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The Work of Harvey Friedman1)
HARVEY FRIEDMAN'S RESEARCH ON THE FOUNDATIONS OF MATHEMATICS, L.A. Harrington et al. (editors) Elsevier Science Publishers B.V. (North-Holland), 1985
1
The Work o f Harvey Friedman A n i l Nerode and Leo A.
1)
Harrington
Mathematical l o g i c i a n Harvey Friedman was r e c e n t l y awarded t h e N a t i o n a l Science F o u n d a t i o n ' s annual Waterman P r i z e , h o n o r i n g t h e most o u t s t a n d i n g American s c i e n t i s t under t h i r t y - f i v e y e a r s o f age i n a l l f i e l d s o f science and engineering.
When a mathematician wins such an award, t h e mathematical community
n a t u r a l l y wishes t o understand t h e u n d e r l y i n g achievements, and t h e i r i m p l i c a tions.
Friedman c o n t i n u e s t h e g r e a t t r a d i t i o n o f Frege, Russel, and G'ddel.
This
can be c h a r a c t e r i z e d as t h e e x e r c i s e o f a c u t e p h i l o s o p h i c a l p e r s p e c t i v e t o d i s t i l l exact mathematical d e f i n i t i o n s and q u e s t i o n s f r o m i m p o r t a n t f o u n d a t i o n a l issues.
These q u e s t i o n s i n t u r n g i v e r i s e t o mathematical s u b j e c t s and theorems
o f depth and beauty.
Friedman's c o n t r i b u t i o n s span a l l branches o f mathematical
l o g i c ( r e c u r s i o n t h e o r y , p r o o f t h e o r y , model t h e o r y , s e t t h e o r y , t h e o r y o f computation).
He i s a g e n e r a l i s t i n an age o f s p e c i a l i z a t i o n , y e t h i s theorems o f t e n
require extraordinary technical v i r t u o s i t y .
We d i s c u s s o n l y a few s e l e c t e d
high1 i g h t s . Friedman's ideas have y i e l d e d r a d i c a l l y new k i n d s o f independence r e s u l t s . The k i n d s o f statements proved independent b e f o r e Friedman were m o s t l y d i s g u i s e d p r o p e r t i e s o f formal systems (such as Godel's theorem on u n p r o v a b i l i t y of cons i s t e n c y ) o r a s s e r t i o n s about a b s t r a c t s e t s (such as t h e continuum h y p o t h e s i s o r S o u s l i n ' s hypothesis).
I n c o n s t r a s t , Friedman's independence r e s u l t s a r e about
q u e s t i o n s o f a more c o n c r e t e n a t u r e i n v o l v i n g , f o r example, Bore1 f u n c t i o n s o r t h e H i l b e r t cube.
R e p r i n t e d from t h e N o t i c e s o f t h e American Mathematical S o c i e t y , (1984) "The Work o f Harvey Friedman",
by A n i l Nerode and Leo H a r r i n g t o n , Volume 31, pp. 563-
566, by p e r m i s s i o n o f t h e American Mathematical Society.
2
A. NERODE. L.A. HARRINCTON I n [l] Friedman showed t h a t B o r e l determinacy cannot be proved i n Zermelo
s e t t h e o r y w i t h t h e axiom o f c h o i c e (ZC), o r any o f t h e usual formal systems a s s o c i a t e d w i t h a t most c o u n t a b l y many i t e r a t i o n s o f t h e power s e t o p e r a t i o n , a f t e r M a r t i n [17] proved B o r e l determinacy from c e r t a i n reasonable e x t e n s i o n s o f Zermelo Fraenkel s e t t h e o r y with t h e axiom of c h o i c e (ZFC).
Subsequently, M a r t i n
[18] gave a p r o o f o f B o r e l determinacy u s i n g uncountably many i t e r a t i o n s o f t h e power s e t o p e r a t i o n . These r e s u l t s reached t h e i r mature f o r m i n [9] THEOREM 1 (FRIEDMAN [9]).
as f o l l o w s :
I t i s necessary and s u f f i c i e n t t o use uncountably
many i t e r a t i o n s o f t h e power s e t o p e r a t i o n t o prove t h e f o l l o w i n g .
Every symme-
t r i c B o r e l subset o f t h e u n i t square c o n t a i n s o r i s d i s j o i n t from t h e graph o f a Borel function.
I n p a r t i c u l a r , t h i s a s s e r t i o n i s p r o v a b l e i n ZFC b u t n o t i n ZC.
I n Friedman [4,9],
t h e c l a s s i c a l theorem o f Cantor t h a t t h e u n i t i n t e r v a l
i s uncountable i s analyzed.
C a n t o r ' s argument produces an
t e r m i n any g i v e n i n f i n i t e sequence
-
Borel diagonalization function y
(here
F(y)
a
= 1'
F : Q
+
i s t h e H i l b e r t cube).
E I
y?,...
yl,
I
.
F
many c o o r d i n a t e s o f
and x
.
x
-
y
F(y)
i s a coordinate o f
Friedman observes t h a t t h e v a l u e o f
coming from C a n t o r ' s argument.
t h e same coordinates,
which i s n o t a
Specifically, there i s a
such t h a t no
depends on t h e o r d e r i n which t h e c o o r d i n a t e s o f
f o r the
x E I
I
Let
mean t h a t
x = y
y
a r e given, a t l e a s t
mean t h a t
x,y E
5
have
i s o b t a i n e d by permuting f i n i t e l y
y
THEOREM 2 (FRIEDMAN [4,9]). t h e invariance condition of
x
.
If
F :q
-t
x = y + F(x) = F(y)
I
i s a Borel function s a t i s f y i n g
, then
some
F(x)
i s a coordinate
Furthermore, t h i s i s p r o v a b l e i n ZC b u t not i n ZFC w i t h t h e power s e t
axiom d e l e t e d (even f o r t h e weaker theorem w i t h Subsequently, Friedman [12]
r e p l a c e d by
-)
.
gave c l o s e l y re1 a t e d one-dimensi onal theorems
w i t h t h e same meta-mathematical p r o p e r t i e s .
K
-
We present two o f these.
s : K + K by (2) D e f i n e t h e "square" x(') f o r x E K by x S(X,) = x Let T n+l * n = x 2 ' n [ O , l ) w i t h a d d i t i o n modulo 1). We say t h a t F : K + K be t h e c i r c l e group (i.e., Let
= {O,l}w
be t h e Cantor s e t .
Define t h e s h i f t
3
The Work of Harvey Friedman
i s shift invariant i f invariant i f
.
F(sx) = F(x)
F(2x) = F(x)
.
THEOREM 3 (FRIEDMAN [12]). somewhere i t s square.
We say t h a t
F : T + T
i s doubling
K
Every s h i f t i n v a r i a n t B o r e l f u n c t i o n on
There i s a B o r e l ( i n f a c t , c o n t i n u o u s ) f u n c t i o n on
which agrees somewhere w i t h every d o u b l i n g i n v a r i a n t B o r e l f u n c t i o n on
T
is T
.
Furthermore, t h e s e ( t h r e e ) theorems a r e p r o v a b l e i n ZC h u t n o t i n ZFC w i t h t h e power s e t axiom d e l e t e d . Friedman [9] develops e x t e n s i o n s o f Theorem 2, which proved u s i n g uncountRecently, Friedman has extended
a b l y many i t e r a t i o n s o f t h e power s e t o p e r a t i o n . t h i s i d e a as f o l l o w s : space o f a l l subsets o f on
Let
Q
Q
.
be t h e r a t i o n a l numbers and Let
G
P(Q)
be t h e Cantor
be t h e B a i r e space of a l l p r o d u c t s d e f i n e d
.
w
THEOREM 4 (FRIEDMAN [ l l ] ) .
