T
H
E
JOURNAL
OF
SYMBOLIC LOGIC
EDITED BY
ALONZO CHURCH
H E R B E R T ENDERTON
C . E . M . YATES
WILLIAM CRAIG
DAGFINN E0LLESDAL
A N I L NERODE
Y . N . MOSCHOVAKIS
M a n a g i n g E d i t o r : A L F O N S BORGERS
Consulting
Editors:
Jiftf B E £ V Ä R
G.
ANDRZEJ BLIKLE
H A N S HERMES
HASENJAEGER
MARTIN DAVIS
H.
EDWARD E .
AZRIEL LEVY
VERENA H .
CALVIN C.
FREDERK: B .
DAWSON
DYSON
ELGOT
FITCH
V
BARKLEY
JEROME KEISLER
KURT
SCHÜTTE
DANTA S C O T T
DONALD M O N K
JOSEPH R .
GERT H . MÜLLER
J. F.
MICHAEL O.
JEAN VAN
GENE F.
ROSSER
A R T O SALOMAA
RABIN
SHOENEIELD
THOMSON
HEIJENQORT
ROSE
V O L U M E 43
NUMBER 1
MARCH
1978
P U B t r I S H E D Q U A R T E R L V B Y T H E A S S O C I A T I O N FOR S Y M B O L I C L O G I C , I N C .
S U P P O R T F R O M I C S U A N D FROM I N S T I T U T I O N A L A N D C O R P O R A T E
v
Reproduction
Copyright
© 1978 by the Association
for Symbolic Logic, I n c .
by photostat,
photo-print,
microfilm,
or like process by permission
?0
G
WITH
MEMBERS
only
TABLE OF CONTENTS
On the inconsistency of Systems simüar to J^* . ByM. W . BUNDER and R . K .
MEYER
.
1
First Steps in intuitionistic model theory. By H . DE SWART......
3
Controlling the dependence degree of.a recursively enumerable vectot Space. By
RICHARD A. SHORE......
Relativized realizability in intuitionistic arithmetic of all finite types. By
NICOLAS D . GOODMAN
'.
Degrees of sensible lambda theories. By HENK BARENDREGT, J A N BERGSTRA,
JAN WILLEM K L O P and HENRI VOLKER....
13
23
45
Projective algebra and the calculus of relations. By A. R . BEDNAREK änd S. M .
ULAM.
.
Saturated ideals. By KENNETH KUNEN................
56
..
65
Separation principles and the axiom of determinateness. By ROBERT A. VAN
WESEP
77
The model theory of ordered differential fields. By MICHAEL F . SINGER
82
Rings which admit elimination of quantifiers. By BRUCE I . ROSE..
92
Nöte on an induction axiom. By J . B . PARIS
113
Provable wellorderings of formal theories for transfinitely iterated inductive
definitions. By W . BUCHHOLZ and W . POHLERS....
118
Existentially cojnplete torsion-free nilpotent groups. By D, SARACINO
126
Sets which do not have subsets of every higher degree. By STEPHEN G . SIMPSON 1 3 5
Reviews.
,
L.;
,
139
The JOURNAL OF SYMBOLIC LOGIC is the official organ of the Association for Symbolic Logic Inc.,
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back
cover
T H E
JOURNAL
OF
SYMBOLIC LOGIC
EDITED
BY
ALONZO CHURCH
HERBERT
ENDERTON
C . E . M . YATES
WILLIAM CRAIG
DAGFINN
F0LLESDAL
ANIL
NERODE
Y. N . MOSCHOVAKIS
Managing Editor: ALFONS BORGERS
Consulting Editors:
JIRI BECVÄR
ANDRZEJ BLIKLE
MARTIN DAVIS
EDWARD E. DAWSON
VERENA H. DYSON
CALVIN C. ELGOT
FREDERIC B. FITCH
G. HASENJAEGER
HANS HERMES
H. JEROME KEISLER
AZRIEL LEVY
DONALD MONK
GERT H. MÜLLER
MICHAEL 0. RABIN
G E N E F . ROSE
VOLUME
BARKLEY ROSSER
ARTO SALOMAA
KURT SCHÜTTE
D A N A SCOTT
JOSEPH R . SHOENFIELD
J. F. THOMSON
JEAN V A N HEIJENOORT
43
1978
PUBLISHED QUARTERLY BY THE ASSOCIATION FOR SYMBOLIC LOGIC, INC. WITH
SUPPORT FROM I C S U AND FROM INSTITUTIONAL AND CORPORATE MEMBERS
The four numbers of Volume 43 were issued at the following dates:
Number 1, pages 1-160, June 27, 1978
Number 2, pages 161-384, August 16, 1978
Number 3, pages 385-622, August 28, 1978
Number 4, pages 623-759, February 14, 1979
Numbers 1—3 of this volume are copyrighted © 197 8 by the Association for Symbolic Logic , Inc.
