- Email: [email protected]
Some Rapidly Growing Functions1), 2)
HARVEY FRIEDMAN'S RESEARCH ON THE FOUNDATIONS OF MATHEMATICS, L.A. Harrington et al. (editors) Elsevier Science Publishers B.V. (North-Holland), 1985
367
Some R a p i d l y Growing F u n c t i o n s
I), 2)
C r a i g Smorynski
The purpose o f t h i s n o t e i s pure iconoclasm. m a t i c a l myths about how l a r g e " l a r g e " i s .
I wish t o debunk a few mathe-
When t h e mathematician says " l a r g e " ,
t h e l o g i c i a n i s sure t o t h i n k " s m a l l " . The f i r s t clich; Skewes number. T(X)
-
li(x)
u s u a l l y r e s o r t e d t o i n d i s c u s s i o n s o f largeness i s t h e
For a l l c a l c u l a t e d values, t h e number t h e o r e t i c f u n c t i o n Thus, when i n 1914 J. E. L i t t l e w o o d n o n c o n s t r u c t i v e -
i s negative.
l y proved t h a t t h i s f u n c t i o n changed s i g n s i n f i n i t e l y o f t e n , c u r i o s i t i e s about
where i t c o u l d become p o s i t i v e were aroused.
I n 1933, on t h e assumption o f t h e
Riemann Hypothesis, S . Skewes gave an upper bound f o r t h e f i r s t change o f sign. T h i s bound was so l a r g e (by t h e standards o f t h e day) t h a t i s achieved i n s t a n t n o t o r i e t y and even a t i t l e
-
t h e Skewes Number: 4.369
s
= 10lo < e
e
ee
The Skewes number has s i n c e been t o p p l e d from i t s p o s i t i o n o f supremacy.
In
1955, Skewes showed how t o l o w e r t h e bound i f one s t i l l assumed t h e Riemann Hypothesis; b u t he saved h i s r e p u t a t i o n by o b t a i n i n g t h e even l a r g e r upper bound,
e S ' = ee
e
7.705
,
"Reprinted from t h e Mathematical I n t e l l i g e n c e r 2 (1980) 149-154, o f S p r i nger-Verlag
.
by permission
"Editors' note: F u r t h e r d i s c u s s i o n o f t h i s t o p i c may be found i n t h e o t h e r s h o r t a r t i c l e s o f C. Smorynski i n t h i s volume, as w e l l as i n t h e c o n t r i b u t i o n s o f S. G. Simpson and R. L. Smith.
c. SMORYNSKI
368
when t h e Riemann Hypothesis was n o t assumed. shrunk,
i t s importance has:
While t h e number
has not
S'
S m a l l e r bounds f o r t h e s i g n change e x i s t today.
For
example, i n 1966 R. S. Lehman gave t h e bound
e L = e
e
2.067
L l a r g e these days:
Not even mathematicians would f i n d
Alan Baker won h i s
F i e l d s medal f o r , among o t h e r t h i n g s , g i v i n g e f f e c t i v e bounds i n number t h e o r y . I n h i s book on Transcendental Numbers, f o r i n s t a n c e , he c i t e s t h e bound 10
f o r a l l i n t e g r a l zeros
x,y
c o e f f i c i e n t s ) o f genus
1
o f an i r r e d u c i b l e polynomial
, degree
n
, and
height
f(x,y)
.
H
O f course, w i t h so much emphasis on t h e e f f e c t i v e n e s s o f t h a t mathematicians do n o t r e g a r d
B as b e i n g very l a r g e .
modern mathematician r e g a r d as l a r g e ?
(with integral
B
, we
must assume
What t h e n does t h e
Well, i n h i s column i n t h e November 1977
i s s u e o f S c i e n t i f i c American, M a r t i n Gardner c i t e s a r e s u l t o f R.
L. Graham.
According t o Gardner, " I n an u n p u b l i s h e d p r o o f , Graham has r e c e n t l y e s t a b l i s h e d an upper bound
... so v a s t t h a t
i t h o l d s t h e r e c o r d f o r t h e l a r g e s t number ever
used i n a s e r i o u s mathematical proof." i s , we f i r s t d e f i n e a f u n c t i o n
K
Intriguing?
To see what Graham's number
by r e c u r s i o n :
K(x,y)
i s "something l i k e " an e x p o n e n t i a l s t a c k o f
From K
, we
d e f i n e another f u n c t i o n
G
y's
by r e c u r s i o n :
o f height
x + 2 :
'
369
Some Rapidly Growing Functions = K(3,3)
G(0)
G(x + 1) = K(G(x),3) The growth o f stack o f
G
.
i s a b i t more d i f f i c u l t t o imagine.
i s something l i k e a
G(1)
G(0) 3 ' s , i.e.
I l e a v e t o t h e reader an e s t i m a t e o f
.
G(2)
G = G(64)
The bound t h a t Graham g i v e s i s
. I will
Now t h i s i s something t h e mathematician o f today regards as l a r g e . concede t h a t i t dwarfs numbers l i k e
f o r reasonably small values o f t h a t it i s
large
-
n
S , B, o r even say,
and
as
G
5
.
100)
I w i l l a l s o concede
a c c o r d i n g t o Gardner, t h e c o n s t a n t t h a t
bound f o r i s g e n e r a l l y b e l i e v e d t o be i s large.
(say
H
6 (!)
.
G
i s an upper
Rut I w i l l n o t concede t h a t
G
How can any number t h a t i s t h e value o f as s l o w l y growing a f u n c t i o n
on so small an argument as
64 be considered l a r g e ?
To g i v e us some standard f o r comparison, l e t me i n t r o d u c e a h i e r a r c h y o f number t h e o r e t i c f u n c t i o n s . F
n
: w + w
F o r each n a t u r a l number
0
Fn+,(x) Fn+l(x)
= Fn(
... ( F n ( x ) ) ...)
= x
a function
+ 1 x+l
= Fn
, with
F o ( x ) = x + 1, F1(x) = 2x + 1, F 2 ( x ) thing l i k e
, define
as f o l l o w s :
F (x)
i.e.
n E w
(XI ,
x + 1
nestings o f
i s something l i k e
2'
F
n
, and
.
Thus,
F (x) 3
some-
c. SMORYNSKI
370
F
4
, like
G(x)
,
i s a l i t t l e harder t o describe.
B e f o r e making any comparisons,
and H a r e something l i k e each o t h e r , 1 2 I do n o t mean t o i m p l y any r e l a t i o n n e a r l y as t i g h t as a s y m p t o t i c i t y . On t h e
t h a t I am g o i n g t o use:
When I say
l e t me q u i c k l y e x p l a i n t h e r u l e o f comparison
H
c o n t r a r y , I have i n mind a l o o s e r , more l i b e r a l , equivalence r e l a t i o n whose looseness grows w i t h t h e s i z e o f t h e elements o f a g i v e n e q u i v a l e n c e c l a s s . Given a f u n c t i o n taking
H
H , one n a t u r a l l y d e f i n e s a f a m i l y o f f u n c t i o n s
along w i t h a few b a s i c f u n c t i o n s (e.g.
5(H) by
a d d i t i o n ) and c l o s i n g under
e x p l i c i t d e f i n a b i l i t y (composition, a d d i t i o n o f dummy v a r i a b l e s , etc.).
I say
i s something l i k e H i f t h e c l a s s e s 3 ( H ) and 5 ( H ) a r e c o f i n a l 1 2 1 2 i f every f u n c t i o n o f 5 ( H ) i s m a j o r i s e d by one o f 5 ( H ) i n each o t h e r , i.e. 1 2
that
H
and v i c e versa.
T h i s i s a v e r y l o o s e measure o f equivalence:
a r e a l l something l i k e i t s own
2
X
, x! , and
n - f o l d composition w i t h i t s e l f ! )
x
X
T h i s looseness
F - H i e r a r c h y t h a t much more i m p r e s s i v e - f o r i t i s a h i e r a r c h y : n F i s n o t h i n g l i k e F , i.e. F i s so l a r g e r e l a t i v e t o F t h a t you n ntl n n+l need something l i k e F t o reach F from F F w i l l not do: F ntl ntl n n ntl e v e n t u a l l y m a j o r i s e s every f u n c t i o n o f Z ( F ) n The f u n c t i o n s K ( o r , r a t h e r , i t s d i a g o n a l ) and G f i t n e a t l y i n t o t h e
o f f i t makes t h e
.
.
.
F -Hierarchy. K(x,x) i s something l i k e F . w h i l e G i s something l i k e F n 3 ' 4 R e f l e c t i n g on t h e r a p i d i t y o f t h e growth of F F , o r even F 5 ' 236 G(64) ' the
reader w i l l see t h a t I was n o t b e i n g e n t i r e l y f a c e t i o u s when I r e f e r r e d e a r l i e r t o t h e slow r a t e o f growth e x h i b i t e d by Graham's f u n c t i o n
G(x)
.
" O f course",
t h e reader might o b j e c t , "anyone can produce ever more r a p i d l y growing f u n c t i o n s . But t h e Graham f u n c t i o n was used i n a " s e r i o u s mathematical proof." that
F
has never been so a p p l i e d .