If
F : G
+
i s a B o r e l f u n c t i o n such t h a t
G
isomorphic elements go t o i s o m o r p h i c values, t h e n some imbeddable i n
.
x
F(x)
i s isomorphically
Furthermore, t h i s theorem i s p r o v a b l e i n ZC b u t n o t i n ZFC
w i t h t h e power s e t axiom d e l e t e d . THEOREM 5 (FRIEDMAN [ l l ] ) . I f
F : P(Q)
+
P(Q)
i s a B o r e l f u n c t i o n such
t h a t o r d e r isomorphic arguments go t o o r d e r isomorphic values, then some i s isomorphic t o an i n t e r v a l i n points i n
A)
.
A
(even o f t h e f o r m
(a,b)
where
a,b
F(A) are
Furthermore, t h i s theorem r e q u i r e s uncountably many i t e r a t i o n s
o f t h e power s e t o p e r a t i o n t o prove.
I n p a r t i c u l a r , i t i s p r o v a b l e i n ZFC b u t
n o t i n ZC. Friedman [9] a l s o develops f a r - r e a c h i n g e x t e n s i o n s o f Theorem 2 by combining t h e s e i d e a s w i t h Ramsey's theorem.
T h i s l e a d s t o independence r e s u l t s from f u l l
ZFC, and i n f a c t t o theorems about B o r e l f u n c t i o n s on t h e H i l b e r t cube f o r which i t i s necessary and s u f f i c i e n t t o use " l a r g e c a r d i n a l s " t o prove.
The l a r g e c a r d i n a l s i n v o l v e d a r e as f o l l o w s :
A < K
.
K
implies We say t h a t
2
A
< K
unbounded subset o f
K
, and
K
i s not t h e sup o f fewer t h a n
i s a Mahlo c a r d i n a l i f K
A c a r d i n a l i s i n a c c e s s i b l e if
K
K
c a r d i n a l s below
i s i n a c c e s s i b l e and every c l o s e d
c o n t a i n s an i n a c c e s s i b l e c a r d i n a l .
A. NERODE. L.A. HARRINGTON
4
The
1-Mahlo c a r d i n a l s a r e t h e Mahlo c a r d i n a l s .
are the
The
(n+l)-Mahlo c a r d i n a l s
n-Mahlo c a r d i n a l s i n which e v e r y c l o s e d unbounded subset c o n t a i n s an
n-Mahlo c a r d i n a l . The group o f a l l p e r m u t a t i o n s o f
a
a c t s on 71
by
Q
that
g x,y
-
by permuting c o o r d i n a t e s . (xl
,...,xn)
which f i x a l l b u t f i n i t e l y many numbers
w
T h i s group a l s o a c t s d i a g o n a l l y on any
,...,gx,) .
= (gxl
11
Q
x,y E
For
let
-y
x
indicate
a r e i n t h e same o r b i t under t h i s a c t i o n .
THEOREM 6 (FRIEDMAN [S]).
x
that if
€6,y,z
F(x
x s' t
, and
E
an i n f i n i t e sequence
{x }
k
,..., x t
1
for a l l
n
,
Let y
F : Q x $ + I
-z
from
, then
be a B o r e l f u n c t i o n such
F(x,y)
= F(x,z)
.
such t h a t f o r a l l i n d i c e s
.
Then t h e r e i s
1 n I n order t o prove t h i s
) i s the f i r s t coordinate o f x s+l n i t i s necessary and s u f f i c i e n t t o use t h e e x i s t e n c e o f
cardinals f o r a l l
<...
s < t
,
n-Mahlo
.
n
These ideas have been extended i n Friedman [ll] t o much l a r g e r c a r d i n a l s . measurable c a r d i n a l i s a c a r d i n a l measure on a l l subsets o f every set
E
x
.
x
of
B
K-additive
A Ramsey c a r d i n a l i s a c a r d i n a l
o f f i n i t e subsets o f
f o r f i n i t e subsets of
K
which c a r r i e s a
K
K
, there
, membership
i s an unbounded
in
{O,l)-valued such t h a t f o r
K
B
L=
such t h a t
K
depends o n l y on t h e c a r d i n a l i t y
E
. Let
G O
be t h e B a i r e space o f f i n i t e l y generated p r o d u c t s on
THEOREM 7 (FRIEDMAN [ l l ] ) .
Let
F : G: + Go
i s i s o m o r p h i c a l l y imbeddable i n t o a c o o r d i n a t e o f
w
.
be a B o r e l f u n c t i o n mapping
p o i n t w i s e isomorphic arguments t o isomorphic values. x
Then f o r some
.
x C G:,
F(x)
Furthermore, t h i s theorem
i s p r o v a b l e i n ZFC b u t n o t i n ZC.
be a B o r e l f u n c t i o n mapping 0 p o i n t w i s e isomorphic arguments t o isomorphic values. Then f o r some x E Gw 0 ' f o r a l l subsequences y o f x, F ( y ) i s i s o m o r p h i c a l l y imbeddable i n t o a THEOREM 8 (FRIEDMAN [ll]). Let
coordinate o f
y
.
F : G:
+
G
T h i s i s p r o v a b l e i n ZFC + " t h e r e i s a measurable c a r d i n a l , "
b u t n o t i n ZFC + " t h e r e i s a Ramsey c a r d i n a l . "
A
5
The Work of Harvey Friedman I n P a r i s - H a r r i n g t o n [21] an i n t e r e s t i n g example o f a theorem s t a t e d i n f i n i t e s e t t h e o r y b u t n o t p r o v a b l e i n f i n i t e s e t t h e o r y , i s given.
A l s o see
C231, C241.
ClSl,
THEOREM 9 (PARIS-HARRINGTON [21]). l a r g e t h a t i f a l l subsets o f then t h e r e i s a s e t
[l,n]
[l,n]
E
E
same c o l o r and t h e s i z e o f
F o r each
o f size
k
, there
k,r,s
s
r
are colored with
such t h a t a l l subsets o f i s at least
i s an
E
of size
and t h e minimum o f
E
n
so
colors,
k
.
have t h e Further-
more, t h i s i s p r o v a b l e i n f i n i t e s e t t h e o r y augmented w i t h d e f i n i t i o n by r e c u r s i o n on
, but
w
not i n f i n i t e set theory.
Subsequently, Friedman [7] found an i n t e r e s t i n g example of a f i n i t e theorem which i s c o n c e p t u a l l y even c l e a r e r and independent o f much s t r o n g e r systems. J u s t as Theorem 9 i s based on Ramsey's theorem,
Friedman's work i s based on t h e
f o l l o w i n g theorem o f J. B. K r u s k a l 1171. A t r e e i s a nonempty p a r t i a l o r d e r i n g w i t h a l e a s t element, such t h a t t h e set o f predecessors o f any element i s l i n e a r l y order. trees then h(a) an
5
h :T h(b)
1
+ T
, and
2
T are f i n i t e 1' 2 i s s a i d t o be a homeomorphic imbedding i f a 5 b iff T-
h(inf(a,b))
.
= inf(h(a),h(b))
If
T
We w r i t e
T2 exists.
h
THEOREM 10 (KRUSKAL [19]), t r e e s t h e n f o r some
i < j, Ti
NASH-WILLIAMS [20]).
5
If
T,I
T
< T
1-
T2,...
2
1 i f such
are f i n i t e
.