Number 4 of this volume is copyrighted © 1979 by the Association for Symbolic Logic, Inc. Reproduction of copyrighted numbers of the Journal by photostat, photoprint, microfilm, or like
process is forbidden, except by written permission, to be obtained through the Business Office of
the Association for Symbolic Logic, P . O . Box 6248, Providence, Rhode Island 02940, U. !S. A.
A U T H O R INDEX
A B R A M S O N , F R E D G. and H A R R I N G T O N , L E O A Models without indiscernibles.. 5 7 2
B A R E N D R E G T , H E N K , B E R G S T R A , J A N , K L O P , J A N W I L L E M and V O L K E N , H E N R I .
Degrees of sensible lambda theories
45
B A R W I S E , J O N and M O S C H O V A K I S , Y I A N N I S N . Global inductive definability
521
B E D N A R E K , A . R. and U L A M , S. M . Projective algebra and the calculus
of relations
56
B E E S O N , M I C H A E L . A type-free Gödel Interpretation
.
213
. Some relations between classical and constructive mathematics
228
B E R G S T R A , J A N . See B A R E N D R E G T , H E N K .
B O T T S , T R U M A N . What is the Conference Board of the Mathematical Sciences?.. 6 2 0
B R U C E , K I M B. Ideal models and some not so ideal problems in the model
theory o f L ( ß )
304
^ B U C H H O L Z , W. and P O H L E R S , W. Provable wellorderings of formal theories for
transfinitely iterated inductive definitions
B U N D E R , M . W. and M E Y E R , R. K .
similar to g r *
118
On the inconsistency of Systems
1
t
B U N D E R , M . W. and S E L D I N , J O N A T H A N P. Some anomalies in Fitch's System
QD
247
B U R G E S S , J O H N P. On the Hanf number of Souslin logics
CUTLAND, NIGEL.
568
Zj-compactness in languages stronger than i£
508
A
D E S W A R T , H . First Steps in intuitionistic model theory
D I P R I S C O , C. A and H E N L E , J .
EPSTEIN, R
See
On the compactness of N
3
t
and N
2
394
P O S N E R , D.
F E R R O , R U G G E R O . Interpolation theorems for h\*
535
G A L V I N , F., J E C H , T. and M A G I D O R , M.
284
k
A n ideal game
G O O D M A N , N I C O L A S D. Relativized realizability in intuitionistic arithmetic of
all finite types
23
. The nonconstructive content of sentences of arithmetic
GREEN, JUDY.
K-Suslin logic
659
H A R R I N G T O N , L E O A . Analytic determinacy and 0
.
See
H E N L E , J . See
497
#
685
A B R A M S O N , F R E D G.
DIPRISCO, C . A .
H I R S C H F E L D , J O R A M . Examples in the theory of existential completeness
650
H O D E S , H A R O L D T. Uniform upper bounds on ideals of Turing degrees
601
iii
iv
AUTHOR INDEX
H O W A R D , P A U L E . , R U B I N , A R T H U R L . and R U B I N , J E A N E . Independence
results for class forms of the axiom of choice
J E C H , T. See
673
GALVIN,F.
J O C K U S C H , C A R L G., J R . and P O S N E R , D A V I D B . Double jumps of minimal
degrees
715
JOHNSON, NANCY.
Classifications of generalized index sets of open classes
694
J O N E S , J A M E S P. Three universal representations of recursively enumerable
sets
335
KALANTARI, IRAJ.
Major subspaces of recursively enumerable vector Spaces
K E C H R I S , A L E X A N D E R S. Minimal upper bounds for sequences of
293
-degrees...
502
. The perfect set theorem and definable wellorderings of the
continuum
630
KEISLER, H. JEROME.
The stability function of a theory
K L E E N E , S T E P H E N C. A n addendum to "The work of Kurt Gödel"
KLOP, J A N WILLEM.
See
. Prime and atomic models
Saturated ideals
65
Type two partial degrees
M A G I D O R , M , See
G A L V I N , F.
M E Y E R , R . K . See
BUNDER, M. W .
270
A model of Peano arithmetic with no elementary end
extension
563
MOSCHOVAKIS, YIANNIS N .
See
BARWISE, J O N .
N O R M A N N , D A G . A continuous functional with noncollapsing hierachy
PARIS, J. B.
456
623
^ M A A S S , W O L F G A N G . The uniform regulär set theorm in a-recursion theory
^MILLS, GEORGE,
331
385
L E G G E T T , A N N E . a-Degrees of maximal a-r.e. sets
L I H , KO-WEI.