G(64)
But
F
w
and
have
FE
It i s t r u e
-
and we
0
h a v e n ' t even reached t h e s e f u n c t i o n s y e t ! To d i s c u s s t h e r a t e s o f growth o f f u n c t i o n s used i n l o g i c , i t i s necessary
F -Hierarchy i n t o t h e t r a n s f i n i t e . n one s i m p l y i t e r a t e s what one d i d a t a :
t o extend t h e
A t successor o r d i n a l s , a
+
1
,
37 1
Some Rapidly Growing Functions
A t l i m i t ordinals,
one d i a g o n a l i s e s on what one has done b e f o r e , e.g.
F (x) w
.
= F (x) X
The f i r s t few values a r e now l o w e r t h a n those one had b e f o r e , but t h e f u n c t i o n s d e f i n e d a t l i m i t o r d i n a l s do c a t c h up and surpass t h e i r predecessors.
M. H. L'db
and S. Wainer have shown how t o c a r r y o u t t h i s c o n s t r u c t i o n t o any preassigned countable ordinal.
O f course, i n l o g i c one d o e s n ' t a c t u a l l y need t o go t h a t f a r
i n t o t h e t r a n s f i n i t e any more t h a n i n a n a l y s i s one needs t o use f u n c t i o n s growing
.
F The f u n c t i o n s we wish t o d i s c u s s a r e t h e F ' s f o r G(64) a a5€ The o r d i n a l t i s , t h e reader m i g h t r e c a l l , t h e l e a s t f i x e d p o i n t o f 0 0 E = min [8 = w 8 1 A more i n t u i t i v e o r d i n a l e x p o n e n t i a t i o n w i t h base w , i.e. 0 B i s g i v e n by v i e w i n g i t as t h e l i m i t o f t h e sequence, picture of E
as f a s t as
.
.
n
w , w
w
The immensity o f t h e s t e p from Moreover, between
and
w
w
w
ww
and
ww
a
w
w
)... .
to
there are not
w
between
, w
a
+ 1 i n c r e a s e s as a increases. 1
, not
w
, but
w
w
such steps:
w
there are
ww
human mind can comprehend t h e growth o f l o g i c a l (and, as we s h a l l see:
such steps; e t c .
FE
.
I don't think that the
Yet, i t i s f u n c t i o n o f d e f i n i t e
0 combinatorial) interest.
The f u n c t i o n s o f t h e
set,
a r e p r e c i s e l y t h o s e p r o v a b l y computable i n formal number t h e o r y .
(Believe it or
n o t , t h e s e f u n c t i o n s a r e ( t h e o r e t i c a l l y , i f not p r a c t i c a l l y ) computable.)
Thus,
i s , i n a sense, t h e f i r s t f u n c t i o n t o e v e n t u a l l y m a j o r i s e a l l f u n c t i o n s 0 p r o v a b l y computable i n formal number t h e o r y .
Ft
L e t me i n t e r p r e t t h i s l a s t f a c t . o f formal number t h e o r y o f t h e f o r m
Suppose we have a sentence o f t h e language
c. SMORYNSKI
312
,
Y x ByA(x,y) where
A
i s some p r o v a b l y d e c i d a b l e r e l a t i o n .
some p r o v a b l y computable
I f ( 1 ) i s provable, then,
for
,
F
a < E
i s a l s o provable; whence, f o r some
i s provable.
(1)
[ I n f a c t , t h e exact
a
0 '
can be determined from t h e p r o o f o f ( 1 ) .
Using such c o n s i d e r a t i o n s , i n 1952 G. K r e i s e l reviewed L i t t l e w o o d ' s 1914 paper
I f , however, (1) i s t r u e
and n o t e d e s s e n t i a l l y t h e same upper bound as Skewes.]
b u t unprovable ( a p o s s i b i l i t y n o t t o be o v e r l o o k e d ) , t h e source o f i t s unprova b i l i t y c o u l d be t h e f a i l u r e o f ( 2 ) f o r a l l p r o v a b l y computable a < E
f a i l u r e o f (3) f o r a l l
.
0
F
, i.e.
I f t h i s i s t h e case, any f u n c t i o n
F
the making
( 2 ) t r u e has moments o f r a p i d growth. Now t h e s i t u a t i o n I have j u s t d e s c r i b e d i s not h y p o t h e t i c a l :
J e f f P a r i s and
Leo H a r r i n g t o n have r e c e n t l y e x h i b i t e d such an independent statement somewhat i n t e r e s t i n g r e l a t i o n
A(x,y)
.
-
with a
Since t h i s i s t h e f i r s t n o t p u r e l y
l o g i c a l example o f a problem w i t h such n e a r l y astronomical bounds ( I comment on a f u n c t i o n o f t r u l y astronomical growth a t t h e end o f t h i s note), I s h a l l d i s c u s s i t i n some d e t a i l .
I suppose we should f i r s t s e t t l e on some n o t a t i o n . n a t u r a l numbers, we w r i t e X
.
A colouring, C
, of
[XI n [XI
n
for the collection of a l l
c
Ramsqy proved t h a t , i f respect t o
Y
n-element subsets o f
: [XIn
+
c
,
i s some p o s i t i v e n a t u r a l number and we i d e n t Fy a n a t u r a l number with c = {O,
i t s s e t o f predecessors:
subsets:
i s a set of
i s s i m p l y a map
c where
If X
c C
1
and n [Y]
n
C
...,c - 11 .
i s a colouring o f
, then
some b i g subset
i s constant.
homogeneous w i t h r e s p e c t t o
I n 1929 t h e economist F. P. n [XI and X i s b i g enough w i t h Y
X
has monochromatic
n-element
(To a v o i d clumsy phrasing, we c a l l such a s e t C .)
More s p e c i f i c a l l y , Ramsey proved t h e
373
Some Rapidly Growing Functions f o l l o w i n g two theorems.
Infinite C : [w]
i.e.
Ramsey Theorem.
n +
'
C
, there
c
[Yl
n
n, c
be p o s i t i v e i n t e g e r s .
i s an i n f i n i t e s e t
Y
F o r any c o l o u r i n g ,
homogeneous w i t h respect t o
w
C
,
i s constant.
F i n i t e Ramsey Theorem.
1. n
Let
Let
s
( f o r s i z e ) , n, c
be p o s i t i v e i n t e g e r s , w i t h
.
There i s a number R(s, n, c ) such t h a t , f o r a l l r 1. R(s, n, c ) n and a l l c o l o u r i n g s C : [ r ] + c , t h e r e i s a homogeneous Y r of cardinality s
+ 1
5 .
(The r e s t r i c t i o n
s
1. n +
1 s i m p l y r u l e s out t r i v i a l cases.)
N e i t h e r o f t h e s e statements i s p a r t i c u l a r l y i n t u i t i v e .
The b e s t way t o view
them i s as h i g h e r dimensional analogues o f D i r i c h l e t ' s Schubfachprinzip:
, the
n = 1
I n f i n i t e Ramsey Theorem j u s t a s s e r t s t h a t ,
If
i f an i n f i n i t e s e t i s
s p l i t i n t o a f i n i t e d i s j o i n t union o f subsets, one o f t h e subsets must be i n -
-
finite
ple:
A f i n i t e union o f f i n i t e sets i s f i n i t e .
i n i t s more usual f o r m u l a t i o n :
n = 1 and
For
s = 2
, the
F i n i t e Ramsey Theorem i s e x a c t l y D i r i c h l e t ' s P r i n c i -
R(2, 1, c ) = c + 1 and t e s t i t y o u r s e l f .
Take
The F i n i t e Ramsey Theorem i s a c e n t e r p i e c e o f f i n i t e c o m b i n a t o r i c s , w i t h much energy b e i n g expended on t h e c a l c u l a t i o n of
R(s, n, c )
and r e l a t e d "Ramsey
For, though such c a l c u l a t i o n i s c o n c e p t u a l l y t r i v i a l (One merely enu-
Numbers".
merates a l l p o s s i b i l i t i e s ...), i t i s i m p r a c t i c a l l y d i f f i c u l t . g i v e easy u p p w bounds: like
F
-
3
The diagonal
R ( x + 1, x, x )
Still,
one can
i s bounded by something
no l o n g e r a very l a r g e number by anyone's reckoning.
However, by
making a s u b t l e ( ? ) change i n t h e statement of t h e theorem, P a r i s and H a r r i n g t o n o b t a i n a v a r i a n t where t h e f u n c t i o n i n q u e s t i o n e x h i b i t s a more r e s p e c t a b l e r a t e o f growth
-
t h e f u n c t i o n i s something l i k e
FE
.
Q We need o n l y one more d e f i n i t i o n t o s t a t e t h e P a r i s - H a r r i n g t o n Theorem:
A
set
mum:
X
w
card(X)
i s r e l a t i v e l y l a r g e i f i t s c a r d i n a l i t y i s not l e s s t h a n i t s m i n i -
1. min(X)
.
P a r i s - H a r r i n g t o n Theorem. There i s a number
Let
H(s, n, c )
s, n, c
be p o s i t i v e i n t e g e r s w i t h
such t h a t , f o r a l l
h
1. H(s,
n, c )
s
1. n +
and a l l
1
.
c. SMORYNSKI
374 n
colourings
, there
C : [hl" + c
Y
8
h
of
.
s
c a r d i n a l i t y at l e a s t
i s a r e l a t i v e l y l a r g e homogeneous
The P a r i s - H a r r i n g t o n Theorem i s t r u e , b u t not p r o v a b l e i n formal number theory.