Tj
The formal system ATR i s o b t a i n e d f r o m f i n i t e s e t t h e o r y by i n t r o d u c i n g c o u n t a b l y i n f i n i t e s e t s w i t h t h e p r i n c i p l e o f d e f i n i t i o n by t r a n s f i n i t e r e c u r s i o n on c o u n t a b l e w e l l o r d e r i n g s .
T h i s system goes j u s t beyond what i s r e f e r r e d t o as
predicative analysis. THEOREM 11 (FRIEDMAN 17,133;
SIMPSON [22]).
large that for a l l f i n i t e trees
T
L...'
T
i <...< 1
i
k
such t h a t
T. 1
,...,T n
1
i
.
and
For a l l card(T.)
T h i s theorem,
1
k
t h e r e i s an
5 i , there
n
are
as w e l l as K r u s k a l ' s
1 k Theorem 10 above, a r e p r o v a b l e i n ZFC w i t h o u t t h e power s e t axiom, b u t n o t i n ATR.
A l s o see [16,23,24].
so
A. NERODE, L.A. HARRINGTON
6
THEOREM 12 (FRIEDMAN [8,13]).
Theorem 11 f o r
k = 12
i s p r o v a b l e i n ZFC
w i t h o u t t h e power s e t axiom i n a few pages, b u t any p r o o f i n ATR must use a t [lOOOl least 2 pages. Friedman
[lo]
has extended K r u s k a l ' s theorem i n an i n t e r e s t i n g way so t h a t
t h e theorem has y e t s t r o n g e r metamathematical p r o p e r t i e s . T r (n)
o f f i n i t e trees with
< T i f and 1-r 2 which i s a l a b e l - p r e s e r v i n g homeomorphic i m -
d i s t i n c t l a b e l s , and d e f i n e s
n
m
o n l y i f t h e r e i s an
He c o n s i d e r s t h e c l a s s
+ T 1 2 bedding, p r e s e r v i n g l e f t t o r i g h t n e s s , w i t h t h e a d d i t i o n a l c r u c i a l c o n d i t i o n t h a t
if
then
b
h : T
T
i s an immediate successor o f l(c)
1. l ( h ( b ) ) ( l ( c )
a
in
T
h(a) < c < h(b)
and
1
.
i s the label o f c)
in
T
2 '
" r " means
The s u b s c r i p t
"restricted". THEOREM 13 (FRIEDMAN i < j
t h e r e are
that for all
T
[lo],
,...,Tm
F o r every
.
< T F o r a l l k,n 1-r 2 E T r (n) w i t h each c a r d ( T . )
such t h a t
1
SIMPSON [22]).
T
a
...
.
T1,
T2,
t h e r e i s an
5
, there
... E T r m
m
(n)
,
so l a r g e
i <...< i 1 k These theorems a r e p r o v a b l e i n t h e s i n g l e i
are
1
T. < T < < T 1 -r i -r -r i 1 2 k s e t q u a n t i f i e r comprehension axiom system w i t h t h e f u l l scheme o f i n d u c t i o n 1 1 (nl-CA) , b u t not i n U1-CA w i t h s e t i n d u c t i o n i n s t e a d o f t h e f u l l scheme o f 1 i n d u c t i o n (nl-CAO)
such t h a t
.
Two o f t h e most fundamental concepts i n mathematical l o g i c a r e t h e model t h e o r e t i c concept o f t r a n s l a t a b i l i t y and t h e p r o o f t h e o r e t i c concept o f r e l a t i v e consistency.
Friedman [6]
t h e s e concepts c o i n c i d e . order theories.
proves t h a t under s u r p r i s i n g l y general c o n d i t i o n s , Specifically, l e t
We say t h a t
S
S
, when
be f i n i t e l y a x i o m a t i z e d f i r s t -
i s translatable into
o r d e r d e f i n i t i o n s o f t h e symbols i n axiom o f
S,T
T
i f there are f i r s t -
i n terms o f t h o s e i n
S
so t r a n s l a t e d , becomes a theorem o f
T
There are a few somewhat d i f f e r e n t ways o f d e f i n i n g relative t o
T
," considered
i n [6].
. "S
T
, such
t h a t every
i s consistent
We focus a t t e n t i o n on t h e f o l l o w i n g one:
L e t €FA ( e x p o n e n t i a l f u n c t i o n a r i t h m e t i c ) be t h e s t a n d a r d weak system o f a r i t h m e t i c based on
0,l
,< , = ,t ,
a p p l i e d t o bounded formulas only.
, and
e x p o n e n t i a t i o n , where i n d u c t i o n i s
The q u a n t i f i e r c o m p l e x i t y o f a f o r m u l a i s a
s t a n d a r d measure o f t h e number o f a l t e r n a t i o n s o f q u a n t i f i e r s t h a t a r e present.
7
The Work of Harvey Friedman
We say t h a t
i f f o r some f i x e d
n
, the
I f t h e r e i s an i n c o n s i s t e n c y p r o o f i n
S
then
i s consistent r e l a t i v e t o
S
f o l l o w i n g i s p r o v a b l e i n EFA.
t h e r e i s an i n c o n s i s t e n c y p r o o f i n c o m p l e x i t y i s a t most
n
T
T
which uses o n l y formulas whose q u a n t i f i e r
more t h a n t h e g r e a t e s t q u a n t i f i e r c o m p l e x i t y o f
formulas used i n t h e g i v e n i n c o n s i s t e n c y p r o o f i n
S
.
I t i s s t r a i g h t f o r w a r d t o see t h a t t r a n s l a t a b i l i t y i m p l i e s r e l a t i v e
consistency
.
THEOREM 14 (FRIEDMAN [S]). c o n t a i n i n g EFA
i f applicable).
Then
i s consistent r e l a t i v e t o
S
S,T
be f i n i t e l y axiomatized t h e o r i e s
and a weak t h e o r y o f f i n i t e sequences o f o b j e c t s o t h e r t h a n
n a t u r a l numbers if
Let
T
S
i s translatable into
T
i f and o n l y
.
Friedman [5] has i n i t i a t e d an i n t e r e s t i n g new branch o f model t h e o r y c a l l e d Borel model t h e o r y .
A t o t a l l y B o r e l model i s a s t r u c t u r e whose domain i s IR and
every r e l a t i o n t h a t i s d e f i n a b l e ( i n t h e language considered) over t h e s t r u c t u r e Friedman [ 5 l c o n s i d e r s t h e t h r e e fundamental q u a n t i f i e r s
i s Borel.
a l l i n t h e sense o f Lebesgue measure), Q
Q
m
(almost
( a l m o s t a l l i n t h e sense o f B a i r e C
category), and quantifiers
( u n c o u n t a b l y many), i n a d d i t i o n , o f course, t o t h e usual
Q
w1 V, P.
Friedman proves t h e f o l l o w i n g completeness and d u a l i t y theorem. THEOREM 15 (FRIEDMAN 151, STEINHORN [25]).
Q
(or m only i f
Qm
0
and
YJ)
.
0
Then
Let
0
be a sentence based on
i s t r u e i n a l l t o t a l l y B o r e l models i f and
can be proved using, r o u g h l y speaking, t h e f o l l o w i n g p r i n c i p l e s : 0
,
union o f two s e t s o f measure
0
s i n g l e t o n s a r e o f measure
subsets o f measure i s o f measure
0
0,R
a r e o f measure
i s not o f measure
almost a l l v e r t i c a l cross s e c t i o n s o f a two-dimensional
If
Qc
0
0
, and 0
(Fubini's
, t h e n t h i s i s t r u e i f "measure 0" m As a consequence we have t h a t 0 i s t r u e i n a l l
i s used i n s t e a d o f
i s replaced by "meager."
, the
s e t a r e o f measure
and o n l y i f almost a l l h o r i z o n t a l cross s e c t i o n s a r e o f measure theorem).