613
BARENDREGT, HENK.
K N I G H T , J U L I A F. A n inelastic model with indiscernibles
KUNEN, KENNETH.
481
Note on an induction axiom
487
113
. Some independence results for Peano arithmetic
/ T I L L A Y , A N A N D . Number of countable models
725
492
P O H L E R S , W. Ordinals connected with formal theories for transfinitely iterated
inductive definitions
.
See
161
BUCHHOLZ, W.
P O S N E R , D A V I D B . See
JOCKUSCH, C A R L G., J R .
P O S N E R , D. and E P S T E I N , R . Diagonalization in degree constructions
280
R E M M E L , J . A r-maximal vector space not contained in any maximal vector
Space
430
RETZLAFF, ALLEN.
ROSE, A L A N .
calculi
Simple and hyperhypersimple vector Spaces
260
Formalisations of further N -valued Lukasiewicz propositional
0
207
V
AUTHOR INDEX
R O S E , B R U C E I.
Rings which admit elimination of quantifiers
92
. The ttj-categoricity of strictly upper triangulär matrix rings over
algebraically closed fields
R U B I N , A R T H U R L.
R U B I N , J E A N E.
SARACINO, D.
See
See
250
H O W A R D , P A U L E.
H O W A R D , P A U L E.
Existentially complete torsion-free nilpotent groups
SCHLIPF, JOHN STEWART.
126
Toward model theory through recursive Saturation
^ S C H Ü M M , G E O R G E F . A n incomplete nonnormal extension of S3
S E L D I N , J O N A T H A N P.
211
A sequent calculus formulation of type assignment with
equality rules for the 50-calculus
.
See
643
B U N D E R , M. W.
SHELAH, SAHARON.
.
183
On the number of minimal models
475
End extensions and numbers of countable models
550
S H O R E , R I C H A R D A Controlling the dependence degree of a recursively
enumerable vector Space
13
. Nowhere simple sets and the lattice of recursively enumerable sets
SIMPSON, S T E P H E N G. Sets which do not have subsets of every higher degree
322
135
S I N G E R , M I C H A E L F . The model theory of ordered differential fields
82
S O L O M O N , M A R T I N K . Some results on measure independent Gödel
speed-ups
667
STEPANEK, PETR.
U L A M , S. M.
See
Cardinal collapsing and ordinal definability
635
B E D N A R E K , A . R.
V A N WESEP, R O B E R T A. Separation principles and the axiom of determinateness
V O L K E N , HENRI.
See
77
BARENDREGT, HENK.
W H E E L E R , W I L L I A M H . A characterization of companionable, universal theories
402
W O O D R O W , R O B E R T E. Theories with a finite number of countable models
442
Meeting of the Association for Symbolic Logic, Campinas, Brazil, 1 9 7 6 . By A Y D A
I. A R R U D A , F R A N C I S C O M I R O Q U E S A D A , N E W T O N C. A. D A C O S T A
and
ROLANDO CHUAQUI
352
Abstracts of papers
352
Annual meeting of the Association for Symbolic Logic, Saint Louis, 1 9 7 7 . By
J O N B A R W I S E , K E N N E T H K U N E N and J O S E P H
ULLIAN
365
Abstracts of papers
365
Meeting of the Association for Symbolic Logic, Chicago, 1 9 7 7 . By C A R L G.
J O C K U S C H , J R . , R O B E R T I. S O A R E , W I L L I A M T A I T
and
GAISI T A K E U T I
614
Abstracts of papers
614
Reviews
139
Reviews
373
List of officers and members of the Association for Symbolic Logic
732
Notice of meeting
759
THE
J O U R N A L O F SYMBOLIC LOGIC
Volume 43, Number 1, March 1978
PROVABLE
WELLORDERINGS OF FORMAL THEORIES
FOR
TRANSFINITELY ITERATED
INDUCTIVE DEFINITIONS
W.
B U C H H O L Z and W. P O H L E R S
Introduction
By [12] we know that transfinite induction up to ©en^iO is not provable in
I D , the theory of N-times iterated inductive definitions. In this paper we will
show that conversely transfinite induction up to any ordinal less than ©en^iO is
provable in IDjy, the intuitionistic version of I D , and extend this result to
theories for transfinitely iterated inductive definitions.
