REALIZABILITY A ND SHANIN'S AL GORITHM
FOR THE CONSTRUCTIVE DECIPHERING
OF MATHEMATICAL SENTENCES
S. C. KLEENE
/. I n [ 2] o r [ 5] § 82 Kleene int erpret ed number-t heoret ic statements t hrough a not ion o f «realizability», wh i c h c an be int roduc ed
in t he f ollowing manner.
Consider a f ormal language lik e t hat i n [ 5] b u t wi t h symbols f or
some f urt her primit iv e rec urs iv e f unc t ions a n d predicates (besides
, + , • , = ) . We merely assume t hat these inc lude a l l t he ones appearing below wi t h notations s imilar t o t he i nf or mal not at ions o f
[5] (index ed bottom p. 538). We also write ( a
o An algorit hm is giv en ([2] t op ID. 120, or § 5 below) wh i c h t o each
f, ormula
E „o)f t his
correlates
anot her f ormula r E o f t he
. . . , a
f language
o r
p
same
language. A closed f ormula E is s aid t o be realiz able, i f r E
o
ais °t rue; a n open f ormula, i f r v E, where v E is t he closure of E, is
.true.
. . .Thus
. p a closed f ormula is int erpret ed b y reading r E i n plac e
of
E itself.
a
Readers
o f I.51 § 82 w i l l not find r E. but only «E is realizable»
r i
in
n t he (inf ormal language. Thenc e r E is t o be obt ained by transla1tion i n t o t he f o r ma l language ( n o w mo r e extensive t h a n i n [ 5] ).
Specifically, r E is De[ e r E] where, i f E is closed, e r E expresses
)
.
«6' realizes E». I f E is open, e r E expresses t hat e realizes E f o r
given nat ural numbers y
yi
i,
and
yy „ ,
E(Yi,
i,a
sy m ) .
general,
,yt In h
e certain constructive properties are made ex plic it i n r E
which,
Kleene
are implic it in E f r om the int uit ionis t ic standy„v
a
l
u argued,
e
point
but
not
f
rom
the
classical. For this reason, r E is not in general
„,s ,
equivalent
t
o
E
classically;
f o r E ma y b e t rue classically wit hout
foo
f
ft
o
h
e
rE
f (
r
e
system
1
o f [ 5 ] , w e c o u ld us e t h e f a m i l i a r proc edures f o r e limin a t in g t h e
t;e
additional
)
symbols (151 § 74 Ex ample 9 p. 415), ex cept (f o r a reas on t o apov
h
a
r
i
pear
T i n § 5 below) us ing V ins tead o f 3 (c f . 151 * 181 p. 408). Th e f o rmu la
ra o b
e
l
e
in t h e p r o o f o f Th e o re m 27 m i d d l e p . 243 c a n b e replac ed b y v c d{ vu
isP3(c,
n
g d, 0, u) Q ( x
n2D e11(i, y, x
u
t,2 twlx & w < ( i ' • d ) ' (c f . [ 5 ] *180b), and i < y by v a t i * y i - a l .
m
hn, bx
e
y u ) ]
re,n154
t&, h
a
leVu e )
1 1
i
<
sri r l
yD e
m
e
Bw
D
xi(
Vi c
nu, t v
p
ro[d h B
l(, o c
e
,
sody u
those properties holding, s o t hat r E is false (ex ample i n [2] § 9 or
151 bot t om p. 513). A f ort iori, r E is not i n general equiv alent t o E
in t he p r i o r int uit ionis t ic f ormal systems o f number theory, wh i c h
are subsystems o f t he classical (t hough i n [ 2] Kleene proposed adjoining r E E t o such systems).