0
t o t a l l y Borel structures f o r
Q
m
Q
( o r (Q ,",a)) m
i f and o n l y i f
$*
i s true i n
if
8
A. NERODE, L.A. HARRINGTON
a l l t o t a l l y Bore1 s t r u c t u r e s f o r
( o r (Q Y,X)) C 1 (duality).
0 by r e p l a c i n g Q
Q
where
m*
i s obtained from
by Q m C Friedman has o b t a i n e d a number o f fundamental r e s u l t s i n i n t u i t i o n i s t i c s e t
theory.
The usual axioms f o r ZF a r e e x t e n s i o n a l i t y , p a i r i n g , union, i n f i n i t y ,
f o u n d a t i o n , power s e t , comprehension, and f i n a l l y replacement.
The axiom scheme
o f c o l l e c t i o n i s an a l t e r n a t i v e t o t h e axiom scheme o f replacement, b u t i t i s a fundamental theorem o f s e t t h e o r y t h a t t h e s e a r e e q u i v a l e n t . However, i f we use i n t u i t i o n i s t i c l o g i c i n s t e a d o f o r d i n a r y l o g i c , t h e n t h e p r o o f t h a t replacement i m p l i e s c o l l e c t i o n breaks down.
Thus we l e t ZFIR be ZF
w i t h i n t u i t i o n i s t i c l o g i c f o r m u l a t e d w i t h t h e scheme o f replacement, and ZFIC be ZF w i t h i n t u i t i o n i s t i c l o g i c f o r m u l a t e d w i t h t h e scheme o f c o l l e c t i o n . THEOREM 16 (FRIEDMAN [2]) ; FRIEDMAN-SCEDROV [15]). ZFIC.
ZFIR does n o t i m p l y
It i s p r o v a b l e i n a weak system o f a r i t h m e t i c t h a t o r d i n a r y ZFC i s
c o n s i s t e n t i f and o n l y i f ZFIC i s c o n s i s t e n t . Two b a s i c d e s i r a b l e p r o p e r t i e s o f i n t u i t i o n i s t i c formal systems (which almost never h o l d f o r o r d i n a r y formal systems) a r e t h e d i s j u n c t i o n p r o p e r t y , which a s s e r t s t h a t i f a d i s j u n c t i o n
A V B
i s p r o v a b l e t h e n one o f t h e d i s j u n c t s
i s provable; and t h e numerical e x i s t e n c e p r o p e r t y , which a s s e r t s t h a t i f (Zn)(Vn)
i s p r o v a b l e t h e n f o r some
n,AG
i s provable.
DP t r i v i a l l y f o l l o w s
f r o m NEP. Friedman proves t h e f o l l o w i n g h i g h l y s u r p r i s i n g theorem v i a a m y s t e r i o u s appl ic a t i o n o f Godel s e l f - r e f erence.
THEOREM 17 (FRIEDMAN [31).
Let
T
be a r e c u r s i v e l y axiomatized i n t u i t i o n -
i s t i c formal system s u b j e c t t o , r o u g h l y , t h e same weak hypotheses commonly used i n Godel's incompleteness theorems. t y i f and o n l y i f
T
Then
T
has t h e numerical e x i s t e n c e p r o p e r -
has t h e d i s j u n c t i o n p r o p e r t y . SELECTED REFERENCES
[l]H. Friedman, Higher s e t t h e o r y and mathematical p r a c t i c e , Ann. Math. L o g i c 2 (1971), pages 326-357.
9
The Work of Harvey Friedman
, The c o n s i s t e n c y o f c l a s s i c a l s e t t h e o r y r e l a t i v e t o a s e t t h e o r y w i t h i n t u i t i o n i s t i c l o g i c , J. Symbolic L o g i c , 38 (1973), pages 315-319.
, The
c31
d i s j u n c t i o n property i m p l i e s t h e numerical existence pro-
p e r t y , Proc. Nat. Acad. Sci.,
72 (1975), pages
communicated b y K u r t G i d e l ,
2877-2878. C43 ______ , H i g h e r s e t t h e o r y and e x i s t e n t i a l s t a t e m e n t s , October 1976, preliminary report.
C5l
--, On
[Sl
-__
C71
, Independence r e s u l t s i n f i n i t e graph t h e o r y March 1981, p r e l i m i n a r y r e p o r t .
C8l
, The e x i s t e n t i a l i n c o m p l e t e n e s s phenomenon preliminary report.
t h e l o g i c o f measure and c a t e g o r y 1, and Addendum, December 1978; September 1979, p r e l i m i n a r y r e p o r t s .
, T r a n s l a t a b i l i t y and r e l a t i v e c o n s i s t e n c y 1-11, November 1976; September 1980, p r e l i m i n a r y r e p o r t s .
, On
c91 Math.,
I-VII,
February-
I, A p r i l 1981,
t h e necessary use o f a b s t r a c t s e t t h e o r y , Advances i n
41 (1981), pages 209-280.
, Beyond
ClOl
K r u s k a l ' s theorem 1-111, J u n e - J u l y 1982, p r e l i m i n a r y
report.
, New
Clll
necessary uses o f a b s t r a c t s e t t h e o r y f o r B o r e l October 1983; J u l y 1983, p r e l i m i n a r y r e p o r t .
functions I - I V ,
, Unary
c121 i n Math.,
C131
B o r e l f u n c t i o n s and second o r d e r a r i t h m e t i c , Advances
50 (1983), pages 155-159.
, An
improved f i n i t e f o r m o f K r u s k a l ' s theorem,
May 1984,
preliminary report.
C141
, The
metamathematics o f K r u s k a l ' s theorem ( i n p r e p a r a t i o n ) .
C151 H. Friedman and A. Scedrov, The l a c k o f d e f i n a b l e w i t n e s s e s and p r o v a b l y recursive functions i n i n t u i t i o n i s t i c set theories, appear).
[16]
Advances i n Math. ( t o
G. K o l a t a , Does G i d e l ' s theorem m a t t e r t o mathematics?, Science, 218 (1982), pages 779-780.
C171 J. 8. K r u s k a l , Well q u a s i - o r d e r i n g , t h e t r e e theorem, and Vazsony's c o n j e c t u r e , Trans. Amer. Math. SOC. 95 (1960), pages 210-225. C181 D. A. M a r t i n , Measurable c a r d i n a l s and a n a l y t i c games, Fund. Math. 66 (1970), pages 287-291. C191
, Borel
d e t e r m i n a c y , Annals o f Math. 102 (1975), pages 363-371.
[201 C. Nash-Williams, On w e l l - q u a s i - o r d e r i n g f i n i t e t r e e s , Proc. Cambridge P h i l . SOC. 59 (1963), pages 833-835.
10 [21]
A. NERODE. L.A. HARRINGTON
J . P a r i s and L. 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 Handbook o f Mathematical L o g i c (Jon Barwise, Ed.), Holland, Amsterdam, 1977, pages 1133-1142.
North-
[22]
S. Simpson, U n p r o v a b i l i t y o f c e r t a i n c o m b i n a t o r i a l p r o p e r t i e s o f f i n i t e t r e e s ( t o appear).
[23]
C.
[241
, " B i g " news f r o m Archimedes t o Friedman, N o t i c e s Amer. Math. S O ~ . 30 (1983), pages 251-256.
[25]
C. Steinhorn, Bore1 s t r u c t u r e s and measure and c a t e g o r y l o g i c , ModelT h e o r e t i c L o g i c s (J. Barwise and S. Feferman, Ed.), North-Holland, Amsterdam ( t o appear).
