n) =
H[ω 2 + ω × 2 + n](n) = 24(n+n) 4(n + n)
H[ω 2 + ω × k](n) =
22 n (2k n)
3
= 28n (8n) = 22 n (23 n) k
H[ω 2 × 2](n) =
H[ω 2 + ω × n](n) = 22
n
n
(2n n)
✓
✏ H[α + β](n) = H[α](H[β](n))
✒
α + β = γ + β, α > γ H[α + β](n)
γ H[α](n)
H[β](n)
α = ω 2 + 2, β = ω α + β = ω2 + 2 + ω = ω2 + ω +2 α = ω2 , β = ω + 3
H[α + β](n) = H[α](H[β](n))
H[α + β](n) = H[ω 2 + ω + 3](n) = 22(n+3) 2(n + 3)
H[α](n)
=
H[ω 2 ](n) = 2n n
H[β](n)
=
H[ω + 3](n) = 2(n + 3)
H[α](H[β](n)) =
H[α × 2](n)
=
22(n+3) 2(n + 3)
H[α + α](n) = H[α](H[α](n)) = H[α]2 (n)
✑
5.2.
129 H[α × 3](n)
H[α × 2 + α](n) = H[α × 2](H[α](n))
= =
H[α]2 (H[α](n)) = H[α]3 (n)
H[α × 4](n)
=
H[α]4 (n)
H[α × k](n)
=
H[α]k (n)
H[α × ω](n)
=
H[α]n (n)
H[α × ω](n)
H[ω](n) = H[ω 2 ](n) =
H[ω 3 ](n)
H[α](n)
n
2n H[ω × ω](n) = H[ω]n (n) = 2n n
=
H[ω 2 × ω](n) = H[ω 2 ]n (n)
f (n) = 2n n H[ω 2 ](n)
=
f (n) = 2n n
H[ω 2 × 2](n)
=
f 2 (n) = 22
H[ω 2 × 3](n)
=
f 3 (n) = 22
n
n
(2n n)
2n n
(2n n) 2n n
2
(2n n)
f (n) = 2n H[ω 2 ](n)
> f (n) = 2n
H[ω 2 × 2](n)
> f 2 (n) = 2
2
n>2
H[ω 2 × 3](n)
> f 3 (n) = 2
2
2
n>2
H[ω × 4](n)
> f (n) = 2
2
2
2
H[ω 3 ](n)
2
2
4
H[ω 2 ](n) > 2n n
>
2 3
n>2
4
n H[ω 3 ](n) > 2
130
5
H[ω 4 ](n) > 2
n
m>1
H[ω m ](n)
>
m−1
2
m=2
n
m=k
m=k+1 H[ω k ](n)
>
k−1
2
n
H[ω k × 2](n) =
H[ω k ]2 (n) > 2
H[ω × 3](n) =
H[ω ] (n) > 2
k
H[ω k+1 ](n) =
k 3
k−1 k−1
H[ω k × ω](n) > 2
2
k−1
2
k−1
k
k
n>2 2
k−1
2
n>2
k
3
n ✷
5.2.2 15
hierarchy) ✓
16
(FGH; Fast-growing
F [α](n)
✏
(FGH)
✒
F [0](n)
=
n+1
F [α + 1](n)
=
F [α]n (n)
F [α](n)
=
F [αn ](n)(α
Wiki
✑
)
fα (n) F [α](n)
15 16
Wiki ,
,
,
http://ja.googology.wikia.com/wiki/ (1997)
5.2.
131
F [α](n) = fα (n) α < ϵ0 F [α](n) = H[ω α ](n) 1 F [0](n) = n + 1 = H[1](n) = H[ω 0 ](n) F [α](n) = H[ω α ](n)
2 F [α + 1](n)
=
F [α]n (n) = H[ω α ]n (n)
=
H[ω α × ω](n)
=
H[ω α+1 ](n)
F [α + 1](n)
α = ϵ0
α
F [α](n) = H[ω ](n) F [ϵ0 ](3)
ω
=
F [ω ω ](3)
=
H[ω ω
≈
H[ϵ0 ](4)
ωω
](3)
F [ϵ0 ](n) ≈ H[ϵ0 ](n + 1) = H[ϵ0 + 1](n) F [ϵ0 ](n) ≈ H[ϵ0 ](n)
+1
f n (n)
1953 gorczyk, 1922–2014) chy)
(Andrzej Grze(Grzegorczyk hierar-
17
17 Grzegorczyk, A. (1953) Some classes of recursive functions, Rozprawy Matematyczne 4: 1–45.
132
5 n
• E0
En
x + 1, x + 2, ...