The f u n c t i o n
.
FG 0
like
H(x + 1, x, x )
( i n f a c t , H(x + 1, x, 3 ) )
i s something
Yet, a t l e a s t f o r t h e novice, t h e d i f f e r e n c e between t h e F i n i t e
Rainsey Theorem and t h e P a r i s - H a r r i n g t o n Theorem i s minimal.
This minimality i s
u n d e r l i n e d by t h e f a c t t h a t t h e s e theorems share a common n o n - e f f e c t i v e p r o o f . It i s i n t h e i r e f f e c t i v e p r o o f s , o f course, t h a t t h e y d i f f e r .
F o r t h e c u r i o u s reader, I o u t l i n e t h e n o n - e f f e c t i v e p r o o f . case o f t h e F i n i t e Ramsey Theorem. t h e theorem were f a l s e . r
1. s , t h e r e
.
Y E r
Suppose f o r f i x e d
.
f i n i t e set
C = U C i i
C : [rln
+
c
w i t h no s i z e
o f the f i r s t
s
s
homogeneous s e t
C
<...
C < C (by K o n i g ' s 0 1 and i n v o k e t h e I n f i n i t e Ramsey Theorem t o o b t a i n an i n -
Choose an i n f i n i t e p a t h
,
homogeneous w i t h r e s p e c t t o
X
+ 1 ,
2n
< C i f f , f o r some 1 0 T h i s p a r t i a l l y ordered s e t t u r n s o u t t o be an i n f i n i t e
f i n i t e l y branching t r e e . Lemma), t a k e
s
By a m o n o t o n i c i t y p r o p e r t y i t f o l l o w s t h a t , f o r every
i s a colouring
[rln
, and
n, c
P a r t i a l l y o r d e r such c o l o u r i n g s by e x t e n s i o n :
r, C0 = C1
Consider t h e
elements o f
g i v e s one a c o n t r a d i c t i o n .
X,r = max(Y)
C :
+
1
Iwln , and
c
+
.
C' = C
Letting
Y
consist
[rln r e a d i l y
The n o n - e f f e c t i v e p r o o f o f t h e P a r i s - H a r r i n g t o n
Theorem i s e n t i r e l y analogous. The o r i g i n a l p r o o f o f t h e independence o f t h e P a r i s - H a r r i n g t o n Theorem over formal number t h e o r y i s very a p p e a l i n g t o t h e l o g i c i a n .
Combinatorial construc-
t i o n s o f nonstandard models o f a r i t h m e t i c i n s i d e g i v e n such models can be used t o
i
prove
. t h e independence o f t h e Theorem,
ii. t h e equivalence o f t h e Theorem
w i t h a s t r o n g expression o f f a i t h i n t h e system (i.e.
a stronger-than-usual
a s s e r t i o n o f c o n s i s t e n c y ) , and iii. t h e eventual m a j o r i s a t i o n o f a l l p r o v a b l y computable f u n c t i o n s by
H(x
+
1, x, x)
.
Combined w i t h a f a m i l i a r p r o o f t h e o r e -
t i c a n a l y s i s o f formal number t h e o r y , t h i s g i v e s i n f o r m a t i o n about t h e growth o f H(x + 1, x, x)
F -Hierarchy. a As I say, t h i s p r o o f appeals t o t h e l o g i c i a n .
others.
i n terms o f t h e
I t might n o t appeal t o
I t a l s o m i g h t n o t do much f o r t h e u n d e r s t a n d i n g by e i t h e r o f what
e x a c t l y makes
H(x + 1, x, x)
grow so r a p i d l y .
E n t e r Robert M. Solovay.
Having
375
Some Rapidly Growing Functions heard about t h e P a r i s - H a r r i n g t o n Theorem and i t s independence, b u t not having seen t h e p r o o f and, consequently,
unaware t h a t one c o u l d read o f f i n f o r m a t i o n on
t h i s growth from t h e i r p r o o f , he s e t o u t t o e s t a b l i s h t h i s growth d i r e c t l y .
He
succeeded w i t h t h e l o w e r bounds; b u t had t o r e s o r t t o t h e p r o o f t h e o r y t o o b t a i n t h e upper bounds.
L a t e r , J u s s i Ketonen gave d i r e c t p r o o f s o f t h e upper bounds
and t h e two o f them went on t o g i v e r a t h e r sharp e s t i m a t e s o f t h e growth o f t h e h
H(x + 1, x,
function
7)
example, t h e y showed t h a t , f o r F (X €0
H(x + 1, x, 7 ) )
and some o f i t s v a r i a n t s (e.g.
-
x
.
For
1. 20 ,
3) < H ( x
+ 1,
X,
X)
-
< F (X €0
1)
.
The Ketonen-Solovay elementary p r o o f , l i k e elementary p r o o f s o f theorems o f a n a l y t i c number t h e o r y , i s somewhat l o n g e r t h a n t h e P a r i s - H a r r i n g t o n p r o o f and I However, I can g i v e a
c e r t a i n l y cannot present i t i n t h e space a l l o t t e d here.
+
H(x
b i t o f t h e f l a v o u r o f t h e i r p r o o f by showing t h a t
1, x, x)
eventually
.
majorises a l l functions
F for finite n n As i s always t h e case i n such m a t t e r , I must f i r s t pause t o g i v e a
d e f i n i t i o n and comment on n o t a t i o n . F i r s t t h e d e f i n i t i o n : L e t us say t h a t a 2 c o l o u r i n g C : ( w I + c c a p t u r e s a f u n c t i o n F i f every n o n - t r i v i a l r e l a t i v e l y large set
Y homogeneous w i t h r e s p e c t t o C s a t i s f i e s
if
Y and x < y , t h e n Fx
x, y E
ii j u s t a s s e r t s t h a t
Y
As f o r n o t a t i o n ,
I shall write
k 1. yo
C(x,y)
C({x,y})
a. F o r each F
n
n
.
Condition
grows a t l e a s t as r a p i d l y as
t i v e l y l a r g e , then instead of
'y
, there
.
Y = y
Further, i f
< y1 <...< x < y
,
. min
i
(Y)
1. 3
and ii.
i i s technical; condition
.
F
.
If Y i s relak-1 i t i s customary t o w r i t e y
.
i s a colouring
C
: [u]* + 4
3n
t h a t captures
.
n
.
Proof:
By i n d u c t i o n on
Basis.
A moment's thought w i l l r e v e a l t h a t c o n d i t i o n ii o f t h e d e f i n i t i o n o f
c a p t u r i n g i s always s a t i s f i e d .
To c a p t u r e c o n d i t i o n
i
,
simply define
c. SMORYNSKI
376
( O , x = O
13, X L 3 .
If C
0
Y = y
0
, then
< y
1
<...< ’k-1
C (y yl) 0 0’
I n d u c t i o n step.
Let C
y )
= Co(yl,
Suppose
entails
2
(uI2 +
:
‘n
4
-
3n
captures
.
Fn
yo
Define
1. 3
C
n+l
. by
<...<
n+l
homogeneous f o r
C
n
-
whence i. C
captures
n+l
i g n o r e t h e f i r s t c o o r d i n a t e and j u s t w r i t e Since
But
C (y , y ) = 3, i.e. 0 0 1
be r e l a t i v e l y l a r g e and homogeneous w i t h r e s p e c t t o < y1 yk-l 0 P r o j e c t i n g on t h e f i r s t c o o r d i n a t e o f C (x,y) , we n o t e t h a t Y i s
Y = y
. n+l
i s r e l a t i v e l y l a r g e and homogeneous w i t h respect t o
C
n+l
captures
F
n
k = card(Y) L m i n ( Y ) = y
0
, we
C
n+l
F
n
(x,y)
, and
ii. we can n o t a t i o n a l l y
= 0,l
, or
2
.
have
and
.
F
n
i s monotone, so
( y ,y ) # 0 [Perhaps I should p o i n t o u t t h a t t h i s i s n+l 0 k-1 t h e p a r t of t h e proof i n which r e l a t i v e largeness i s used.] and we see t h a t
Since
C
n+l
C
(yi,y.)
# J
0
.
Some Rapidly Growing Functions
311
But
and
F
i s monotone, whence
[Here we use c o n d i t i o n
i : yo
e q u a l i t y , we conclude
C(y.,y.) 1
i.e.
captures
Cntl
x
For a g i v e n
24
jn
< y
Y = y
n+l
H(x + 1, 2 , x)
Theorem.
Proof:
F
captures
n
= 2
'y
0
1 .] From t h i s i n -
t
thus
J
.
(I.E.D.
e v e n t u a l l y m a j o r i s e s each
, let
2 Cn : [ w ]
+
4
-
3n
F
n
.
capture
Fn
.
.