Subsequently ( i n [ 6] ) Kleene receded f r o m t he pos it ion t hat t his
realiz abilit y algorit hm r i s wh o l l y appropriat e as a n int uit ionis t ic
interpretation o f number-t heoret ic f ormulas ; i t enforces constructivity i n such a drastic f or m as t o leave no room f or relat iv iz ed c onstructivity. I n [ 6] h e int roduc ed anot her realiz abilit y not ion r'. Th e
algorit hm r' takes any f ormula E of int uit ionis t ic analysis (construed
to inc lude a sufficient c ollec t ion o f symbols f o r primit iv e recursive
functionals a n d predicates), a n d hence any f ormula o f t he pres ent
language, int o a f ormula r ' E of int uit ionis t ic analysis. Mos t of what
is done below f or r wi l l be applic able t o r ' also (
2
) . 2. I n [17] (
standing
3
o f mat hemat ic al sentences presented b y Kleene hav e, i n
our) opinion, r e a l shortcomings o f a f undament al character. Thes e
shortcomings
S h a n i c an be c learly seen i n t he s imples t examples.» Let F
benv x y z (x =y & y = z x = z ) . Should we writ e out r F, (4we would
obtain
w r an
i alt
t oget her c umbers ome sentence F
and
1 e c somplex t han t he original sentence F. Furt hermore, i n t he n e w
sentence
, « m oT we
r edo not reveal any features t hat would testify t o the fact
that,
c hu i nmesome
b erespect,
r s ot hemn eew sentence is s impler t han t he original
and
p may
r be
i taken as an int erpret at ion of the original one. I f now
wen s hould
c i apply t he dec iphering rules t o F
sentence
muc h more inv olv ed t han F
1 p l F2
e
1
f or
sentences
by Kleene are not idempotent.» (F
,rules
w
e
w o upresented
l d
s
F2
rn r dF,
,g1 is
a
i se
r t etc.)
sF ,o
o
n
.
a
o
proposes
T
t fShanin
hh
ie r
da dif f erent algorit hm, wh i c h w e c a l l s , wh i c h
dconnects
e c tihe pconstructive
h e r problems
i n g not wi t h a l l sentences but only
c
o
wit h some. Alt ering i n t his direc t ion t he basic theoretical apparatus
s mak es f o r great er int elligibilit y o f t he problem o f c onofnKleene
t
r
structive unders tanding of mat hemat ic al sentences t han has hit hert o
u atct ained i n ot her theories.»
been
t
i
v3. K l ee e n e aimed at s implic it y i n t he statement o f t he algorit hm,
u
while
Shanin aims at s implic it y in the results of its application. Each
n
d
(
indicated
e
in 141 and [61 is in preparation.
2
r)(') I t hank A . Alex ander Khoury f o r his t rans lat ion o f t his paper.
-A
m
o
n
o
g
r
a
p
h
[
7
155
k ind of s implic it y has it s uses, and t he t wo k inds are mut ually opposed.
Kleene's r E is g e [ e r E
seven
s imple clauses, eac h apply ing unif ormly t o a l l f ormulas t hat
]
have
a giv
, w
h eenr logic
e al s y mbol (or none) outermost.
Shanin
writ esr «The algorit hm o f exposition o f constructive probe
lems
E a n d t h e algorit hm o f int erpret at ion o f element ary f ormulas
[ whic
constitute s] a r e rat her unwieldy . » I n [171 h e only
i h together
s
illustrates
t
h
e
ac
d
e
f
i t ion
n oef s , f o r t h e det ailed des c ript ion o f wh i c h
thed reader is ref erred t o t he longer paper [16].
r
e
c
u
r
s
4. Kleene' s objectives did not relat e t o t he f or m of t he f ormulas
i
v
e
l
y
r E. Theref ore i t was permis s ible i n his context t o s implif y the f orb
mulas r E by use of any accepted constructive principles of inference.
y
In fact, r is idempotent, not «immediat ely » i n the sense t hat r r E
is (the same f ormula as) E, but «essentially» i n t he sense that r r E
is equivalent (int uit ionis t ic ally ) t o r E. Tak e the case (of primary interest f or t he int erpret at ion) o f a closed E. By Nelson's [14] Corollary Theorem 2 p. 323, r E D E is realizable (
E
E ) , i f r E is realizable, so is E; i.e.