Smorynski, The v a r i e t i e s o f a r b o r e a l experience, Math. I n t e l l i g e n c e r , 4 (1982), pages 182-189.
1
The Work o f Harvey Friedman A n i l Nerode and Leo A.
1)
Harrington
Mathematical l o g i c i a n Harvey Friedman was r e c e n t l y awarded t h e N a t i o n a l Science F o u n d a t i o n ' s annual Waterman P r i z e , h o n o r i n g t h e most o u t s t a n d i n g American s c i e n t i s t under t h i r t y - f i v e y e a r s o f age i n a l l f i e l d s o f science and engineering.
When a mathematician wins such an award, t h e mathematical community
n a t u r a l l y wishes t o understand t h e u n d e r l y i n g achievements, and t h e i r i m p l i c a tions.
Friedman c o n t i n u e s t h e g r e a t t r a d i t i o n o f Frege, Russel, and G'ddel.
This
can be c h a r a c t e r i z e d as t h e e x e r c i s e o f a c u t e p h i l o s o p h i c a l p e r s p e c t i v e t o d i s t i l l exact mathematical d e f i n i t i o n s and q u e s t i o n s f r o m i m p o r t a n t f o u n d a t i o n a l issues.
These q u e s t i o n s i n t u r n g i v e r i s e t o mathematical s u b j e c t s and theorems
o f depth and beauty.
Friedman's c o n t r i b u t i o n s span a l l branches o f mathematical
l o g i c ( r e c u r s i o n t h e o r y , p r o o f t h e o r y , model t h e o r y , s e t t h e o r y , t h e o r y o f computation).
He i s a g e n e r a l i s t i n an age o f s p e c i a l i z a t i o n , y e t h i s theorems o f t e n
require extraordinary technical v i r t u o s i t y .
We d i s c u s s o n l y a few s e l e c t e d
high1 i g h t s . Friedman's ideas have y i e l d e d r a d i c a l l y new k i n d s o f independence r e s u l t s . The k i n d s o f statements proved independent b e f o r e Friedman were m o s t l y d i s g u i s e d p r o p e r t i e s o f formal systems (such as Godel's theorem on u n p r o v a b i l i t y of cons i s t e n c y ) o r a s s e r t i o n s about a b s t r a c t s e t s (such as t h e continuum h y p o t h e s i s o r S o u s l i n ' s hypothesis).
I n c o n s t r a s t , Friedman's independence r e s u l t s a r e about
q u e s t i o n s o f a more c o n c r e t e n a t u r e i n v o l v i n g , f o r example, Bore1 f u n c t i o n s o r t h e H i l b e r t cube.
R e p r i n t e d from t h e N o t i c e s o f t h e American Mathematical S o c i e t y , (1984) "The Work o f Harvey Friedman",
by A n i l Nerode and Leo H a r r i n g t o n , Volume 31, pp. 563-
566, by p e r m i s s i o n o f t h e American Mathematical Society.
2
A. NERODE. L.A. HARRINCTON I n [l] Friedman showed t h a t B o r e l determinacy cannot be proved i n Zermelo
s e t t h e o r y w i t h t h e axiom o f c h o i c e (ZC), o r any o f t h e usual formal systems a s s o c i a t e d w i t h a t most c o u n t a b l y many i t e r a t i o n s o f t h e power s e t o p e r a t i o n , a f t e r M a r t i n [17] proved B o r e l determinacy from c e r t a i n reasonable e x t e n s i o n s o f Zermelo Fraenkel s e t t h e o r y with t h e axiom of c h o i c e (ZFC).
Subsequently, M a r t i n
[18] gave a p r o o f o f B o r e l determinacy u s i n g uncountably many i t e r a t i o n s o f t h e power s e t o p e r a t i o n . These r e s u l t s reached t h e i r mature f o r m i n [9] THEOREM 1 (FRIEDMAN [9]).
as f o l l o w s :
I t i s necessary and s u f f i c i e n t t o use uncountably
many i t e r a t i o n s o f t h e power s e t o p e r a t i o n t o prove t h e f o l l o w i n g .
Every symme-
t r i c B o r e l subset o f t h e u n i t square c o n t a i n s o r i s d i s j o i n t from t h e graph o f a Borel function.
I n p a r t i c u l a r , t h i s a s s e r t i o n i s p r o v a b l e i n ZFC b u t n o t i n ZC.
I n Friedman [4,9],
t h e c l a s s i c a l theorem o f Cantor t h a t t h e u n i t i n t e r v a l
i s uncountable i s analyzed.
C a n t o r ' s argument produces an
t e r m i n any g i v e n i n f i n i t e sequence
-
Borel diagonalization function y
(here
F(y)
a
= 1'
F : Q
+
i s t h e H i l b e r t cube).
E I
y?,...
yl,
I
.
F
many c o o r d i n a t e s o f
and x
.
x
-
y
F(y)
i s a coordinate o f
Friedman observes t h a t t h e v a l u e o f
coming from C a n t o r ' s argument.
t h e same coordinates,
which i s n o t a
Specifically, there i s a
such t h a t no
depends on t h e o r d e r i n which t h e c o o r d i n a t e s o f
f o r the
x E I
I
Let
mean t h a t
x = y
y
a r e given, a t l e a s t
mean t h a t
x,y E
5
have
i s o b t a i n e d by permuting f i n i t e l y
y
THEOREM 2 (FRIEDMAN [4,9]). t h e invariance condition of
x
.
If
F :q
-t
x = y + F(x) = F(y)
I
i s a Borel function s a t i s f y i n g
, then
some
F(x)
i s a coordinate
Furthermore, t h i s i s p r o v a b l e i n ZC b u t not i n ZFC w i t h t h e power s e t
axiom d e l e t e d (even f o r t h e weaker theorem w i t h Subsequently, Friedman [12]
r e p l a c e d by
-)
.
gave c l o s e l y re1 a t e d one-dimensi onal theorems
w i t h t h e same meta-mathematical p r o p e r t i e s .
K
-
We present two o f these.
s : K + K by (2) D e f i n e t h e "square" x(') f o r x E K by x S(X,) = x Let T n+l * n = x 2 ' n [ O , l ) w i t h a d d i t i o n modulo 1). We say t h a t F : K + K be t h e c i r c l e group (i.e., Let
= {O,l}w
be t h e Cantor s e t .
Define t h e s h i f t
3
The Work of Harvey Friedman
i s shift invariant i f invariant i f
.
F(sx) = F(x)
F(2x) = F(x)
.
THEOREM 3 (FRIEDMAN [12]). somewhere i t s square.
We say t h a t
F : T + T
i s doubling
K
Every s h i f t i n v a r i a n t B o r e l f u n c t i o n on
There i s a B o r e l ( i n f a c t , c o n t i n u o u s ) f u n c t i o n on
which agrees somewhere w i t h every d o u b l i n g i n v a r i a n t B o r e l f u n c t i o n on
T
is T
.
Furthermore, t h e s e ( t h r e e ) theorems a r e p r o v a b l e i n ZC h u t n o t i n ZFC w i t h t h e power s e t axiom d e l e t e d . Friedman [9] develops e x t e n s i o n s o f Theorem 2, which proved u s i n g uncountRecently, Friedman has extended
a b l y many i t e r a t i o n s o f t h e power s e t o p e r a t i o n . t h i s i d e a as f o l l o w s : space o f a l l subsets o f on
Let
Q
Q
.
be t h e r a t i o n a l numbers and Let
G
P(Q)
be t h e Cantor
be t h e B a i r e space of a l l p r o d u c t s d e f i n e d
.
w
THEOREM 4 (FRIEDMAN [ l l ] ) .