Assume a l s o t h a t
as mapping i n t o x) Now suppose n < H(x t 1, 2, x) i s r e l a t i v e l y l a r g e , homogeneous w i t h
<...< 'k-1 , and
- 2 , and
2y0
implies
= C(yo,y2)
( s o t h a t we can view
0 1 respect t o C Since
n
1. 3
C
-
x t 1
has c a r d i n a l i t y a t l e a s t
.
.
.
> 3 , we have y > 3 + i I n particular, y > x But C 0ix-1 n ) > F (x) C o l l e c t i n g o u r i n e q u a l i t i e s , we F , whence y x Fn(yx-l n n y
.
have
for all
x
1. 4
3
n
.
We remark t h a t t h i s p r o o f shows Since
H(x + 1, 2, 4
3')
H(x + 1, 2, 4
i s something l i k e
-
3')
t o majorise
H(x + 1, 2, x)
, we
F
w
.
can conclude
c. SMORYNSKI
378
H(x t 1, 2, x)
t o l i e at least at level H(x + 1, 3 , x)
i s exactly i t s level.
of the
w
F -Hierarchy.
I n fact, t h i s
a
i s something l i k e
F
: H(x t 1, 4 , x )
w
i s something l i k e
F
; etc. w
w
Proponents o f r a p i d growth do n o t wish t o s t o p here.
They seek ever m r e r a p i d l y
growing f u n c t i o n s and ever more powerful p r i n c i p l e s t o produce such f u n c t i o n s . D e f i n i n g , f o r example, a f i n i t e s e t card(X)
F(min(X))
, one
t o be r e l a t i v e l y
X
F-large i f
can i t e r a t e t h e P a r i s - H a r r i n g t o n c o n s t r u c t i o n and
generate more r a p i d l y growing f u n c t i o n s .
T h i s i s , however, a p e d e s t r i a n
A seemingly more e x c i t i n g approach i s t o l o o k f o r a new statement o f
approach. t h e form,
Vx ByA(x.y)
,
which i s independent over a c o n s i d e r a b l y s t r o n g e r t h e o r y t h a n formal number t h e o r y and hope t h a t t h e c o r r e s p o n d i n g c h o i c e f u n c t i o n
F
,
This
e v e n t u a l l y m a j o r i s e s a l l p r o v a b l y computable f u n c t i o n s o f t h e new t h e o r y . has been done, b u t n o t w i t h v e r y i n t e r e s t i n g r e l a t i o n s
.
A(x,y)
I n all, the
problem o f g e t t i n g a f u n c t i o n which i s a t once more r a p i d l y growing t h a n
FE 0
H(x + 1, x, x )
and o f as g r e a t an i n t e r e s t as t h a t o f
seems t o be open.
A c t u a l l y , what I should have s a i d was t h a t t h e problem o f g e t t i n g i n t e r e s t i n g computable f u n c t i o n s o f more r a p i d growth t h a n t h a t o f
FE
I f we
i s open.
0 drop t h e requirement o f c o m p u t a b i l i t y Function.
-
w e l l , t h e r e i s T i b o r Rado's Busy Beaver
T h i s i r r e p r e s s i b l e l i t t l e f e l l o w grows more r a p i d l y t h a n any t h e o r e -
t i c a l l y computable f u n c t i o n
-
a f e a t o f n o t p a r t i c u l a r l y g r e a t magnitude:
It i s
an easy m a t t e r t o e v e n t u a l l y m a j o r i s e a l l computable f u n c t i o n s by d i a g o n a l i s a t i o n ; and t h e Busy Beaver i s t h e r e s u l t o f such a d i a g o n a l i s a t i o n .
But i t i s a
d e l i g h t f u l d i a g o n a l i s a t i o n and I must comment on i t here. A T u r i n g machine (named a f t e r Alan Turing, an e n i g m a t i c c r y p t a n a l y s t ) i s an imaginary d e v i c e f o r computing f u n c t i o n s . number
x
Each T u r i n g machine
M
has a f i n i t e
o f s t a t e s and a two-way i n f i n i t e t a p e d i v i d e d i n t o c o n s e c u t i v e con-
g r u e n t squares, each o f which can e i t h e r bear t h e symbol 1 o r be blank.
M
has
Some Rapidly Growing Functions
379
t h e g r e a t f l e x i b i l i t y o f , on r e a d i n g what i s on t h e p a r t i c u l a r square i t i s scann i n g and on c o n s i d e r i n g t h e s t a t e i t i s i n , b e i n g a b l e t o erase t h e
1 i f it i s
1 i f t h e r e i s none, o r l e t i t stand; t o move t h e t a p e one square
there, p r i n t a
t o t h e l e f t o r t o t h e r i g h t , o r t o c o n t i n u e s t a r i n g a t t h e same one; and t o change s t a t e s or remain i n t h e same s t a t e .
-
n o t a l l o w e d any f r e e w i l l
Being a machine
M
i s , o f course,
i t s a c t i o n s a r e c o m p l e t e l y determined by i t s b a s i c
program. T u r i n g machines do n o t sound p a r t i c u l a r l y u s e f u l ; b u t t h e y a r e powerful enough t o compute a l l t h e o r e t i c a l l y computable f u n c t i o n s . 1 ' s and blanks, t h i s r e q u i r e s a coding. x
by a s t r i n g o f
x
+
1 1's.)
(Of course, w i t h o n l y
One u s u a l l y r e p r e s e n t s a n a t u r a l number
Moreover, t h e y have a l l t h e i n h e r e n t d i f f i c u l -
t i e s a s s o c i a t e d w i t h more s o p h i s t i c a t e d machines
-
most n o t a b l y t h e d i f f i c u l t y i n
d e t e r m i n i n g from t h e machine's program whether i t s computations w i l l ever f i n i s h . T h i s problem
has no e f f e c t i v e s o l u t i o n .
The Busy Beaver poses h i m s e l f t h e f o l l o w i n g problem: machine w i t h number o f
x
1's
I f one feeds
states. that
M
Suppose
M
is a
a blank tape, what i s t h e maximum
M
can p r i n t on t h e t a p e b e f o r e h a l t i n g ?
BB(x) = m a x ( { # ( l ' s
M
can p r i n t ) :
M
has e x a c t l y
x
This function,
,
states})
i s not computable and, i n f a c t , i t e v e n t u a l l y m a j o r i s e s every computable function.
[The obvious computation procedure i s n o t e f f e c t i v e :
t o s i m p l y enumerate t h o s e machines w i t h
x
w a i t u n t i l t h e y f i n i s h t h e i r computations, prints.
One would l i k e
s t a t e s , f e e d them a l l b l a n k tapes, 1's
and t h e n see how many
The problem i s t h a t some o f t h e s e machines m i g h t never f i n i s h
t h e r e i s no way o f knowing which ones t h e s e are.
-
and
But r a t h e r t h a n e x p l a i n t h e
The p r o o f i s q u i t e simple:
r e s u l t , l e t me prove it.]
each one
Let
F be any monotone
computable f u n c t i o n and l e t F ' ( x ) = F(2x + 3) Suppose
M
machine
M
s t a t e s . F o r each x , we can f i n d a 0 x + 1 s t a t e s t h a t w i l l p r i n t x + 1 1 ' s and h a l t when f e d
computes with
.
F'
and has
x
X
a b l a n k tape.
The hook-up o f
M
with
M
r e s u l t s i n a machine w i t h
X
s t a t e s which, when f e d a b l a n k t a p e w i l l o u t p u t
F'(x) + 1 1's
.
If
x + x x
1. x
0
+ 2
0 '
c. SMORYNSKI
380
F(2x + 2)
5 F(2x
+ 3) = F ' ( x )
< BB(2x + 2)
5 BB(2x +
3)
,
which was t o be proven. A b i t more on T u r i n g machines and t h e Busy Beaver can be found i n E n d e r t o n ' s c o n t r i b u t i o n t o t h e Handbook o f Mathematical L o S .
We n o t e merely t h a t t h e few
known values o f t h e f u n c t i o n , BB(0) = 0, BB(1) = 1, BB(2) = 4 BB(3) = 6, BB(4) = 13 do n o t e x a c t l y r e f l e c t l a t e r developments
-
,
,
even Graham's f u n c t i o n s t a r t s out
b e t t e r than t h i s . REFERENCES [l]Herb Enderton, Elements o f r e c u r s i o n t h e o r y , i n : J. Barwise, ed. Handbook o f Mathematical L o g i c (North-Holland, Amsterdam 1977). [2]
Paul Erdos and George M i l l s , Some bounds f o r t h e Ramsey-Paris-Harrington numbers, t o appear.
[31
J u s s i Ketonen and Robert M. Solovay, R a p i d l y growing Ramsey f u n c t i o n s , t o appear.
[4]
M a r t i n H. L i b and Stan Wainer, H i e r a r c h i e s o f number-theoretic f u n c t i o n s , I , 11, A r c h i v f. math. L o g i k 13 (1970), pp. 39-51, 97-113.
[5l
J e f f P a r i s and Leo H a r r i n g t o n , A mathematical incompleteness i n Peano a r i t h m e t i c , i n : J. Barwise, ed., Handbook o f Mathematical Logic ( N o r t h H o l l and, Amsterdam 1977)
.
[6]
Frank P. Ramsey, On a problem o f formal l o g i c , Proc. London Math. SOC. ( 2 ) 30 (1929), pp. 264-286.