41 p. 313 (since r E, r E
)r. r ES implies
o
br E.y S i mi l[a r l y1 , r4 E ]implies r r E (
5 h e 1 is
t o t he res ult o f an arbit rary substituTTheorem
o applied
r e instead
m
tion
of, E.) Anot her proof wi l l be
) • (ofF numerals
o r
o f orp t he
e free
n variables
E
given
[
1 i n §4 6 below.
]
The proof s o f s uc h equivalences mi g h t be c arried out i n stated
f ormal systems, as a c ont rol t hat t hey use only princ iples o f inf erence whic h are accepted as constructive, and also t o def ine any dif ferences i n t he extensions o f constructiveness.
It is k nown t hat a f ormal system S consisting of t he int uit ionis t ic
f ormal number t heory (as i n [ 5] ) wi t h t he rec urs ion equations f o r
a suitable f init e lis t of addit ional primit iv e recursive functions wi l l
suffice f o r f ormaliz ing t he us ual int uit ionis t ic ally developed t heory
of general and part ial recursive functions (
tion
i n this direction was Nelson's [14]. From wo r k in a preliminary
1
stage,
w approach
t o recursive functions may
) . T we
h ebelieve
o ra irecent
g i nn e a
l
offer
some
advantages
i
n
t
he
f
ormaliz
at
ion.
i n v e s t i g a (
ollary
4 Theorem 5 are stated in [2] as (I ), (I I ), (I I I ), and (I V), respectively (f or
the) notation, c f. p . 120). Theorem 1 is als o giv en w i t h a s imp le r p ro o f i n
[5] N
Theorem 62 (a) p.504.
(e
is (es
5l s entially ) idempotent. A proof, i n our notes since September 1952, was
intended
f o r public at ion i n F L
)s
E
o
156
D
n
r'
's
E
[
i1
s4
n]
oT
th
ae
lo
In t h i s s hor t c ommunic at ion w e mu s t c ont ent ours elv es w i t h
proofs presented inf ormally , alt hough t he results ma y be stated i n
the f ormal language. But it s hould be c lear t hat the inf ormal proofs
can be f ormaliz ed i n an S o f t he s ort described, o r (where stated)
in S wi t h Mark ov 's princ iple (§ 5 below) adjoined.
Howev er f o r t he (essential) idernpotence o f r, t h e f ormaliz at ion
of t he above proof is already av ailable i n Nelson's v ers ion S3 of S.
By [14] Corollary Theorem 5 p. 365, H r ( r E D E) (
4ollary 4.1 P. 361, r r E, r ( r E D E) r E. Thence by the deduction
)theorem,
. B y H [ r 1r E4 ]r E. CSimilarly
o r , - r E D r r E. Thus H r r E
r E.
5. S h a n i n does have an objective relat ing t o the f o r m of the f ormulas s E. He f inds i n Lorenzen [ 12] ideas by wh i c h i n his v i e w
formulas c ont aining only t he logic al symbols &, ,
y
can be
constructively understood. Un d e r Brouwer's int erpret at ion of V a n d
D, f ormulas composed by apply ing only these t wo symbols t o c omponents already constructively understood c an be constructively understood (
6formulas by f ormulas of t his sort (whic h, wh e n only g is applied,
he
) • calls completely regular; when V is applied only after 3, regular).
Acceptance
o f »Markov's princ iple» ([131, a n d lectures i n 1952-53),
T h i s
whic
g i h vwe c an f ormaliz e b y t he f ormula
M
e s
gives
IS
hh i m t he means f o r doing so (
t
7that
:a
nhe same essentially can be done us ing r.