If
F : G
+
i s a B o r e l f u n c t i o n such t h a t
G
isomorphic elements go t o i s o m o r p h i c values, t h e n some imbeddable i n
.
x
F(x)
i s isomorphically
Furthermore, t h i s theorem i s p r o v a b l e i n ZC b u t n o t i n ZFC
w i t h t h e power s e t axiom d e l e t e d . THEOREM 5 (FRIEDMAN [ l l ] ) . I f
F : P(Q)
+
P(Q)
i s a B o r e l f u n c t i o n such
t h a t o r d e r isomorphic arguments go t o o r d e r isomorphic values, then some i s isomorphic t o an i n t e r v a l i n points i n
A)
.
A
(even o f t h e f o r m
(a,b)
where
a,b
F(A) are
Furthermore, t h i s theorem r e q u i r e s uncountably many i t e r a t i o n s
o f t h e power s e t o p e r a t i o n t o prove.
I n p a r t i c u l a r , i t i s p r o v a b l e i n ZFC b u t
n o t i n ZC. Friedman [9] a l s o develops f a r - r e a c h i n g e x t e n s i o n s o f Theorem 2 by combining t h e s e i d e a s w i t h Ramsey's theorem.
T h i s l e a d s t o independence r e s u l t s from f u l l
ZFC, and i n f a c t t o theorems about B o r e l f u n c t i o n s on t h e H i l b e r t cube f o r which i t i s necessary and s u f f i c i e n t t o use " l a r g e c a r d i n a l s " t o prove.
The l a r g e c a r d i n a l s i n v o l v e d a r e as f o l l o w s :
A < K
.
K
implies We say t h a t
2
A
< K
unbounded subset o f
K
, and
K
i s not t h e sup o f fewer t h a n
i s a Mahlo c a r d i n a l i f K
A c a r d i n a l i s i n a c c e s s i b l e if
K
K
c a r d i n a l s below
i s i n a c c e s s i b l e and every c l o s e d
c o n t a i n s an i n a c c e s s i b l e c a r d i n a l .
A. NERODE. L.A. HARRINGTON
4
The
1-Mahlo c a r d i n a l s a r e t h e Mahlo c a r d i n a l s .
are the
The
(n+l)-Mahlo c a r d i n a l s
n-Mahlo c a r d i n a l s i n which e v e r y c l o s e d unbounded subset c o n t a i n s an
n-Mahlo c a r d i n a l . The group o f a l l p e r m u t a t i o n s o f
a
a c t s on 71
by
Q
that
g x,y
-
by permuting c o o r d i n a t e s . (xl
,...,xn)
which f i x a l l b u t f i n i t e l y many numbers
w
T h i s group a l s o a c t s d i a g o n a l l y on any
,...,gx,) .
= (gxl
11
Q
x,y E
For
let
-y
x
indicate
a r e i n t h e same o r b i t under t h i s a c t i o n .
THEOREM 6 (FRIEDMAN [S]).
x
that if
€6,y,z
F(x
x s' t
, and
E
an i n f i n i t e sequence
{x }
k
,..., x t
1
for a l l
n
,
Let y
F : Q x $ + I
-z
from
, then
be a B o r e l f u n c t i o n such
F(x,y)
= F(x,z)
.
such t h a t f o r a l l i n d i c e s
.
Then t h e r e i s
1 n I n order t o prove t h i s
) i s the f i r s t coordinate o f x s+l n i t i s necessary and s u f f i c i e n t t o use t h e e x i s t e n c e o f
cardinals f o r a l l
<...
s < t
,
n-Mahlo
.
n
These ideas have been extended i n Friedman [ll] t o much l a r g e r c a r d i n a l s . measurable c a r d i n a l i s a c a r d i n a l measure on a l l subsets o f every set
E
x
.
x
of
B
K-additive
A Ramsey c a r d i n a l i s a c a r d i n a l
o f f i n i t e subsets o f
f o r f i n i t e subsets of
K
which c a r r i e s a
K
K
, there
, membership
i s an unbounded
in
{O,l)-valued such t h a t f o r
K
B
L=
such t h a t
K
depends o n l y on t h e c a r d i n a l i t y
E
. Let
G O
be t h e B a i r e space o f f i n i t e l y generated p r o d u c t s on
THEOREM 7 (FRIEDMAN [ l l ] ) .
Let
F : G: + Go
i s i s o m o r p h i c a l l y imbeddable i n t o a c o o r d i n a t e o f
w
.
be a B o r e l f u n c t i o n mapping
p o i n t w i s e isomorphic arguments t o isomorphic values. x
Then f o r some
.
x C G:,
F(x)
Furthermore, t h i s theorem
i s p r o v a b l e i n ZFC b u t n o t i n ZC.
be a B o r e l f u n c t i o n mapping 0 p o i n t w i s e isomorphic arguments t o isomorphic values. Then f o r some x E Gw 0 ' f o r a l l subsequences y o f x, F ( y ) i s i s o m o r p h i c a l l y imbeddable i n t o a THEOREM 8 (FRIEDMAN [ll]). Let
coordinate o f
y
.
F : G:
+
G
T h i s i s p r o v a b l e i n ZFC + " t h e r e i s a measurable c a r d i n a l , "
b u t n o t i n ZFC + " t h e r e i s a Ramsey c a r d i n a l . "
A
5
The Work of Harvey Friedman I n P a r i s - H a r r i n g t o n [21] an i n t e r e s t i n g example o f a theorem s t a t e d i n f i n i t e s e t t h e o r y b u t n o t p r o v a b l e i n f i n i t e s e t t h e o r y , i s given.
A l s o see
C231, C241.
ClSl,
THEOREM 9 (PARIS-HARRINGTON [21]). l a r g e t h a t i f a l l subsets o f then t h e r e i s a s e t
[l,n]
[l,n]
E
E
same c o l o r and t h e s i z e o f
F o r each
o f size
k
, there
k,r,s
s
r
are colored with
such t h a t a l l subsets o f i s at least
i s an
E
of size
and t h e minimum o f
E
n
so
colors,
k
.
have t h e Further-
more, t h i s i s p r o v a b l e i n f i n i t e s e t t h e o r y augmented w i t h d e f i n i t i o n by r e c u r s i o n on
, but
w
not i n f i n i t e set theory.
Subsequently, Friedman [7] found an i n t e r e s t i n g example of a f i n i t e theorem which i s c o n c e p t u a l l y even c l e a r e r and independent o f much s t r o n g e r systems. J u s t as Theorem 9 i s based on Ramsey's theorem,
Friedman's work i s based on t h e
f o l l o w i n g theorem o f J. B. K r u s k a l 1171. A t r e e i s a nonempty p a r t i a l o r d e r i n g w i t h a l e a s t element, such t h a t t h e set o f predecessors o f any element i s l i n e a r l y order. trees then h(a) an
5
h :T h(b)
1
+ T
, and
2
T are f i n i t e 1' 2 i s s a i d t o be a homeomorphic imbedding i f a 5 b iff T-
h(inf(a,b))
.
= inf(h(a),h(b))
If
T
We w r i t e
T2 exists.
h
THEOREM 10 (KRUSKAL [19]), t r e e s t h e n f o r some
i < j, Ti
NASH-WILLIAMS [20]).
5
If
T,I
T
< T
1-
T2,...
2
1 i f such
are f i n i t e
.
Tj
The formal system ATR i s o b t a i n e d f r o m f i n i t e s e t t h e o r y by i n t r o d u c i n g c o u n t a b l y i n f i n i t e s e t s w i t h t h e p r i n c i p l e o f d e f i n i t i o n by t r a n s f i n i t e r e c u r s i o n on c o u n t a b l e w e l l o r d e r i n g s .