• E1
x + y, 3x, ...
• E2
xy, x3 , ...
• E3
x y , 22
2x
, elementary recursive
function
• E4 En
n
E n−1
(extended Grzegorczyk hierarchy)
18
1904
1953 (John Edensor Littlewood, 1885–1977) 19
18
(2016) J. E. (1953) A mathematician’s miscellany. Methuen
19 Littlewood,
5.2.
133 20
H[ω m ](n) > 2 F [m](n) = H[ω m ](n) > 2
m−1 m−1
n
n 6
m=3
F [α](n) F [α + 1](n)
m
F [m](n)
n F [n](n)
2 F [ω](n) = F [n](n) F [ω](n)
2
n−1
F [ω](n) > 2
f (n) = 3
n
n = A(n + 1, n − 3) + 3 > A(n, n)
f 64 (4)
3
F [ω + 1](n) = F [ω + 1](64)
=
F [ω]n (n) ≈ f n (n) f 64 (64) ≈
f (n) = 3 ω+1 1 1 ω
1
2 3 20
( )
3
n
≈
( ) (1990)
f (n) ≈ F [ω](n)
F [ω](n)
n
3
ω
f n (n)
+1
134
5
3
3
3
3
n
≈
F [ω + ω](n) = F [ω × 2](n)
3
3
3
n
≈
F [ω × 2 + ω](n) = F [ω × 3](n)
✓
✏
3
3
2n
≈
3
23
2n
≈
3
23
23
2n
≈
23
23
23
2n
≈
3
3n
≈
3
4n
≈
3
nn
≈
✒
F [ω × ω](n) = F [ω 2 ](n) F [ω 2 + ω](n)
F [ω 2 + ω × 2](n) F [ω 2 + ω × 3](n)
F [ω 2 + ω × ω](n) = F [ω 2 × 2](n) F [ω 2 × 3](n)
F [ω 2 × ω](n) = F [ω 3 ](n) 1
2
ω
2
ω
3
n ω n−1
• n • n
F [ω n−1 an−1 + ω n−2 an−2 + ... + a0 ](n)
n
3 F3
s(n)
fb (n) = A(X, b, n) (X
0
fb+1 (n)
=
A(X, b + 1, n)
=
A(X, b, A(X, b + 1, n − 1))
=
fb (A(X, b + 1, n − 1))
=
0
)
fb2 (A(X, b + 1, n − 2))
✑
5.2.
135 = fbn (A(X, b + 1, 0))
= ≈
fbn (n) s(1)f (x) = f x (x)
s(1) 3 1.
fb (n) = A(X, b, n)
fb+1 (n) ≈ fbn (n)
s(1)f (n) = f n (n)
2. s(1)
F [α + 1](n) = F [α]n (n)
3.