N
N
In [14] Schütte proves the wellordering of his notational Systems X ( N ) using
predicates 93 (a) : e ( a £ M A {X E M : x < a } is wellordered) with M : = {x E
2(N): ^ ( K o x ) A * • • A Sk-iCKfc-jx)} and 0 < k < N. Obviously the predicates
93o,.. .,93N-I are definable in I D ' with the defining axioms:
k
k
k
k
1
N
(83*1)
Prog[M ,93 ],
(93 2)
Prog [ M , m - > Vx (93 (x) - + g[x ]),
k
k
k
k
k
where P r o g [ M , X ] means that X is progressive with respect to M , i.e.
k
k
Prog [ M , X ] : ~ V x E M ( V y E M ( y < x -> X ( y ) ) - > X ( x ) ) .
k
k
k
The crucial point in Schütte's wellordering proof is L e m m a 19 [14, p. 130]
which can be modified to
(I)
TI[M
k + 1
, a ] , Sb = k, 93 (fc)=> 93 ((a,fc)),
k
k
for 0 < k < N - l ,
2
where T I [ M i , a ] is the scheme of transfinite induction over M
up to a .
Checking the proof of (I) it turns out that besides (93 l) and (23 2)
(0 < k < N - 1) only finitary methods (including mathematical induction) are
used. Since the proof uses "excluded middle" only for decidable formulas it is
formalizable i n I D ' . Following the proof of Lemma 17 in [14] one gets
k +
k + i
k
k
N
(II)
(III)
IDiv h 93 (1) A • • • A ® - i ( n - , )
0
N
N
and
IDJvhTI[M ,a ].
N
N
F r o m (III) one derives in the well-known way (due to Gentzen [5])
(IV)
ID'„ h T I [ M N , c ]
n
for each n £ J V ,
Received January 5, 1976.
K x is a finite set of subterms of x . 93„(K„x) means Vy € X„x(93„(y)).
For an exact definition see notational Convention (5), page 3 of the present paper.
1
n
2
118
© 1978, Association for Symbolic Logic
WELLORDERINGS OF FORMAL THEORIES
119
where C : = { 1 N , c : = ( l , c „ ) . B y (I), (II), (IV) and the facts that M = S ( N )
and, 93 (a) implies T I [ M , a ] one gets
0
n + 1
0
fc
k
(V)
I D ' h T I [2(N), ft[c , 0]]
N
for each n E JV,
n
where ft[c„, 0] :=((•• • (c„, n _i), . . . , Hj), 1).
Since
s u p „ fi[c„, 0] =
n [ ( l # l , f t ) , 0 ] and the order type of {JC E £ ( N ) : X < f t [ ( l # l,n ),0]} is
0 £ n ^ i O , t r a n s f i n i t e i n d u c t i o n up to any o r d i n a l less than 0 £ n ^ i O is p r o v a b l e i n
IDjs/, which we will abbreviate by I D N I - T I [ < ©en^iO].
Similar considerations apply to the wellordering proof of the System @({g})
given in [2]. W e will prove the following
results:
N
e N
N
N
3
(A)
I D > T I [ < ©erv-iO]
for any countable v E @({g})\
,
(B)
ID *hTI[<0en
<
5
ni+
iO] ,
where I D t and ID<. are the intuitionistic versions of the theories I D and ID<. defined in [4, pp. 307-308] (see also the last paragraph of page 3 of the present paper).
ID<„ is defined to be the theory U < I D . B y ID^we mean the theory of
autonomously iterated inductive definitions (i.e. if I D h T I [ y ] thert.ID„ C I D ) .
For a theory T h we take | T h | := sup { £ E O n : T h h T I [ £ ] } . There are the
following ordinal theoretic relations:
(1) ©£n +i0 = © £ n i 0 and @en, i0 = @€n, i0 for v<@aill0.
(So the
above derived results on transfinite induction in I D ' ( N < a>) are special cases
of (A).)
(2) eSl 0
= sups
Öen^iO for limit v < @a 0.
(3) 01X0 = v for v = © a o .
(4) @ ü 0 = s u p „ ^ with v : = l , ivn := 0ft»* 0.
By (A),
and [13] (cf. footnote 4) we get the equations:
V
€
ai
n | +
V
{
+
+
N
v
<v
ftl
n i
n i
e N
(AI)
n
0
| I D , | = |IDt| = ee ^i0
n
(A2)
| ID<„ | = | ID!c„| = 0^,0
(A3)
_
HD l&n^ol = l
I D ( 0
Preliminaries.
l = ®nn,0
_
and I D
for v < @fl 0.
ni
for limit v < ©ft^O.
( 0
has the same theorems as IDi&n^o.
In the sequel we assume an arithmetization of the notational
System @({g}), such that all relevant ordinal sets, functions and relations of [2]
(as S£,
K a , S, + , 0, < , etc.) become primitive recursive . W e will identify
ordinal notations and their arithmetizations.