[7]
T i b o r Rado, On non-computable f u n c t i o n s , B e l l System Tech. J. 41 (1962), pp. 877-884.
367
Some R a p i d l y Growing F u n c t i o n s
I), 2)
C r a i g Smorynski
The purpose o f t h i s n o t e i s pure iconoclasm. m a t i c a l myths about how l a r g e " l a r g e " i s .
I wish t o debunk a few mathe-
When t h e mathematician says " l a r g e " ,
t h e l o g i c i a n i s sure t o t h i n k " s m a l l " . The f i r s t clich; Skewes number. T(X)
-
li(x)
u s u a l l y r e s o r t e d t o i n d i s c u s s i o n s o f largeness i s t h e
For a l l c a l c u l a t e d values, t h e number t h e o r e t i c f u n c t i o n Thus, when i n 1914 J. E. L i t t l e w o o d n o n c o n s t r u c t i v e -
i s negative.
l y proved t h a t t h i s f u n c t i o n changed s i g n s i n f i n i t e l y o f t e n , c u r i o s i t i e s about
where i t c o u l d become p o s i t i v e were aroused.
I n 1933, on t h e assumption o f t h e
Riemann Hypothesis, S . Skewes gave an upper bound f o r t h e f i r s t change o f sign. T h i s bound was so l a r g e (by t h e standards o f t h e day) t h a t i s achieved i n s t a n t n o t o r i e t y and even a t i t l e
-
t h e Skewes Number: 4.369
s
= 10lo < e
e
ee
The Skewes number has s i n c e been t o p p l e d from i t s p o s i t i o n o f supremacy.
In
1955, Skewes showed how t o l o w e r t h e bound i f one s t i l l assumed t h e Riemann Hypothesis; b u t he saved h i s r e p u t a t i o n by o b t a i n i n g t h e even l a r g e r upper bound,
e S ' = ee
e
7.705
,
"Reprinted from t h e Mathematical I n t e l l i g e n c e r 2 (1980) 149-154, o f S p r i nger-Verlag
.
by permission
"Editors' note: F u r t h e r d i s c u s s i o n o f t h i s t o p i c may be found i n t h e o t h e r s h o r t a r t i c l e s o f C. Smorynski i n t h i s volume, as w e l l as i n t h e c o n t r i b u t i o n s o f S. G. Simpson and R. L. Smith.
c. SMORYNSKI
368
when t h e Riemann Hypothesis was n o t assumed. shrunk,
i t s importance has:
While t h e number
has not
S'
S m a l l e r bounds f o r t h e s i g n change e x i s t today.
For
example, i n 1966 R. S. Lehman gave t h e bound
e L = e
e
2.067
L l a r g e these days:
Not even mathematicians would f i n d
Alan Baker won h i s
F i e l d s medal f o r , among o t h e r t h i n g s , g i v i n g e f f e c t i v e bounds i n number t h e o r y . I n h i s book on Transcendental Numbers, f o r i n s t a n c e , he c i t e s t h e bound 10
f o r a l l i n t e g r a l zeros
x,y
c o e f f i c i e n t s ) o f genus
1
o f an i r r e d u c i b l e polynomial
, degree
n
, and
height
f(x,y)
.
H
O f course, w i t h so much emphasis on t h e e f f e c t i v e n e s s o f t h a t mathematicians do n o t r e g a r d
B as b e i n g very l a r g e .
modern mathematician r e g a r d as l a r g e ?
(with integral
B
, we
must assume
What t h e n does t h e
Well, i n h i s column i n t h e November 1977
i s s u e o f S c i e n t i f i c American, M a r t i n Gardner c i t e s a r e s u l t o f R.
L. Graham.
According t o Gardner, " I n an u n p u b l i s h e d p r o o f , Graham has r e c e n t l y e s t a b l i s h e d an upper bound
... so v a s t t h a t
i t h o l d s t h e r e c o r d f o r t h e l a r g e s t number ever
used i n a s e r i o u s mathematical proof." i s , we f i r s t d e f i n e a f u n c t i o n
K
Intriguing?
To see what Graham's number
by r e c u r s i o n :
K(x,y)
i s "something l i k e " an e x p o n e n t i a l s t a c k o f
From K
, we
d e f i n e another f u n c t i o n
G
y's
by r e c u r s i o n :
o f height
x + 2 :
'
369
Some Rapidly Growing Functions = K(3,3)
G(0)
G(x + 1) = K(G(x),3) The growth o f stack o f
G
.
i s a b i t more d i f f i c u l t t o imagine.
i s something l i k e a
G(1)
G(0) 3 ' s , i.e.
I l e a v e t o t h e reader an e s t i m a t e o f
.
G(2)
G = G(64)
The bound t h a t Graham g i v e s i s
. I will
Now t h i s i s something t h e mathematician o f today regards as l a r g e . concede t h a t i t dwarfs numbers l i k e
f o r reasonably small values o f t h a t it i s
large
-
n
S , B, o r even say,
and
as
G
5
.
100)
I w i l l a l s o concede
a c c o r d i n g t o Gardner, t h e c o n s t a n t t h a t
bound f o r i s g e n e r a l l y b e l i e v e d t o be i s large.
(say
H
6 (!)
.
G
i s an upper
Rut I w i l l n o t concede t h a t
G
How can any number t h a t i s t h e value o f as s l o w l y growing a f u n c t i o n
on so small an argument as
64 be considered l a r g e ?
To g i v e us some standard f o r comparison, l e t me i n t r o d u c e a h i e r a r c h y o f number t h e o r e t i c f u n c t i o n s . F
n
: w + w
F o r each n a t u r a l number
0
Fn+,(x) Fn+l(x)
= Fn(
... ( F n ( x ) ) ...)
= x
a function
+ 1 x+l
= Fn
, with
F o ( x ) = x + 1, F1(x) = 2x + 1, F 2 ( x ) thing l i k e
, define
as f o l l o w s :
F (x)
i.e.
n E w
(XI ,
x + 1
nestings o f
i s something l i k e
2'
F
n
, and
.
Thus,
F (x) 3
some-
c. SMORYNSKI
370
F
4
, like
G(x)
,
i s a l i t t l e harder t o describe.
B e f o r e making any comparisons,
and H a r e something l i k e each o t h e r , 1 2 I do n o t mean t o i m p l y any r e l a t i o n n e a r l y as t i g h t as a s y m p t o t i c i t y . On t h e
t h a t I am g o i n g t o use:
When I say
l e t me q u i c k l y e x p l a i n t h e r u l e o f comparison
H
c o n t r a r y , I have i n mind a l o o s e r , more l i b e r a l , equivalence r e l a t i o n whose looseness grows w i t h t h e s i z e o f t h e elements o f a g i v e n e q u i v a l e n c e c l a s s . Given a f u n c t i o n taking
H
H , one n a t u r a l l y d e f i n e s a f a m i l y o f f u n c t i o n s
along w i t h a few b a s i c f u n c t i o n s (e.g.
5(H) by
a d d i t i o n ) and c l o s i n g under
e x p l i c i t d e f i n a b i l i t y (composition, a d d i t i o n o f dummy v a r i a b l e s , etc.).
I say
i s something l i k e H i f t h e c l a s s e s 3 ( H ) and 5 ( H ) a r e c o f i n a l 1 2 1 2 i f every f u n c t i o n o f 5 ( H ) i s m a j o r i s e d by one o f 5 ( H ) i n each o t h e r , i.e. 1 2
that
H
and v i c e versa.
T h i s i s a v e r y l o o s e measure o f equivalence:
a r e a l l something l i k e i t s own
2
X
, x! , and
n - f o l d composition w i t h i t s e l f ! )
x
X
T h i s looseness
F - H i e r a r c h y t h a t much more i m p r e s s i v e - f o r i t i s a h i e r a r c h y : n F i s n o t h i n g l i k e F , i.e. F i s so l a r g e r e l a t i v e t o F t h a t you n ntl n n+l need something l i k e F t o reach F from F F w i l l not do: F ntl ntl n n ntl e v e n t u a l l y m a j o r i s e s every f u n c t i o n o f Z ( F ) n The f u n c t i o n s K ( o r , r a t h e r , i t s d i a g o n a l ) and G f i t n e a t l y i n t o t h e
o f f i t makes t h e
.
.
.
F -Hierarchy. K(x,x) i s something l i k e F . w h i l e G i s something l i k e F n 3 ' 4 R e f l e c t i n g on t h e r a p i d i t y o f t h e growth of F F , o r even F 5 ' 236 G(64) ' the
reader w i l l see t h a t I was n o t b e i n g e n t i r e l y f a c e t i o u s when I r e f e r r e d e a r l i e r t o t h e slow r a t e o f growth e x h i b i t e d by Graham's f u n c t i o n
G(x)
.
" O f course",
t h e reader might o b j e c t , "anyone can produce ever more r a p i d l y growing f u n c t i o n s . But t h e Graham f u n c t i o n was used i n a " s e r i o u s mathematical proof." that
F
has never been so a p p l i e d .