)vi • nT h i s
se
g eno sreason
t s wh y & s hould not be inc luded wit h the sy mbols y
t (a)u Wge see
3
applic
able
t
o
t
he
components.
th
o
x
(
ue
s
[
0,
7 2, ...).oA ll these formulas M „ are imp lie d by M
ti- 1,
n
for
P (expressing a p rimit iv e rec urs iv e predic ate P). A n
Jv ) anye p rimer f ormula
i
f
y
7
c
equivalent
ion in different s y mbolis m is : —
, Ra se
if ormulat
s
—
v primit
A
iv e( xrecurs
7n
)
(iv e E(Kreis
x el).
) A ( x ) ,
?
e( x t ) A
( It
was
f
ound
f
ro
m
t
he
iN
v
pi y P
D beginning o f t he inv es tigations o f realiz abilit y i n
e1941
9e
l t hat t here a r e
y casesl i n w )h i c h t h e realiz abilit y o f a f o rmu la wa s
o n ly classically, inc luding cases wh e re wh a t is lac k ing int uit ionis t it,proved
a
xcally
c is t his princ iple t o in f e r t hat a v alue o f a p a rt ia l rec urs iv e f unc t ion
o
(p(x)
, i = 1.tyP (x, y) i s def ined ( y P ( x , y ) ) w h e n o n ly t h e abs urdit y o f its
d
indefinition
3
yP(x, y) o r - 7 t e y P ( x , y)) is g iv e n . O n e c las s o f
yn
e
examples dat ing f ro m February 1951 i s i n G . F. Rose's [ 15] p. 11. Cf . Re mark
c) g 2.5 below.
T
iD Especially i n c onnec tion w i t h t he author's inv es tigations o f r', i t bec ame
Di
apparent
t o h im (before he learned of Markov's 1 3 D t hat considerable int erp
(
y
h
e
157
T
e
,
ri x
(a ,
e
r y
,b )
xi b
t, y
ry T
) „
a
(
r]
e
Let us wr i t e t he def init ion o f e r E i n what seems n o w t he mos t
convenient f orm, c hanging it f rom [ 2] p. 120 or the direc t f ormaliz ation o f [ 5] § 82 i n respects immat erial f o r t he def init ion o f r E as
3 e[e r El (
sions
(s imilarly lat er).
8
) . 1. e r P is P (f or P a p r i me f ormula).
T 2.h ee r ( A & B) is (e)
r ur (AA V B) &is [ ( e )
b 03.o e
(0d ee= )r ( A 0D B ) i s v a[a r A D 3 y [ T
n 4.
&a e r r —7A
( r e i s) V a
v 15.
(
6.
[
e
,
e
r
a
v
T
ta b . ,x Ar ( x )y i s ) v x 3 y [ &
i B
7.
e
r
3
x
A
(
x
)
i
s
(
e
)
U
i
7
a
(
y
r
)
r
l Ae s l
1
A
&
Ve , rx] ,l .A y( l ( )e ).
a (B
THEOREM
1.
Fo
r
each
f ormula E, t he f ormula e r E is equiv alent
0
U
(
y
)
(
e
)
r [
to r)0a . f o r mu l a eAr
e
t3y(*TE x i ) n ]
.
t
or
not occur. (Then r E is equiv alent t o j e[e r
w1 0yh) and
i cV does
h
o
tg( tPROOF.
E
We def ine e r, E lik e e r E, c hanging three clauses (cf. [ 5]
&
b*158
and
]o,
c
ce§ 63 (61a)),
)
e (
.ux) 3. re r s
c t
o, ir
n
h attaches t o int uit ionis t ic systems w i t h t h is p rin c ip le adjoined, bec aus e
est
ly (B A y
o a v ariet y o f results wh ic h i t then ( o r only then) becomes possible t o obcf
i) V
. rec ently come to lig h t is Gbdel's res ult that, i f s trong c omtain.
s l A n ex ample
n( B
pleteness
o f the int uit ionis t ic predic ate c alc ulus is prov able int uit ionis t ic ally ,
e
) --7 (x
px is
a )A (x ) —4 (E x )--so
n
[91
andr [101).