Though we presume some familiarity with [2], we will give a short
d e s c r i p t i o n of the System
0({g}). ©({g}) is a set ST of ordinal notations ordered
by a r e l a t i o n < . Each element of £ has the shape 0, a + b, @ab or gab with
6
u
3
Cf. [3]. Note that the System 2(N) in [3] is a slight modification of that in [14]. In [3] the first
element of X(N) is 0 instead of 1.
Recently the second author [13] was able to show I D „ / T I [ Ö e „ + i 0 ] .
Kino's wellordering proof for her ordinal diagrams Od(J) [8] is formalizable in ID'<.. Hence
4
n
5
IOlc-hTI[||Od(/), <-||]- But as remarked in [2] ||Od(/),<~||<Oa (r + 1)<0(n , + 1)0<
n|
0e ^ , 0
6
for || / | | = 1 + r
< e(nn, + 1)0.
For a Subsystem of 0({g}) such an arithmetization will be carried out in [15].
n
120
W. B U C H H O L Z A N D
W. P O H L E R S
a, b E ST. The Symbols + (ordinal sum), 0 and g denote 2-place ordinal
functions. So each term a E St canonically represents an ordinal | a |, for
example | © 0 0 | = 1, |©01| = a>, | ©101 = e . F o r the order relation < we have
a < b * * \ a \ E \ b \ . Terms of the shape & a b or g a b a r e called main terms; they
represent ordinals closed under + . Ä is the set of all terms g a b E St\ the
elements of Si represent initial ordinals > o). There is a primitive recursive
order isomorphism a »-» C l from 2T onto Ä := {0} U Ä with f l = 0 and | ft | =
f l | | for a ^ 0. For each a E ST there is exactly one x E St with C l < a < H ;
we dehne Sa: = C l and call it the level (Stufe) of a . For all a , b E St we have
S@ab = Sfc, h e n c e 0 a O < f t , . W e dehne M :={x EST: Sx =0} = {x E S~: x <
fli}. M represents a Segment of the countable ordinals, i.e.
(1) { £ E O n : £ < | a | } = {|x|: x E M A X < a ) for a E M .
For a E S" and w E Ä ,
is a finite set of main terms with levels < u. The
sets K a have the following properties:
(2) If Sa < u, then iC a is the set of components of a .
(3) K b C K ( a + b ) C K a \ J K b .
(4) w < ü A c e K a -» K c C X a .
0
a
0
0
a
a
x
x + ]
x
0
0
0
0
0
u
7
M
u
u
u
v
(5)
u
w
w
v<(l -*K a, CK aCK n U{l}.
a
v
a
v
v
a
We fix t h e f o l l o w i n g n o t a t i o n a l C o n v e n t i o n s :
(1) a, b, c, x, y, z denote elements of 3T.
(2) M, u, w denote elements of Ä .
(3) SR,3E serve as syntactical variables for sets {x: $ [ x ] } c S " , where $ [ x ] is a
formula of the theory considered.
(4) X D a : = { x G X : x < a } , X n w := {x E X : Sx < w}.
(5) Prog [SR, X] abbreviates the formula V x E »(SR n x C £ -> x E £ ) .
TI[SR,£, a] abbreviates the fromula a E SR A (Prog [9t, £]-»SR D a C X \ and
TI[SR, a] denotes the scheme {TI[SR,
a ] } * expressing the principle of transfinite induction over SR up to a .
T r a n s f i n i t e i n d u c t i o n s p r o v a b l e i n ID'„ a n d ID<*. Our main tool in proving
transfinite inductions will be the concept of the a c c e s s i b l e p a r t W[SR] of a set SR,
usually defined by W[SR] = {x E SR: SR H x is wellordered}, which is a secondorder definition. This definition however can be replaced by an inductive
definition, which is expressible in a first-order language by the infinite liSt of
axioms:
(i) Prog [SR, W[SR]] and
(ii) Prog [SR, £ ] - > W [ S R ] C £ for each X .
The t h e o r i e s IDj, ( w i t h v E M ) a n d ID<- are formal theories for iterations of
such inductive definitions. They are first-order extensions of Heyting's arithmetic, where ID'„ allows iteration of monotone inductive definitions along the
segment M f l v, while ID<» allows iteration along the accessible part
W : = W [ M ] of M . Besides the axioms for iteration of inductive definitions
(cf. [4, p. 307, (i), (ii)]) there are the axioms:
0
+
0
0
0
0
0
(Tl.)
Prog [Mo, X )
M fl v C X
0
for each 3E, in I D . ,
7
F o r each a^0 there are uniquely determined main terms a > ••• > a„ ( n > 1) such that
a = ÜI + • • • + a . We call a . . . , a the components of a . 0 is defined to have no components.
x
n
u
n
121
WELLORDERINGS O F F O R M A L THEORIES
asserting the wellordering of M ,
0
(W 1)
Prog [Mo, Wo]
0
(W 2)
and
Prog [Mo, £]-» W o C l
0
for each 3E, in IDÜ-,
defining the accessible part W of M .