T h i s system goes j u s t beyond what i s r e f e r r e d t o as
predicative analysis. THEOREM 11 (FRIEDMAN 17,133;
SIMPSON [22]).
large that for a l l f i n i t e trees
T
L...'
T
i <...< 1
i
k
such t h a t
T. 1
,...,T n
1
i
.
and
For a l l card(T.)
T h i s theorem,
1
k
t h e r e i s an
5 i , there
n
are
as w e l l as K r u s k a l ' s
1 k Theorem 10 above, a r e p r o v a b l e i n ZFC w i t h o u t t h e power s e t axiom, b u t n o t i n ATR.
A l s o see [16,23,24].
so
A. NERODE, L.A. HARRINGTON
6
THEOREM 12 (FRIEDMAN [8,13]).
Theorem 11 f o r
k = 12
i s p r o v a b l e i n ZFC
w i t h o u t t h e power s e t axiom i n a few pages, b u t any p r o o f i n ATR must use a t [lOOOl least 2 pages. Friedman
[lo]
has extended K r u s k a l ' s theorem i n an i n t e r e s t i n g way so t h a t
t h e theorem has y e t s t r o n g e r metamathematical p r o p e r t i e s . T r (n)
o f f i n i t e trees with
< T i f and 1-r 2 which i s a l a b e l - p r e s e r v i n g homeomorphic i m -
d i s t i n c t l a b e l s , and d e f i n e s
n
m
o n l y i f t h e r e i s an
He c o n s i d e r s t h e c l a s s
+ T 1 2 bedding, p r e s e r v i n g l e f t t o r i g h t n e s s , w i t h t h e a d d i t i o n a l c r u c i a l c o n d i t i o n t h a t
if
then
b
h : T
T
i s an immediate successor o f l(c)
1. l ( h ( b ) ) ( l ( c )
a
in
T
h(a) < c < h(b)
and
1
.
i s the label o f c)
in
T
2 '
" r " means
The s u b s c r i p t
"restricted". THEOREM 13 (FRIEDMAN i < j
t h e r e are
that for all
T
[lo],
,...,Tm
F o r every
.
< T F o r a l l k,n 1-r 2 E T r (n) w i t h each c a r d ( T . )
such t h a t
1
SIMPSON [22]).
T
a
...
.
T1,
T2,
t h e r e i s an
5
, there
... E T r m
m
(n)
,
so l a r g e
i <...< i 1 k These theorems a r e p r o v a b l e i n t h e s i n g l e i
are
1
T. < T < < T 1 -r i -r -r i 1 2 k s e t q u a n t i f i e r comprehension axiom system w i t h t h e f u l l scheme o f i n d u c t i o n 1 1 (nl-CA) , b u t not i n U1-CA w i t h s e t i n d u c t i o n i n s t e a d o f t h e f u l l scheme o f 1 i n d u c t i o n (nl-CAO)
such t h a t
.
Two o f t h e most fundamental concepts i n mathematical l o g i c a r e t h e model t h e o r e t i c concept o f t r a n s l a t a b i l i t y and t h e p r o o f t h e o r e t i c concept o f r e l a t i v e consistency.
Friedman [6]
t h e s e concepts c o i n c i d e . order theories.
proves t h a t under s u r p r i s i n g l y general c o n d i t i o n s , Specifically, l e t
We say t h a t
S
S
, when
be f i n i t e l y a x i o m a t i z e d f i r s t -
i s translatable into
o r d e r d e f i n i t i o n s o f t h e symbols i n axiom o f
S,T
T
i f there are f i r s t -
i n terms o f t h o s e i n
S
so t r a n s l a t e d , becomes a theorem o f
T
There are a few somewhat d i f f e r e n t ways o f d e f i n i n g relative t o
T
," considered
i n [6].
. "S
T
, such
t h a t every
i s consistent
We focus a t t e n t i o n on t h e f o l l o w i n g one:
L e t €FA ( e x p o n e n t i a l f u n c t i o n a r i t h m e t i c ) be t h e s t a n d a r d weak system o f a r i t h m e t i c based on
0,l
,< , = ,t ,
a p p l i e d t o bounded formulas only.
, and
e x p o n e n t i a t i o n , where i n d u c t i o n i s
The q u a n t i f i e r c o m p l e x i t y o f a f o r m u l a i s a
s t a n d a r d measure o f t h e number o f a l t e r n a t i o n s o f q u a n t i f i e r s t h a t a r e present.
7
The Work of Harvey Friedman
We say t h a t
i f f o r some f i x e d
n
, the
I f t h e r e i s an i n c o n s i s t e n c y p r o o f i n
S
then
i s consistent r e l a t i v e t o
S
f o l l o w i n g i s p r o v a b l e i n EFA.
t h e r e i s an i n c o n s i s t e n c y p r o o f i n c o m p l e x i t y i s a t most
n
T
T
which uses o n l y formulas whose q u a n t i f i e r
more t h a n t h e g r e a t e s t q u a n t i f i e r c o m p l e x i t y o f
formulas used i n t h e g i v e n i n c o n s i s t e n c y p r o o f i n
S
.
I t i s s t r a i g h t f o r w a r d t o see t h a t t r a n s l a t a b i l i t y i m p l i e s r e l a t i v e
consistency
.
THEOREM 14 (FRIEDMAN [S]). c o n t a i n i n g EFA
i f applicable).
Then
i s consistent r e l a t i v e t o
S
S,T
be f i n i t e l y axiomatized t h e o r i e s
and a weak t h e o r y o f f i n i t e sequences o f o b j e c t s o t h e r t h a n
n a t u r a l numbers if
Let
T
S
i s translatable into
T
i f and o n l y
.
Friedman [5] has i n i t i a t e d an i n t e r e s t i n g new branch o f model t h e o r y c a l l e d Borel model t h e o r y .
A t o t a l l y B o r e l model i s a s t r u c t u r e whose domain i s IR and
every r e l a t i o n t h a t i s d e f i n a b l e ( i n t h e language considered) over t h e s t r u c t u r e Friedman [ 5 l c o n s i d e r s t h e t h r e e fundamental q u a n t i f i e r s
i s Borel.
a l l i n t h e sense o f Lebesgue measure), Q
Q
m
(almost
( a l m o s t a l l i n t h e sense o f B a i r e C
category), and quantifiers
( u n c o u n t a b l y many), i n a d d i t i o n , o f course, t o t h e usual
Q
w1 V, P.
Friedman proves t h e f o l l o w i n g completeness and d u a l i t y theorem. THEOREM 15 (FRIEDMAN 151, STEINHORN [25]).
Q
(or m only i f
Qm
0
and
YJ)
.
0
Then
Let
0
be a sentence based on
i s t r u e i n a l l t o t a l l y B o r e l models i f and
can be proved using, r o u g h l y speaking, t h e f o l l o w i n g p r i n c i p l e s : 0
,
union o f two s e t s o f measure
0
s i n g l e t o n s a r e o f measure
subsets o f measure i s o f measure
0
0,R
a r e o f measure
i s not o f measure
almost a l l v e r t i c a l cross s e c t i o n s o f a two-dimensional
If
Qc
0
0
, and 0
(Fubini's
, t h e n t h i s i s t r u e i f "measure 0" m As a consequence we have t h a t 0 i s t r u e i n a l l
i s used i n s t e a d o f
i s replaced by "meager."
, the
s e t a r e o f measure
and o n l y i f almost a l l h o r i z o n t a l cross s e c t i o n s a r e o f measure theorem).