7ra iARemark
( x t) 2.1
f o
t
s a c 110]
esn Kreisel
h s hows t hat t h e princ iple is n o t prov able i n t h e ex is ting i n o
tuitionis
p
i t l systems.
i v
od [ r (i ticm f o rma
A
dif
f
erent
b
u
t
relat
ed ex tens ion o f int uit ionis t ic systems, ex pres s ible b y
ea
e
)
fy
xA(x)
D s J x -7/ k (x )) f o r x ranging ov er t he nat ural numbers
rv
e
c
u
r
0
td
o vA (x )en o t necessarily prime, was proposed b y Kuroda i n [ 11] , k n o wn t o
ibut
hus
i =
i o n ly through Ohnis hi's rev iew.
A
es 0
(d (
c
f
D p.120; e.g. then e r (A & B) is equiv alent to e = ( ( e )
ft 8[2]
.in
c
(
i )(e)
°o
K
r
e
o Some
1
o f the differences between t he present def init ion a n d [ 2] 13.120 are
T
i,rln (ee )s
e
l
due
t o Nels on [14] p. 356. Th e equiv alenc e o f t he pres ent a n d f o rme r r E
tm
c hr)
l eB t hem r E and r
(call
)t t
&
(
ie rp.)Erimit
o
c•f
, iv e rec urs iv e func tions t E(e) a n d q
r
the
r-0vee res
E
s pulte E(Y)
cr t i ovf esubstitutingAany numerals Y f o r the f ree v ariables Y o f E,
i
(laE(e)
eya) ) res a liz
u ecs h
t h a t :
example,
ir° dAE h(s Y )(e;e ) n = F t
w
.Au
pia el Brn d
e
o
v
detail.)
{er&) ed h a
((rea
(w
l
i
z
e
s
[ n
0e
bbe}w
)(y158
, k
l•et h(h ( 0 ,
((dere(oe)eee) a f
1)isl )p), in z i
n
)e
ge)r0 s
,
+tf ° e*
s g
ctE(f e )
(o
0
0.ra( e
D
)(etY•r ( 1 , t
(
1S
ae) 3
s
ee )
(o
cr, (m
m
T
I
1hma)
4. e r
]
vy[T
6.
( Ae r
i
iD V
VYETI(e,
x, Y) D U(Y) r1 A(x)11•
(e,
xB A
a ,
()COROLLARY
x
1.1. U s i n g Markov's princyiple M
is )iiequiv
( aalent
n d t o a[ f ormula
5 ]
e r
)
2 is* E 8r E3i i snaequiv
(Then
) alent
,
t o 3e[ ee r
D
w sV
2
r Eh i c h
E
U
wit hout c larif y ing, s ome
]V V
a6. Shanin' s objec t ion t o r t hat i t alters,
(
simple
,a x[ w] nfhormulas
i d
c hsuggests t o us t o v erifyy that, f o r a c ert ain class o f
E's
ig g
s luding all regular f ormulas ), r E) is equiv alent t o E. The (esa (inc
sential)
idempot
cd yr o m
op l enc
e e to f er wil l l ybec ome ar n applic at ion o f t his result.
rn T
e
i
g
o
u
tl
a
r
i
i
f ormula E(Y) c1ont aining f ree only t he v ar.o (ALEMMA
)e c 2.1a.c T o each
u
iables
r ,D .Y and not c ont aining V or g, t here
3 is a number e
for
E x3 each
s u cchoice
h
t o fh nat
a ural
t , numbers Y ] ( wi t h c orres ponding numerals,) Y):
]
(i) I f E(Y) is realizable, t hen E(Y) i s true.
Y
1
.
)1(ii)a I f E(Y) is t rue, t hen e
PROOF,
E r e bayl induc
i z e tsion oEn t he n u mb e r o f logic al s y mbols i n E(Y)
&
.
(briefly,
as E is of the f orm P (a prime f ormula), A & B,
( Y E).