G(64)
But
F
w
and
have
FE
It i s t r u e
-
and we
0
h a v e n ' t even reached t h e s e f u n c t i o n s y e t ! To d i s c u s s t h e r a t e s o f growth o f f u n c t i o n s used i n l o g i c , i t i s necessary
F -Hierarchy i n t o t h e t r a n s f i n i t e . n one s i m p l y i t e r a t e s what one d i d a t a :
t o extend t h e
A t successor o r d i n a l s , a
+
1
,
37 1
Some Rapidly Growing Functions
A t l i m i t ordinals,
one d i a g o n a l i s e s on what one has done b e f o r e , e.g.
F (x) w
.
= F (x) X
The f i r s t few values a r e now l o w e r t h a n those one had b e f o r e , but t h e f u n c t i o n s d e f i n e d a t l i m i t o r d i n a l s do c a t c h up and surpass t h e i r predecessors.
M. H. L'db
and S. Wainer have shown how t o c a r r y o u t t h i s c o n s t r u c t i o n t o any preassigned countable ordinal.
O f course, i n l o g i c one d o e s n ' t a c t u a l l y need t o go t h a t f a r
i n t o t h e t r a n s f i n i t e any more t h a n i n a n a l y s i s one needs t o use f u n c t i o n s growing
.
F The f u n c t i o n s we wish t o d i s c u s s a r e t h e F ' s f o r G(64) a a5€ The o r d i n a l t i s , t h e reader m i g h t r e c a l l , t h e l e a s t f i x e d p o i n t o f 0 0 E = min [8 = w 8 1 A more i n t u i t i v e o r d i n a l e x p o n e n t i a t i o n w i t h base w , i.e. 0 B i s g i v e n by v i e w i n g i t as t h e l i m i t o f t h e sequence, picture of E
as f a s t as
.
.
n
w , w
w
The immensity o f t h e s t e p from Moreover, between
and
w
w
w
ww
and
ww
a
w
w
)... .
to
there are not
w
between
, w
a
+ 1 i n c r e a s e s as a increases. 1
, not
w
, but
w
w
such steps:
w
there are
ww
human mind can comprehend t h e growth o f l o g i c a l (and, as we s h a l l see:
such steps; e t c .
FE
.
I don't think that the
Yet, i t i s f u n c t i o n o f d e f i n i t e
0 combinatorial) interest.
The f u n c t i o n s o f t h e
set,
a r e p r e c i s e l y t h o s e p r o v a b l y computable i n formal number t h e o r y .
(Believe it or
n o t , t h e s e f u n c t i o n s a r e ( t h e o r e t i c a l l y , i f not p r a c t i c a l l y ) computable.)
Thus,
i s , i n a sense, t h e f i r s t f u n c t i o n t o e v e n t u a l l y m a j o r i s e a l l f u n c t i o n s 0 p r o v a b l y computable i n formal number t h e o r y .
Ft
L e t me i n t e r p r e t t h i s l a s t f a c t . o f formal number t h e o r y o f t h e f o r m
Suppose we have a sentence o f t h e language
c. SMORYNSKI
312
,
Y x ByA(x,y) where
A
i s some p r o v a b l y d e c i d a b l e r e l a t i o n .
some p r o v a b l y computable
I f ( 1 ) i s provable, then,
for
,
F
a < E
i s a l s o provable; whence, f o r some
i s provable.
(1)
[ I n f a c t , t h e exact
a
0 '
can be determined from t h e p r o o f o f ( 1 ) .
Using such c o n s i d e r a t i o n s , i n 1952 G. K r e i s e l reviewed L i t t l e w o o d ' s 1914 paper
I f , however, (1) i s t r u e
and n o t e d e s s e n t i a l l y t h e same upper bound as Skewes.]
b u t unprovable ( a p o s s i b i l i t y n o t t o be o v e r l o o k e d ) , t h e source o f i t s unprova b i l i t y c o u l d be t h e f a i l u r e o f ( 2 ) f o r a l l p r o v a b l y computable a < E
f a i l u r e o f (3) f o r a l l
.
0
F
, i.e.
I f t h i s i s t h e case, any f u n c t i o n
F
the making
( 2 ) t r u e has moments o f r a p i d growth. Now t h e s i t u a t i o n I have j u s t d e s c r i b e d i s not h y p o t h e t i c a l :
J e f f P a r i s and
Leo H a r r i n g t o n have r e c e n t l y e x h i b i t e d such an independent statement somewhat i n t e r e s t i n g r e l a t i o n
A(x,y)
.
-
with a
Since t h i s i s t h e f i r s t n o t p u r e l y
l o g i c a l example o f a problem w i t h such n e a r l y astronomical bounds ( I comment on a f u n c t i o n o f t r u l y astronomical growth a t t h e end o f t h i s note), I s h a l l d i s c u s s i t i n some d e t a i l .
I suppose we should f i r s t s e t t l e on some n o t a t i o n . n a t u r a l numbers, we w r i t e X
.
A colouring, C
, of
[XI n [XI
n
for the collection of a l l
c
Ramsqy proved t h a t , i f respect t o
Y
n-element subsets o f
: [XIn
+
c
,
i s some p o s i t i v e n a t u r a l number and we i d e n t Fy a n a t u r a l number with c = {O,
i t s s e t o f predecessors:
subsets:
i s a set of
i s s i m p l y a map
c where
If X
c C
1
and n [Y]
n
C
...,c - 11 .
i s a colouring o f
, then
some b i g subset
i s constant.
homogeneous w i t h r e s p e c t t o
I n 1929 t h e economist F. P. n [XI and X i s b i g enough w i t h Y
X
has monochromatic
n-element
(To a v o i d clumsy phrasing, we c a l l such a s e t C .)
More s p e c i f i c a l l y , Ramsey proved t h e
373
Some Rapidly Growing Functions f o l l o w i n g two theorems.
Infinite C : [w]
i.e.
Ramsey Theorem.
n +
'
C
, there
c
[Yl
n
n, c
be p o s i t i v e i n t e g e r s .
i s an i n f i n i t e s e t
Y
F o r any c o l o u r i n g ,
homogeneous w i t h respect t o
w
C
,
i s constant.
F i n i t e Ramsey Theorem.
1. n
Let
Let
s
( f o r s i z e ) , n, c
be p o s i t i v e i n t e g e r s , w i t h
.
There i s a number R(s, n, c ) such t h a t , f o r a l l r 1. R(s, n, c ) n and a l l c o l o u r i n g s C : [ r ] + c , t h e r e i s a homogeneous Y r of cardinality s
+ 1
5 .
(The r e s t r i c t i o n
s
1. n +
1 s i m p l y r u l e s out t r i v i a l cases.)
N e i t h e r o f t h e s e statements i s p a r t i c u l a r l y i n t u i t i v e .
The b e s t way t o view
them i s as h i g h e r dimensional analogues o f D i r i c h l e t ' s Schubfachprinzip:
, the
n = 1
I n f i n i t e Ramsey Theorem j u s t a s s e r t s t h a t ,
If
i f an i n f i n i t e s e t i s
s p l i t i n t o a f i n i t e d i s j o i n t union o f subsets, one o f t h e subsets must be i n -
-
finite
ple:
A f i n i t e union o f f i n i t e sets i s f i n i t e .
i n i t s more usual f o r m u l a t i o n :
n = 1 and
For
s = 2
, the
F i n i t e Ramsey Theorem i s e x a c t l y D i r i c h l e t ' s P r i n c i -
R(2, 1, c ) = c + 1 and t e s t i t y o u r s e l f .
Take
The F i n i t e Ramsey Theorem i s a c e n t e r p i e c e o f f i n i t e c o m b i n a t o r i c s , w i t h much energy b e i n g expended on t h e c a l c u l a t i o n of
R(s, n, c )
and r e l a t e d "Ramsey
For, though such c a l c u l a t i o n i s c o n c e p t u a l l y t r i v i a l (One merely enu-
Numbers".
merates a l l p o s s i b i l i t i e s ...), i t i s i m p r a c t i c a l l y d i f f i c u l t . g i v e easy u p p w bounds: like
F
-
3
The diagonal
R ( x + 1, x, x )
Still,
one can
i s bounded by something
no l o n g e r a very l a r g e number by anyone's reckoning.
However, by
making a s u b t l e ( ? ) change i n t h e statement of t h e theorem, P a r i s and H a r r i n g t o n o b t a i n a v a r i a n t where t h e f u n c t i o n i n q u e s t i o n e x h i b i t s a more r e s p e c t a b l e r a t e o f growth
-
t h e f u n c t i o n i s something l i k e
FE
.
Q We need o n l y one more d e f i n i t i o n t o s t a t e t h e P a r i s - H a r r i n g t o n Theorem:
A
set
mum:
X
w
card(X)
i s r e l a t i v e l y l a r g e i f i t s c a r d i n a l i t y i s not l e s s t h a n i t s m i n i -
1. min(X)
.
P a r i s - H a r r i n g t o n Theorem. There i s a number
Let
H(s, n, c )
s, n, c
be p o s i t i v e i n t e g e r s w i t h
such t h a t , f o r a l l
h
1. H(s,
n, c )
s
1. n +
and a l l
1
.
c. SMORYNSKI
374 n
colourings
, there
C : [hl" + c
Y
8
h
of
.
s
c a r d i n a l i t y at l e a s t
i s a r e l a t i v e l y l a r g e homogeneous
The P a r i s - H a r r i n g t o n Theorem i s t r u e , b u t not p r o v a b l e i n formal number theory.