In order to treat I D l and ID<* simultaneously as far as possible, we refer to
both as ID* and define A to be the set { C l : x < v ) in the case of I D l and the set
{ i l : x E Wo} in the case of ID<*. Then A is a segment of Ä H (In, with 0 E A .
I n t h e sequel w, v a r e r e s e r v e d t o denote
elements
of A !
Define 2I[X, Y , x , y ] to be the formula g [ x ] A V x < x ( g [ x ] - » x E X ) ,
where g [ x ] Stands for Sx < ü A V Z , < y({(z , z,}: ZoE K < i x } C Y ) . Then
2l[X, Y, x, y] is an arithmetic formula such that each occurrence of X is
positive. T o 21 corresponds a set constant P (cf. [4, p. 307]). W e define
0
0
x
x
0
0
y
0
0
0
z
%
91
W : = { x : ( x , y > E P } and
a y
M :={x: Sx<w A V V < u(K x
u
CW )}.
v
V
Then the axioms (i) and (ii) of [4, p. 307] become
(Wl)
Vw E A ( P r o g [ M , W ])
u
(W2)
and
u
V K 6 A(Prog[M ,.J?]-> W C X )
M
for each £ .
u
(Wl) and (W2) assert that W i s t h e a c c e s s i b l e p a r t of M . Clearly for u =0, M
coincides with the previously defined set M = iT Pl C l and in the case of ID<*
the set Wo defined by (Wl), (W2) coincides with the set W defined by (W 1),
(W 2). A s immediate consequences of (Wl), (W2) the following formulas are
provable in I D ' :
(6) V x E W ( x E M A M D x = W H x ) , i.e. W is a segment of M .
(7) ö G W ^ T I [ M , J , a ] .
By (3) and the definition of M we get
(8) a,b£M -*a
+b E M
and a + fcEM —>fcE M .
The following lemmata 1-3 are straightforward modifications of corresponding lemmata in [9], [10], [11] and [14].
u
u
0
u
u
0
0
0
u
u
U
u
u
u
u
U
u
u
LEMMA
1.
u
u
(a) a, b E W —> a + b E W
u
M
u
arcd
(b) Sa<w A i£ a C W —» a E W are p r o v a b l e i n I D ' .
M
PROOF.
M
B y (6) and
u
(8).
a,
E W AV X E M
u
f l b(a + x E W )-> a + 6 E
u
M
M A M„ n ( a + fc)C W . Hence by ( W l ) , a E W - > P r o g [ M , { x : x + a E W }]
and thence by (W2), a, E W —> a + 6 E W . Part (b) is an immediate
consequence of (a) and (2).
u
u
u
u
L E M M A 2.
(a) a E W —> K a
u
v
u
u
u
C W
v
and
+
(b) v < u ^ > W = W r \ v
a r e p r o v a b l e i n ID'.
P R O O F . Suppose a E W A v = H . Using ( T I ) or (W 2) resp. we prove
KuöCW, by transfinite induction on x. F o r u < u we have K„flCW„ by
aE.W CM .
From w < i ; we get M C \ u C M ,
hence by (Wl),
Prog [M , {x: x E M„ - * x E WJ] and thence by (W2), W C{x: x E M - » x E
WJ, i.e. W n M u C W„. By the induction hypothesis we have K a C W
for all
v
u
u
x
P
0
+
u
u
v
M
u
u
u
v
w
w
W. BUCHHOLZ AND W. POHLERS
122
w < v and hence a E W n M C W^. B y (2), (4) and (6) we then get K a CM„ f l
(a + 1 ) C W , Part (b) follows from (a) by L e m m a l(b).
LEMMA 3. w E W is p r o v a b l e i n I D \
PROOF. W e have {x: £l E A } C W , which is trivial for I D < « and proved by
( T L ) and ( W l ) in ID'„. So for u = ( l we get x E Wo and thence by (5) and
L e m m a 2(a), K u C K x C W for all v < w, which implies u E M « . N o w suppose
a E M n w, then Sa E A D w, X a C W„ and by the lemmata l(b), 2(b),
a E W . Hence M D u C W and by (Wl), u E W .
DEFINITION. Q : = { X : 3 U ( I G W ) } = U
W ; M : = { x : V v ( K x C W )}.
Consequences.