) According
.
1
A D B, —7A, o r v xA(x), let e
(
respectively.
O n( e case
wi lAl suffice f o r illus t rat ion.
E
0 ,
e
e
, CASE 4: Ee is A D B.1 Then
3 e
,E =
say
A
e it. Suppose A(Y) is true. By t he hypothesis
)realizable;
, ea realizes
of a
Bt he induc
t ion (ii)a,
A
a
e e
in
3y[T
A ,.t rhee faormal
( l i zi language,
e )s
B
y
ind.
(
i
)
,
B
(
Y
)
i
s
t
rue.
Th
1
,
o s eu s A ( Y ) D 13(Y) i s t rue. ( i i ) a Suppose
,A S ( uY p) p
)
A(Y)
D
B(Y)
is
true.
Suppose
a realizes A(Y); t hen by hyp. ind. (i),
(0
e
,
e
s A
o (,
Y
)
&
A(Y)
is
true,
so
B(Y)
is
true,
and
by hyp. ind. (ii)a, 61
A
{ De } (r e
o
Thus
A
a
C
3
r
e
a
l
i
z
e
s
B
(
Y
)
.
,A
y
)
&
AB
x
B
U
(
y
)
) (
Y
)
e
rr eiLEMMA
ael i z2.1b.
ar Bl T
s o ieach f ormula E(Y) c ont aining f ree only t he v arA
(
e(ziables
s Ye Y,) a
n o 3, o t h e r t h a n i n part s o f t h e f o r m
,
s njd c )ont aining
)
,
3xP(x)
wi
t
h
P(x)
prime,
and
no V, t here is a part ial recursive funcA
a
n
d
B
tion
E
(b
Y
y
(
Y
(i) I f y E(Y) is prealizable, t hen E(Y) is true.
)E
h
)
I f E(Y) is true, t hen E
)
D
.(( Y(ii)b
i
E
s. u
B
e
( hYIY)n t h(e ic orres
s ponding le m m a f o r r ' , i s admit t ed als o i n part s o f
(c. (g)
thedf o rm
ac
tP(a)
e
f
i
n wei t hd u a f unc t ion v ariable and 1)(a) p rime .
)t h .
n
d
)
a at
159
r
e
a
l
i
z
e
,
E
f s
(
Y
)
.
o
r
e
a
c
h
Y
(
PROOF, by ind. Ac c ording as E(Y) is of the f orm P, A & B, A D B,
—7A, y x A( x ) , o r 3x P(x ), let EE(Y) 0 , ( W Y )
0,
, A
ExMsI
pressed
iTk ( x ) b y t he p r i me f ormula P(x).
() x) ,,
A
a
E B ( Y ) ,
2.2. F o r each f ormula E composed wit hout us ing D o r y
Y LEMMA
)
f, rom components C
to
r
C
lo t he f ormula composed correspondingly f rom r CI,
1
( (" ) n. Boy induc
t ion, i t w i l l s uf f ic e t o v erif y t h a t r ( A & B),
,r PROOF.
t
A V
r j x A ( x ) a r e equiv alent t o r A & r B,
n
c B),
e sr sa n d
(r ([e
ri Axr V i r lB, y 7 r A, and 3 x r A(x ), respectively. For example:
a
p
r 3: i E ismA V B. P A RT 1. As s ume r ( A V B), i.e., f or some e,
P CASE
[( (e)x )
e
,
0
r,(e)
(e)
=
1
E
Y
r0 A, i.e., f o r some a, a r A. Then ( 0 , a) r ( A V B), s o r ( A V B).
i)i)
CASE
B: r B. Similarly .
&
s,*A
0
e
q
u
0.(
e THEOREM
T
i)&
v
a2. F o r eac h f o r mu l a E i n wh i c h n o 3 , ot her t han i n
parts
(h
)
lw
eof then f orm 3x P(x ) wi t h P(x) prime, and no V lies in the scope
of
any
D or v :
e
th1
(iii)
r
E is equiv alent t o E, i. e. E(Y) i s realiz able i f and only i f
)n
er
E(Y)
i
s
true.