2
1
s(1)
1
f (n) = n + 1 A(2, n)
≈
A(3, n)
≈
A(4, n)
≈
A(m + 1, n)
≈
A(n, n)
≈
s(1)f (n) = f n (n) = F [1](n) = 2n s(1)2 f (n) = F [2](n) = 2n n s(1)3 f (n) = F [3](n) ≈ 2 ↑2 n
s(1)m f (n) = F [m](n) ≈ 2 ↑m−1 n s(1)n f (n) = F [n](n)
s(1) A(1, 0, n) = A(n, n), s(2)f (n) = s(1)n f (n), F [ω](n) = F [n](n) A(1, 0, n)
≈
s(2)f (n) = F [ω](n)
3 fb+1 (n) ≈ fbn (n), s(1)
F [α]n (n) A(1, 1, n)
≈
A(1, 2, n)
≈
s(2) ω , F [α + 1](n) =
s(1)s(2)f (n) = F [ω + 1](n) s(1)2 s(2)f (n) = F [ω + 2](n)
136
5 A(1, 3, n)
≈
A(1, n, n)
≈
≈
s(2)2 f (n) = F [ω × 2](n)
A(1, 0, n)
≈
s(2)f (n) = F [ω](n)
A(2, 0, n)
≈
≈
A(4, 0, n)
≈
A(n, 0, n)
≈
A(1, 0, 0, n)
≈
A(2, 0, 0, n)
≈
A(n, 0, 0, n)
≈
A(1, 0, 0, 0, n)
≈
≈
s(2)2 f (n) = F [ω × 2](n)
s(2)3 f (n) = F [ω × 3](n)
s(2)4 f (n) = F [ω × 4](n)
s(2)n f (n) = F [ω × n](n) = F [ω 2 ](n) s(3)f (n) = F [ω 2 ](n)
s(3)2 f (n) = F [ω 2 × 2](n)
s(3)n f (n) = F [ω 2 × n](n) = F [ω 3 ](n) s(4)f (n) = F [ω 3 ](n)
s(2)s(3)2 s(4)3 f (n) = F [ω 3 × 3 + ω 2 × 2 + ω](n)
✓
✏
s(n)
A(ak , ..., a2 , a1 , a0 , n)
≈ =
✒
s(1)n s(2)f (n) = F [ω + ω](n)
A(2, 0, n)
A(3, 0, n)
A(3, 2, 1, 0, n)
s(1)3 s(2)f (n) = F [ω + 3](n)
A(1, 1, ..., 1) % &' (
≈
s(1)a0 s(2)a1 ...s(k + 1)ak f (n) F [ω k × ak + ... + ω 2 × a2 + ω × a1 + a0 ](n) s(n)f (n) ≈ F [ω ω ](n)
n
s(n)
ω
F [ω ](n)
✑
5.2.
137 F3
ss(n)
ss(1)f (n)
=
ss(1)2 f (x)
≈
ss(2)f (n)
=
ss(3)f (n)
=
ss(4)f (n)
=
ss(n)f (n)
≈
s(n)f (n) ≈ F [ω ω ](n)
F [ω ω + ω ω ](n) = F [ω ω × 2](n)
ss(1)n f (n) ≈ F [ω ω × ω](x) = F [ω ω+1 ](x) ss(2)n f (n) ≈ F [ω ω+2 ](n) ss(3)n f (n) ≈ F [ω ω+3 ](n) F [ω ω×2 ](n)
F3 (n)
=
F3
=
ss(2)63 f (n) ≈ F [ω ω+1 × 63](n) F363 (3) ≈ F [ω ω+1 × 63 + 1](63)
BEAF {n + 1, m + 1, a0 + 1, a1 + 1, a2 + 1, ..., ak + 1} ≈
f m (n),
f (n) = A(ak , ..., a2 , a1 , a0 , n) (n > 1) ✓
✏
n>1 {n + 1, m + 1, a0 + 1, a1 + 1, a2 + 1, ..., ak + 1} ≈ =
✒
{3, 3, ..., 3} ≈ % &' (
F [ω k × ak + ... + ω 2 × a2 + ω × a1 + a0 ]m (n) H[ω ω
k
F [ω ω ](n)
× m](n) ✑
n
1 F3
×ak +...+ω 2 ×a2 +ω×a1 +a0
2
138
5
✓
✒
3
F1
≈
F2
≈
F3
≈
✏
A(1, 0, 1, 63) ≈ {4, 64, 1, 1, 2} ≈ F [ω 2 + 1](63) A(63, 0, 0, 63) ≈ {4, 2, 1, 1, 64} ≈ F [ω 3 ](63) F [ω ω+1 × 63 + 1](63)
✑
5.2.3 21
1 ✓
✒
✏
(SGH)
G[0](n)
=
0
G[α + 1](n)
=
G[α](n) + 1
G[α](n)
=
G[αn ](n)(α
0
G[1](n) =
1
G[2](n) =
2
G[m](n) =
m
G[ω](n) =
n
G[ω2](n) =
2n
G[ω ](n) =
n2
G[ω ω ](n) =
nn
G[ω ω + ω](n) = Wiki -
)
G[0](n) =
2
21
(slow-growing hierarchy)
nn + n
http://ja.googology.wikia.com/wiki/
✑
5.3.
139 ω
G[ω ω ](n) = ω
nn
n
n G[ϵ0 ](n) = n ↑↑ n
3