1. Obviously Sa E A for all a E Q. B y Lemma 3 we then
have V x ( x E Q - » Sx E Q ) and Q H Ä = A . Hence by Lemma 2(b) for all
u E O
u
v
v
u
x
0
x
v
v
v
u
S a
tt
u
u
u
u
u e A
u
v
v
0
(9)
O H M
+
=
W
U
and M = { x : Sx < w A V W ( W E Q n u -> K x C Q)}. That means the set Q is
"ausgezeichnet" in the sense of [2, p. 18] with M ? H M = M
and W ? =
U
w
+
u
w [ M ? n
+
M
] =
w .
u
2. B y (6) and (9) P r o g [ Q , £ ] - > Prog [ M , { x : x E W —» x E £ } ] and thence
by ( W 2 )
(10) P r o g [ 0 , ^ ] ^ Q c £ for each £ ,
which is the first-order formulation of the fact that Q is w e l l o r d e r e d .
3. Since Q is "ausgezeichnet" (provable in I D ' ) we may follow the proof of
Theorem 15(b) in [2, p. 19] and get the formula
u
a E M A V x G M n
u
a ( Q C R ) ^ > Prog[Q, R ] \
x
a
where R : = { y : @ay E iT—>©ay E Q } . Here besides the premise " Q ausgezeichnet" only methods formalizable in Heyting's arithmetic are used. By
(10) it follows
(11) P r o g [ M , { x : V y E Q(@xy E 5T-^@xy E (?)}].
4. F r o m outside we know that ®({g}) = ( & , < ) is wellordered and hence
W = M = {x: Sx < u } and M = ST which implies Q = M D C l ^ er defined by:
DEFINITION.
a
u
U
v,
in the case of I D l ,
Hi,
in the case of ID<..
Of course W = M = {x: Sx < w} is not provable in I D ' , but the weaker
assertion Q = M C\ f l is provable as the following theorem shows.
THEOREM 1. C l ^ E M a n d Q = M f l C l a r e p r o v a b l e i n I D ' .
PROOF. B y Lemma 2(a) we have Q C M D
H a G M and Sa E A we get
K
CL CW
and by Lemma l(b), a E W C Q . So we just have to prove
a E M D
Sa E A and O e M . The proofs differ for I D l , I D U
1. I D l Then A = { w : w <Ü } and trivially aEMnQ, -*SaEA
holds.
By ( T L ) and ( W l ) we get a = v E W . Hence by Lemma 2(a) and (5)
V v i K v S l t r C K u O - C W ) which means
EM .
U
U
a
a
S a
Sa
S a
f f
<r
a
0
v
8
In [2] this formula is proved with Q in place of M, but an analysis of the proof shows that it is
enough to have the premise a E M A V X G M fl a ( Q C R ) .
x
WELLORDERINGS OF FORMAL THEORIES
123
2. ID<*. Suppose a E M H O r . Then K a C W
and Sa = O for some
x < 0 = (7. B y (5) K x CK £l
U{1}. Further K S a C K a and 1 E Wo. W e
therefore get by Lemma l(b) x E W and thence Sa E A . For 0 < u, 2C ft
0
and K f t =
Obviously ftG A and hence O E W , by Lemma 3. B y
Lemma 2(b) it follows V u ( K S l C W ), which means fl«, = ft ,E M
Now by (10) and Theorem 1 we get
0
0
0
0
x
0
0
=
0
u
0
a i
ni
n
u
a i
u
n
(12) T I [ M , 0 , ] .
Hence by (11) V y E Q(0O,y E JT-> © f ^ y E Q ) . Since 0 E Q A 00,0 E
ST H ft, we obtain 00,0 E Q n O , Hence by (9) and (7) T I [ M , 00,0]. This
means we are able to c o l l a p s e the wellordering M fl O , to the provable ordinal
©OrO . This is a special case of the following theorem, which is proved by the
above considerations with c in place of (!<,.
0
9
T H E O R E M 2 (COLLAPSING PROPERTY).
I f TI [M,c]
is provable
in
ID'
and
1
0 c O E 5 T , then T I [ M , @ c O ] i s p r o v a b l e i n I D .
Starting from (12) we now prove T I [ M , c ] for each c E M f l ö l O r using
Gentzen's [5] method for proving T I [ < s ] in number theory.
DEFINITION.
X : = { x : 01O, < x v V y ( M n y C l ^ M
f l (y +_©0jt)C£)}.
L E M M A 4. Vy ( M 0 y C X - » M fl (y + O,) C £)-H> Prog [M, #] is p r o v a b l e
i n ID'.
P R O O F . W e have to prove M H (6 + ©Oa) Cäf under the assumptions
(1) V y ( M H y C l ^ M f l ( y + n . ) C f ) , (2) M f l a C l ,
(3) a < © 1 0 „
(4) M H
c l
By (1) and (4) we get M fl (ft + O , • n) C X for all n E N using mathematical
induction. Hence M PI (fr 4- ©OOr) C.3f because of sup vOr • n = ©Oft,,.. Supposez G M n ( H ©Oa). We may assume b + ©00, < z. B y (3) z < + ©Oa <
b + © l O , . Hence z = b + ©Oa,- n + z, with 1 < n E AT, O , < a, < a, z, < ©Oa,.