1
r(A
r] PROOF. Cons ider t he mi n i ma l components C
ee
stand
ins ide n o D o r y . Thes e c omponent s c ont ain n o 3 , ot her
I
B
F)V
i
n
,
C parts of t he f o r m 3x P(x ), a n d n o V. B y L e mma 2. 1a ( o r
.than
(,[Lemma 2.1b i f C
i
o
f
E
S
xr(i= 1 . . . . . .
But cE ishcomposed f r o m C
w
h
i
iA
,e
ycI . oB ny tLemma
a i n s2.2, ( i i i ) holds .
m
,)
Y
,ip aC r t
0
)w
w i 2.3.
t hT h oe cuondit
t ion on E i n t his t heorem does n o t c ov er
sl*i REMARK
ih
u
all
cases
s
i
n
i
whic
n
h
(iii)
g
holds.
For, i f E, is equivalent int uit ionis t ic ally
3
a
0 x P
se
to
E
by
inferences
f
ormaliz
able
i n S, t hen r E
D
r(& x ) )
tn
l,o i s Nels
using
e q
on'
us i [ 1v4 ]a Th
l ee onr etm 1 o r f o r ma l l y [ 1 4 ] Co r o l l a r y 4 . 2
r
l
hc(trp. 362 o(
r
y
ee
to
4
E
E
,
C
.
pr)then
li) ;
E
P
,part j x
rA
l.ia i ns
A
e,rmight
1
(sd Cq hav e V's and j ' s wh i c h c ould b y int uit ionis t ic equivalences
e
R
B
dw
.u
o
h. i.ex
e
u
T
l„ A ( xq
ih
vn
n (")
ac vI n thea corresponding lemma f or r', —
ie
2
c.1
a
lvlemus
e , t als o be excluded.
g
a
.C 7 ea n dn
an
,n
it frt o r
e
a
A
tcxtstf160
f u n c t i o n
sS
ee
„o
vl
a
r
i
a
b
l
e
e
E
o
s
erE
)a
yC
s ,
u
xA
in
b
a
im
:s
-V
y(t i
E
e
r(e q
sT
s
irB
e
u
hf ii
a
A
.)v a
tee
V
0
C
lios e
rA
=
n
t
srt
B
be brought int o 3 -prenex parts, o r be remov ed by advance across
7
' COROLLARY 2.4. F o r each f ormula E:
s (iv) r r E is equiv alent to r E.
(
[ REMARK 2.5. W e hav e s tated t h e t heorem s o t h a t applic at ions
5do not depend on accepting Markov's princ iple M
M
]I
tplied
by, it st o wn
h erealiz abilit
t h ye(by
o reas
r eoning
m ,f ormaliz able i n S).
*. B y
i
6
s 7. S h a n i n gives n o indic at ion i n [17] t hat he has considered t he
3question whet her s is «essentially» dis t inc t f r o m r i n t he sense t hat
e
,s E is not alway s equivalent to r E. The c urrent situation, i n whic h
q
*several pairwis e es s ent ially dis t inc t int erpret at ions hav e appeared
u
8(the ident ic al interpretation, r Kleene [2], r ' Kleene [6], Gadel's int eri
6pretation [1], [8], and others in Kreisel 181), surely makes it of interest
v
)to k n o w whet her Shanin's s is s t ill another. I n § 8 we s hall s how
a
.that, us ing Markov's princ iple, s is «essentially» equiv alent to r, i.e.
l
s E is alway s equivalent to r E. Two preliminary difficulties face us.
e
First, we mus t arrange t hat s operate i n t he same language as r.