0
t o t a l l y Borel structures f o r
Q
m
Q
( o r (Q ,",a)) m
i f and o n l y i f
$*
i s true i n
if
8
A. NERODE, L.A. HARRINGTON
a l l t o t a l l y Bore1 s t r u c t u r e s f o r
( o r (Q Y,X)) C 1 (duality).
0 by r e p l a c i n g Q
Q
where
m*
i s obtained from
by Q m C Friedman has o b t a i n e d a number o f fundamental r e s u l t s i n i n t u i t i o n i s t i c s e t
theory.
The usual axioms f o r ZF a r e e x t e n s i o n a l i t y , p a i r i n g , union, i n f i n i t y ,
f o u n d a t i o n , power s e t , comprehension, and f i n a l l y replacement.
The axiom scheme
o f c o l l e c t i o n i s an a l t e r n a t i v e t o t h e axiom scheme o f replacement, b u t i t i s a fundamental theorem o f s e t t h e o r y t h a t t h e s e a r e e q u i v a l e n t . However, i f we use i n t u i t i o n i s t i c l o g i c i n s t e a d o f o r d i n a r y l o g i c , t h e n t h e p r o o f t h a t replacement i m p l i e s c o l l e c t i o n breaks down.
Thus we l e t ZFIR be ZF
w i t h i n t u i t i o n i s t i c l o g i c f o r m u l a t e d w i t h t h e scheme o f replacement, and ZFIC be ZF w i t h i n t u i t i o n i s t i c l o g i c f o r m u l a t e d w i t h t h e scheme o f c o l l e c t i o n . THEOREM 16 (FRIEDMAN [2]) ; FRIEDMAN-SCEDROV [15]). ZFIC.
ZFIR does n o t i m p l y
It i s p r o v a b l e i n a weak system o f a r i t h m e t i c t h a t o r d i n a r y ZFC i s
c o n s i s t e n t i f and o n l y i f ZFIC i s c o n s i s t e n t . Two b a s i c d e s i r a b l e p r o p e r t i e s o f i n t u i t i o n i s t i c formal systems (which almost never h o l d f o r o r d i n a r y formal systems) a r e t h e d i s j u n c t i o n p r o p e r t y , which a s s e r t s t h a t i f a d i s j u n c t i o n
A V B
i s p r o v a b l e t h e n one o f t h e d i s j u n c t s
i s provable; and t h e numerical e x i s t e n c e p r o p e r t y , which a s s e r t s t h a t i f (Zn)(Vn)
i s p r o v a b l e t h e n f o r some
n,AG
i s provable.
DP t r i v i a l l y f o l l o w s
f r o m NEP. Friedman proves t h e f o l l o w i n g h i g h l y s u r p r i s i n g theorem v i a a m y s t e r i o u s appl ic a t i o n o f Godel s e l f - r e f erence.
THEOREM 17 (FRIEDMAN [31).
Let
T
be a r e c u r s i v e l y axiomatized i n t u i t i o n -
i s t i c formal system s u b j e c t t o , r o u g h l y , t h e same weak hypotheses commonly used i n Godel's incompleteness theorems. t y i f and o n l y i f
T
Then
T
has t h e numerical e x i s t e n c e p r o p e r -
has t h e d i s j u n c t i o n p r o p e r t y . SELECTED REFERENCES
[l]H. Friedman, Higher s e t t h e o r y and mathematical p r a c t i c e , Ann. Math. L o g i c 2 (1971), pages 326-357.
9
The Work of Harvey Friedman
, The c o n s i s t e n c y o f c l a s s i c a l s e t t h e o r y r e l a t i v e t o a s e t t h e o r y w i t h i n t u i t i o n i s t i c l o g i c , J. Symbolic L o g i c , 38 (1973), pages 315-319.
, The
c31
d i s j u n c t i o n property i m p l i e s t h e numerical existence pro-
p e r t y , Proc. Nat. Acad. Sci.,
72 (1975), pages
communicated b y K u r t G i d e l ,
2877-2878. C43 ______ , H i g h e r s e t t h e o r y and e x i s t e n t i a l s t a t e m e n t s , October 1976, preliminary report.
C5l
--, On
[Sl
-__
C71
, Independence r e s u l t s i n f i n i t e graph t h e o r y March 1981, p r e l i m i n a r y r e p o r t .
C8l
, The e x i s t e n t i a l i n c o m p l e t e n e s s phenomenon preliminary report.
t h e l o g i c o f measure and c a t e g o r y 1, and Addendum, December 1978; September 1979, p r e l i m i n a r y r e p o r t s .
, T r a n s l a t a b i l i t y and r e l a t i v e c o n s i s t e n c y 1-11, November 1976; September 1980, p r e l i m i n a r y r e p o r t s .
, On
c91 Math.,
I-VII,
February-
I, A p r i l 1981,
t h e necessary use o f a b s t r a c t s e t t h e o r y , Advances i n
41 (1981), pages 209-280.
, Beyond
ClOl
K r u s k a l ' s theorem 1-111, J u n e - J u l y 1982, p r e l i m i n a r y
report.
, New
Clll
necessary uses o f a b s t r a c t s e t t h e o r y f o r B o r e l October 1983; J u l y 1983, p r e l i m i n a r y r e p o r t .
functions I - I V ,
, Unary
c121 i n Math.,
C131
B o r e l f u n c t i o n s and second o r d e r a r i t h m e t i c , Advances
50 (1983), pages 155-159.
, An
improved f i n i t e f o r m o f K r u s k a l ' s theorem,
May 1984,
preliminary report.
C141
, The
metamathematics o f K r u s k a l ' s theorem ( i n p r e p a r a t i o n ) .
C151 H. Friedman and A. Scedrov, The l a c k o f d e f i n a b l e w i t n e s s e s and p r o v a b l y recursive functions i n i n t u i t i o n i s t i c set theories, appear).
[16]
Advances i n Math. ( t o
G. K o l a t a , Does G i d e l ' s theorem m a t t e r t o mathematics?, Science, 218 (1982), pages 779-780.
C171 J. 8. K r u s k a l , Well q u a s i - o r d e r i n g , t h e t r e e theorem, and Vazsony's c o n j e c t u r e , Trans. Amer. Math. SOC. 95 (1960), pages 210-225. C181 D. A. M a r t i n , Measurable c a r d i n a l s and a n a l y t i c games, Fund. Math. 66 (1970), pages 287-291. C191
, Borel
d e t e r m i n a c y , Annals o f Math. 102 (1975), pages 363-371.
[201 C. Nash-Williams, On w e l l - q u a s i - o r d e r i n g f i n i t e t r e e s , Proc. Cambridge P h i l . SOC. 59 (1963), pages 833-835.
10 [21]
A. NERODE. L.A. HARRINGTON
J . P a r i s and L. 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 Handbook o f Mathematical L o g i c (Jon Barwise, Ed.), Holland, Amsterdam, 1977, pages 1133-1142.
North-
[22]
S. Simpson, U n p r o v a b i l i t y o f c e r t a i n c o m b i n a t o r i a l p r o p e r t i e s o f f i n i t e t r e e s ( t o appear).
[23]
C.
[241
, " B i g " news f r o m Archimedes t o Friedman, N o t i c e s Amer. Math. S O ~ . 30 (1983), pages 251-256.
[25]
C. Steinhorn, Bore1 s t r u c t u r e s and measure and c a t e g o r y l o g i c , ModelT h e o r e t i c L o g i c s (J. Barwise and S. Feferman, Ed.), North-Holland, Amsterdam ( t o appear).
Smorynski, The v a r i e t i e s o f a r b o r e a l experience, Math. I n t e l l i g e n c e r , 4 (1982), pages 182-189.