By V u ( u < Or = Sa ) and the definition of K —it
is the case that K a =
J ^ Ö O a , C K z . So we get a E M P i a since z G M i s assumed. B y (2), (3), (4)
we get M fl (fr + ©Oai • (n + l))C.3f using mathematical induction. Hence
0
0
neJ
(
v
v
v
x
x
Z E X .
C : = C l ^ c„+i: = @0c„.
One easily proves c E M , c < © l O r = s u p c and @c„0 E 2".
T H E O R E M 3. T I [M,c„] is p r o v a b l e i n I D ' f o r each n E N .
P R O O F . W e prove the theorem by 'metainduction' on n . B y (3)
it
follows
that
a +b E M ^ b E M .
Hence
Prog [ M , £ ] A M
H a C
J-H>Prog[M,{jt: a + x E M - + a + x E X } ] and thence by (12)
DEFINITION.
0
n
(*)
n
Prog [M, X]
Vy(M
k e N
k
fl y C X - > M PI (y + O,) C £ ) .
By (*) and c = Or E M we have T I [ M , c ]. For n > 0 we have the induction
hypothesis T I [ M , c„_,]. Hence P r o g [ M , X] —> c _i E
which implies
Prog [M, l ] - ^ M D c C if.
By
(*)
and
Lemma
4
we
get
Prog[M, J ] ^ P r o g [ M , X ] and therefore P r o g f M , X ] - * M fl c„ CX. Hence
TI[M,c].
T H E O R E M 4.
I n ID', T I [ M , a] is p r o v a b l e f o r each a < ©(01O,)O.
0
0
n
n
0
9
Remember that by (1) M n0n O represents the ordinal ©("UO.
o
CT
W. BUCHHOLZ AND W. POHLERS
124
PROOF.
For a < © ( © l f t ^ O there is an n G JV with a < ©c„0. B y Theorem 3
we have T I [ M , c„], which can be collapsed to T I [ M , @c„0] by Theorem 2. So
T I [ M , a ] holds._
Since M f l © ( © l f l ^ O represents the segment { £ G O n : £<0£n<,+iO}, the
results ( A ) and (B) stated in the introduction follow from Theorem 4.
F I N A L R E M A R K S . T h e above proof of transfinite induction admits the
following generalization. Let T h be a theory containing Heyting's arithmetic
and axioms for iterations of inductive definitions along a provably wellordered
subset
A
of Ä . Then
the sets
W := W [ M ] ,
M :={x:Sx<
0
0
0
0
u
n « ( ^ C ^ ) }
u A V U G A
( H £ A ) ,
U
u
Q : = { X : 3U G A ( X G W ) }
U
and
M : = {x: Vw G A ( K x C Wu)} are definable in T h , and if the formula V u G
A ( u G W A V x G W (5x G A ) ) is provable in T h , one gets:
I. Q is w e l l o r d e r e d , i.e.
u
u
II.
u
Collapsing property.
c] ^> T h h T I [ M , 0 c O ]
Th h T I [ M ,
III. Extension
o
t o the next
(for ©cO G 3T).
e-number.
T h h T I [ M , n ] ^> T h h T I [ M , c ]
for each c G M f l 01O,.
a
From I, II, III it follows:
IV.
Thhft GMAMnft
f l
f l
CO
ThhTI[<0(0in )O]
a
for 0(010, )0G£:
A s an example we regard the following definition by transfinite recursion on
j/G^rno:
A(0):=0,
\(v
A :=0.
o
+ 1) := ß „ „
A (
A „ := A U { O : A ( v ) < x G W[M"]},
with M : = { x : x < \ ( v + 1) A VW G A ( K X
and W defined as above by iteration of
inductive definitions along A .
) +
+ 1
v
v
V
U
C W )}
H
u
v
10
A (*>): = sup^< ,A(^) ,
Av:= U^
i
< V
A^
for limit ordinals v.
Let I D ! be the theory, which allows to define A and to iterate inductive
definitions along A (ID* for example is ID<.). Then by the above considerations we get:
I D t h T I [ < 0(010^)0].
v
v
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v = o>(l + a ) it is ft
A(p)
= \ (v) = g l a .
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WELLORDERINGS O F F O R M A L THEORIES
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Systeme
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MATHEMATISCHES INSTITUT D E R LUDWIG-MAXIMILIANS-UNIVERSITÄT
MÜNCHEN, FEDERAL REPUBLIC OF GERMANY