The f u n c t i o n
.
FG 0
like
H(x + 1, x, x )
( i n f a c t , H(x + 1, x, 3 ) )
i s something
Yet, a t l e a s t f o r t h e novice, t h e d i f f e r e n c e between t h e F i n i t e
Rainsey Theorem and t h e P a r i s - H a r r i n g t o n Theorem i s minimal.
This minimality i s
u n d e r l i n e d by t h e f a c t t h a t t h e s e theorems share a common n o n - e f f e c t i v e p r o o f . It i s i n t h e i r e f f e c t i v e p r o o f s , o f course, t h a t t h e y d i f f e r .
F o r t h e c u r i o u s reader, I o u t l i n e t h e n o n - e f f e c t i v e p r o o f . case o f t h e F i n i t e Ramsey Theorem. t h e theorem were f a l s e . r
1. s , t h e r e
.
Y E r
Suppose f o r f i x e d
.
f i n i t e set
C = U C i i
C : [rln
+
c
w i t h no s i z e
o f the f i r s t
s
s
homogeneous s e t
C
<...
C < C (by K o n i g ' s 0 1 and i n v o k e t h e I n f i n i t e Ramsey Theorem t o o b t a i n an i n -
Choose an i n f i n i t e p a t h
,
homogeneous w i t h r e s p e c t t o
X
+ 1 ,
2n
< C i f f , f o r some 1 0 T h i s p a r t i a l l y ordered s e t t u r n s o u t t o be an i n f i n i t e
f i n i t e l y branching t r e e . Lemma), t a k e
s
By a m o n o t o n i c i t y p r o p e r t y i t f o l l o w s t h a t , f o r every
i s a colouring
[rln
, and
n, c
P a r t i a l l y o r d e r such c o l o u r i n g s by e x t e n s i o n :
r, C0 = C1
Consider t h e
elements o f
g i v e s one a c o n t r a d i c t i o n .
X,r = max(Y)
C :
+
1
Iwln , and
c
+
.
C' = C
Letting
Y
consist
[rln r e a d i l y
The n o n - e f f e c t i v e p r o o f o f t h e P a r i s - H a r r i n g t o n
Theorem i s e n t i r e l y analogous. The o r i g i n a l p r o o f o f t h e independence o f t h e P a r i s - H a r r i n g t o n Theorem over formal number t h e o r y i s very a p p e a l i n g t o t h e l o g i c i a n .
Combinatorial construc-
t i o n s o f nonstandard models o f a r i t h m e t i c i n s i d e g i v e n such models can be used t o
i
prove
. t h e independence o f t h e Theorem,
ii. t h e equivalence o f t h e Theorem
w i t h a s t r o n g expression o f f a i t h i n t h e system (i.e.
a stronger-than-usual
a s s e r t i o n o f c o n s i s t e n c y ) , and iii. t h e eventual m a j o r i s a t i o n o f a l l p r o v a b l y computable f u n c t i o n s by
H(x
+
1, x, x)
.
Combined w i t h a f a m i l i a r p r o o f t h e o r e -
t i c a n a l y s i s o f formal number t h e o r y , t h i s g i v e s i n f o r m a t i o n about t h e growth o f H(x + 1, x, x)
F -Hierarchy. a As I say, t h i s p r o o f appeals t o t h e l o g i c i a n .
others.
i n terms o f t h e
I t might n o t appeal t o
I t a l s o m i g h t n o t do much f o r t h e u n d e r s t a n d i n g by e i t h e r o f what
e x a c t l y makes
H(x + 1, x, x)
grow so r a p i d l y .
E n t e r Robert M. Solovay.
Having
375
Some Rapidly Growing Functions heard about t h e P a r i s - H a r r i n g t o n Theorem and i t s independence, b u t not having seen t h e p r o o f and, consequently,
unaware t h a t one c o u l d read o f f i n f o r m a t i o n on
t h i s growth from t h e i r p r o o f , he s e t o u t t o e s t a b l i s h t h i s growth d i r e c t l y .
He
succeeded w i t h t h e l o w e r bounds; b u t had t o r e s o r t t o t h e p r o o f t h e o r y t o o b t a i n t h e upper bounds.
L a t e r , J u s s i Ketonen gave d i r e c t p r o o f s o f t h e upper bounds
and t h e two o f them went on t o g i v e r a t h e r sharp e s t i m a t e s o f t h e growth o f t h e h
H(x + 1, x,
function
7)
example, t h e y showed t h a t , f o r F (X €0
H(x + 1, x, 7 ) )
and some o f i t s v a r i a n t s (e.g.
-
x
.
For
1. 20 ,
3) < H ( x
+ 1,
X,
X)
-
< F (X €0
1)
.
The Ketonen-Solovay elementary p r o o f , l i k e elementary p r o o f s o f theorems o f a n a l y t i c number t h e o r y , i s somewhat l o n g e r t h a n t h e P a r i s - H a r r i n g t o n p r o o f and I However, I can g i v e a
c e r t a i n l y cannot present i t i n t h e space a l l o t t e d here.
+
H(x
b i t o f t h e f l a v o u r o f t h e i r p r o o f by showing t h a t
1, x, x)
eventually
.
majorises a l l functions
F for finite n n As i s always t h e case i n such m a t t e r , I must f i r s t pause t o g i v e a
d e f i n i t i o n and comment on n o t a t i o n . F i r s t t h e d e f i n i t i o n : L e t us say t h a t a 2 c o l o u r i n g C : ( w I + c c a p t u r e s a f u n c t i o n F i f every n o n - t r i v i a l r e l a t i v e l y large set
Y homogeneous w i t h r e s p e c t t o C s a t i s f i e s
if
Y and x < y , t h e n Fx
x, y E
ii j u s t a s s e r t s t h a t
Y
As f o r n o t a t i o n ,
I shall write
k 1. yo
C(x,y)
C({x,y})
a. F o r each F
n
n
.
Condition
grows a t l e a s t as r a p i d l y as
t i v e l y l a r g e , then instead of
'y
, there
.
Y = y
Further, i f
< y1 <...< x < y
,
. min
i
(Y)
1. 3
and ii.
i i s technical; condition
.
F
.
If Y i s relak-1 i t i s customary t o w r i t e y
.
i s a colouring
C
: [u]* + 4
3n
t h a t captures
.
n
.
Proof:
By i n d u c t i o n on
Basis.
A moment's thought w i l l r e v e a l t h a t c o n d i t i o n ii o f t h e d e f i n i t i o n o f
c a p t u r i n g i s always s a t i s f i e d .
To c a p t u r e c o n d i t i o n
i
,
simply define
c. SMORYNSKI
376
( O , x = O
13, X L 3 .
If C
0
Y = y
0
, then
< y
1
<...< ’k-1
C (y yl) 0 0’
I n d u c t i o n step.
Let C
y )
= Co(yl,
Suppose
entails
2
(uI2 +
:
‘n
4
-
3n
captures
.
Fn
yo
Define
1. 3
C
n+l
. by
<...<
n+l
homogeneous f o r
C
n
-
whence i. C
captures
n+l
i g n o r e t h e f i r s t c o o r d i n a t e and j u s t w r i t e Since
But
C (y , y ) = 3, i.e. 0 0 1
be r e l a t i v e l y l a r g e and homogeneous w i t h r e s p e c t t o < y1 yk-l 0 P r o j e c t i n g on t h e f i r s t c o o r d i n a t e o f C (x,y) , we n o t e t h a t Y i s
Y = y
. n+l
i s r e l a t i v e l y l a r g e and homogeneous w i t h respect t o
C
n+l
captures
F
n
k = card(Y) L m i n ( Y ) = y
0
, we
C
n+l
F
n
(x,y)
, and
ii. we can n o t a t i o n a l l y
= 0,l
, or
2
.
have
and
.
F
n
i s monotone, so
( y ,y ) # 0 [Perhaps I should p o i n t o u t t h a t t h i s i s n+l 0 k-1 t h e p a r t of t h e proof i n which r e l a t i v e largeness i s used.] and we see t h a t
Since
C
n+l
C
(yi,y.)
# J
0
.
Some Rapidly Growing Functions
311
But
and
F
i s monotone, whence
[Here we use c o n d i t i o n
i : yo
e q u a l i t y , we conclude
C(y.,y.) 1
i.e.
captures
Cntl
x
For a g i v e n
24
jn
< y
Y = y
n+l
H(x + 1, 2 , x)
Theorem.
Proof:
F
captures
n
= 2
'y
0
1 .] From t h i s i n -
t
thus
J
.
(I.E.D.
e v e n t u a l l y m a j o r i s e s each
, let
2 Cn : [ w ]
+
4
-
3n
F
n
.
capture
Fn
.
.
Assume a l s o t h a t
as mapping i n t o x) Now suppose n < H(x t 1, 2, x) i s r e l a t i v e l y l a r g e , homogeneous w i t h
<...< 'k-1 , and
- 2 , and
2y0
implies
= C(yo,y2)
( s o t h a t we can view
0 1 respect t o C Since
n
1. 3
C
-
x t 1
has c a r d i n a l i t y a t l e a s t
.
.
.
> 3 , we have y > 3 + i I n particular, y > x But C 0ix-1 n ) > F (x) C o l l e c t i n g o u r i n e q u a l i t i e s , we F , whence y x Fn(yx-l n n y
.
have
for all
x
1. 4
3
n
.
We remark t h a t t h i s p r o o f shows Since
H(x + 1, 2, 4
3')
H(x + 1, 2, 4
i s something l i k e
-
3')
t o majorise
H(x + 1, 2, x)
, we
F
w
.
can conclude
c. SMORYNSKI
378
H(x t 1, 2, x)
t o l i e at least at level H(x + 1, 3 , x)
i s exactly i t s level.
of the
w
F -Hierarchy.
I n fact, t h i s
a
i s something l i k e
F
: H(x t 1, 4 , x )
w
i s something l i k e
F
; etc. w
w
Proponents o f r a p i d growth do n o t wish t o s t o p here.
They seek ever m r e r a p i d l y
growing f u n c t i o n s and ever more powerful p r i n c i p l e s t o produce such f u n c t i o n s . D e f i n i n g , f o r example, a f i n i t e s e t card(X)
F(min(X))
, one
t o be r e l a t i v e l y
X
F-large i f
can i t e r a t e t h e P a r i s - H a r r i n g t o n c o n s t r u c t i o n and
generate more r a p i d l y growing f u n c t i o n s .
T h i s i s , however, a p e d e s t r i a n
A seemingly more e x c i t i n g approach i s t o l o o k f o r a new statement o f
approach. t h e form,
Vx ByA(x.y)
,
which i s independent over a c o n s i d e r a b l y s t r o n g e r t h e o r y t h a n formal number t h e o r y and hope t h a t t h e c o r r e s p o n d i n g c h o i c e f u n c t i o n
F
,
This
e v e n t u a l l y m a j o r i s e s a l l p r o v a b l y computable f u n c t i o n s o f t h e new t h e o r y . has been done, b u t n o t w i t h v e r y i n t e r e s t i n g r e l a t i o n s
.
A(x,y)
I n all, the
problem o f g e t t i n g a f u n c t i o n which i s a t once more r a p i d l y growing t h a n
FE 0
H(x + 1, x, x )
and o f as g r e a t an i n t e r e s t as t h a t o f
seems t o be open.
A c t u a l l y , what I should have s a i d was t h a t t h e problem o f g e t t i n g i n t e r e s t i n g computable f u n c t i o n s o f more r a p i d growth t h a n t h a t o f
FE
I f we
i s open.
0 drop t h e requirement o f c o m p u t a b i l i t y Function.
-
w e l l , t h e r e i s T i b o r Rado's Busy Beaver
T h i s i r r e p r e s s i b l e l i t t l e f e l l o w grows more r a p i d l y t h a n any t h e o r e -
t i c a l l y computable f u n c t i o n
-
a f e a t o f n o t p a r t i c u l a r l y g r e a t magnitude:
It i s
an easy m a t t e r t o e v e n t u a l l y m a j o r i s e a l l computable f u n c t i o n s by d i a g o n a l i s a t i o n ; and t h e Busy Beaver i s t h e r e s u l t o f such a d i a g o n a l i s a t i o n .
But i t i s a
d e l i g h t f u l d i a g o n a l i s a t i o n and I must comment on i t here. A T u r i n g machine (named a f t e r Alan Turing, an e n i g m a t i c c r y p t a n a l y s t ) i s an imaginary d e v i c e f o r computing f u n c t i o n s . number
x
Each T u r i n g machine
M
has a f i n i t e
o f s t a t e s and a two-way i n f i n i t e t a p e d i v i d e d i n t o c o n s e c u t i v e con-
g r u e n t squares, each o f which can e i t h e r bear t h e symbol 1 o r be blank.
M
has
Some Rapidly Growing Functions
379
t h e g r e a t f l e x i b i l i t y o f , on r e a d i n g what i s on t h e p a r t i c u l a r square i t i s scann i n g and on c o n s i d e r i n g t h e s t a t e i t i s i n , b e i n g a b l e t o erase t h e
1 i f it i s
1 i f t h e r e i s none, o r l e t i t stand; t o move t h e t a p e one square
there, p r i n t a
t o t h e l e f t o r t o t h e r i g h t , o r t o c o n t i n u e s t a r i n g a t t h e same one; and t o change s t a t e s or remain i n t h e same s t a t e .
-
n o t a l l o w e d any f r e e w i l l
Being a machine
M
i s , o f course,
i t s a c t i o n s a r e c o m p l e t e l y determined by i t s b a s i c
program. T u r i n g machines do n o t sound p a r t i c u l a r l y u s e f u l ; b u t t h e y a r e powerful enough t o compute a l l t h e o r e t i c a l l y computable f u n c t i o n s . 1 ' s and blanks, t h i s r e q u i r e s a coding. x
by a s t r i n g o f
x
+
1 1's.)
(Of course, w i t h o n l y
One u s u a l l y r e p r e s e n t s a n a t u r a l number
Moreover, t h e y have a l l t h e i n h e r e n t d i f f i c u l -
t i e s a s s o c i a t e d w i t h more s o p h i s t i c a t e d machines
-
most n o t a b l y t h e d i f f i c u l t y i n
d e t e r m i n i n g from t h e machine's program whether i t s computations w i l l ever f i n i s h . T h i s problem
has no e f f e c t i v e s o l u t i o n .
The Busy Beaver poses h i m s e l f t h e f o l l o w i n g problem: machine w i t h number o f
x
1's
I f one feeds
states. that
M
Suppose
M
is a
a blank tape, what i s t h e maximum
M
can p r i n t on t h e t a p e b e f o r e h a l t i n g ?
BB(x) = m a x ( { # ( l ' s
M
can p r i n t ) :
M
has e x a c t l y
x
This function,
,
states})
i s not computable and, i n f a c t , i t e v e n t u a l l y m a j o r i s e s every computable function.
[The obvious computation procedure i s n o t e f f e c t i v e :
t o s i m p l y enumerate t h o s e machines w i t h
x
w a i t u n t i l t h e y f i n i s h t h e i r computations, prints.
One would l i k e
s t a t e s , f e e d them a l l b l a n k tapes, 1's
and t h e n see how many
The problem i s t h a t some o f t h e s e machines m i g h t never f i n i s h
t h e r e i s no way o f knowing which ones t h e s e are.
-
and
But r a t h e r t h a n e x p l a i n t h e
The p r o o f i s q u i t e simple:
r e s u l t , l e t me prove it.]
each one
Let
F be any monotone
computable f u n c t i o n and l e t F ' ( x ) = F(2x + 3) Suppose
M
machine
M
s t a t e s . F o r each x , we can f i n d a 0 x + 1 s t a t e s t h a t w i l l p r i n t x + 1 1 ' s and h a l t when f e d
computes with
.
F'
and has
x
X
a b l a n k tape.
The hook-up o f
M
with
M
r e s u l t s i n a machine w i t h
X
s t a t e s which, when f e d a b l a n k t a p e w i l l o u t p u t
F'(x) + 1 1's
.
If
x + x x
1. x
0
+ 2
0 '
c. SMORYNSKI
380
F(2x + 2)
5 F(2x
+ 3) = F ' ( x )
< BB(2x + 2)
5 BB(2x +
3)
,
which was t o be proven. A b i t more on T u r i n g machines and t h e Busy Beaver can be found i n E n d e r t o n ' s c o n t r i b u t i o n t o t h e Handbook o f Mathematical L o S .
We n o t e merely t h a t t h e few
known values o f t h e f u n c t i o n , BB(0) = 0, BB(1) = 1, BB(2) = 4 BB(3) = 6, BB(4) = 13 do n o t e x a c t l y r e f l e c t l a t e r developments
-
,
,
even Graham's f u n c t i o n s t a r t s out
b e t t e r than t h i s . REFERENCES [l]Herb Enderton, Elements o f r e c u r s i o n t h e o r y , i n : J. Barwise, ed. Handbook o f Mathematical L o g i c (North-Holland, Amsterdam 1977). [2]
Paul Erdos and George M i l l s , Some bounds f o r t h e Ramsey-Paris-Harrington numbers, t o appear.
[31
J u s s i Ketonen and Robert M. Solovay, R a p i d l y growing Ramsey f u n c t i o n s , t o appear.
[4]
M a r t i n H. L i b and Stan Wainer, H i e r a r c h i e s o f number-theoretic f u n c t i o n s , I , 11, A r c h i v f. math. L o g i k 13 (1970), pp. 39-51, 97-113.
[5l
J e f f P a r i s and Leo H a r r i n g t o n , A mathematical incompleteness i n Peano a r i t h m e t i c , i n : J. Barwise, ed., Handbook o f Mathematical Logic ( N o r t h H o l l and, Amsterdam 1977)
.
[6]
Frank P. Ramsey, On a problem o f formal l o g i c , Proc. London Math. SOC. ( 2 ) 30 (1929), pp. 264-286.
[7]
T i b o r Rado, On non-computable f u n c t i o n s , B e l l System Tech. J. 41 (1962), pp. 877-884.