n
Shanin illustrates s starting i n anot her language i n whic h t he t erms
t
are b u i l t f r o m numerals a n d v ariables b y repeat ed us e o f (essent
tially) Kleene's universal part ial recursive functions { z } ( x
o
t
1_1([1yT
r
mulas
, x are ! t («t is defined») and t
n
M
i(z,
where
7 = - x• tt, t
t
l e t e problems » wi t h i n t his language, a n d t hen
1(position
, )c o mo f pconstructive
n
,
e
applies
q
«t
u
he
a
algorit
l i hmt o fyint erpret at ion o f element ary f ormulas » t o
,, t ,
iexpress ! t and t
[2y )
5
]
.i In
t
Shanin's
f
irs
t language o f course he c an express a l l primit iv e
§
a) r) e
erecursive
functions
2
i
n
6
3
)and predicates. The fact t hat he mus t i n his f irs t
a[
n
.algorit
hm
so
express
ev en such s imple predicates as = a n d < rea
y5
,quires
more steps i n his second algorit hm. There c an be
lt] ae ni n general
g
M
u
a g t ion
e f r o m t he standpoint o f t he int erpret at ion t o adding t o
rno
§ objec
m
aShanin's
ls6 . i f irs t language a l l our symbols f or primit iv e recursive f unc rtions
and
k5
e predicates.
H
ko No wu we have a language wi t h bot h his symbols and ours. But his
e.
} ( ), ! , w h i c h ours lacks c an be expressed i n t erms
orsymbols
s
fH
vof
ours.
I
n
the case of no nesting of 1 ( ), equivalents i n ours of ! t
i.i
' and t
rsExample 2 nex t page. Th e R' s whic h these equivalents c ont ain c an
si t
sp
2 remov ed as i n t he p r o o f o f Theorem 1 a n d Corollary abov e,
pbe
tr
ra
ai
161
ir
pm
ne
pe
cg
l(
i
ii
eo
pv
sr
le
««
en
te
ib
hl
y
m
ee
p[
granting Markov's principle. Nested uses of ( ) ( ) c an be eliminat ed
progressively f r o m inside, s imilarly t o [ 5] § 74, but us ing v instead
of 3 ( c f . [ 5] *181). This essentially is Shanin's second algorit hm (").
We c an t hus int erpret eac h f o r mu l a i n t he language hav ing bot h
his symbols and ours as s t anding f or a f ormula i n ours.
In s ome steps his f irs t a l g o r i t h m int roduc es n e w instances o f
}( ), ! , W e c an ins t ead pass direc t ly t o t e r e s u l t o f t h e i r
eliminat ion (as indeed is done i n def ining e r E, e r
compared
E(Y)» v erbally ).
i E
o t o r def ining
e «e realizes
r
2 In brief , hEe applies his t wo algorit hms successively. Instead, we
apply h i s second algorit hm int ermit t ent ly , s o as t o r e ma i n i n o u r
language t hroughout t he applic at ion o f his f irs t algorit hm.
Our second preliminary dif f ic ult y is t hat his t wo algorit hms are
described i n det ail only i n [16], o f whic h we hav e seen only Hey ting's rev iew. But t he illus t rat ions o f t heir ac t ion i n 1171 a r e s uf f icient t o enable us, we believe, t o set up an algorit hm s whic h mus t
agree wi t h his t o wi t h i n inessenfial det ails wh e n t he languages are
brought int o correspondence i n t he manner jus t indicated. We use
the s light s implif ic at ions wh i c h o u r s y mbolis m af f ords wi t h i n t he
spirit of his procedure.
8. N o w we f i n d f o u r general differences bet ween Shanin's procedure and o u r r. (1) Hi s ut iliz es Mark ov 's princ iple t o end u p i n
[16], ac c ording t o Hey t ing's rev iew, wi t h c omplet ely regular f ormulas ( i n [17] wi t h regular formulas). (2) His i n [16] leaves completely regular f ormulas ( i n [ 17] r e g u l a r f ormulas ) unalt ered, a n d a l ters regular components o f irregular f ormulas less t han r does. (3)
His handles consecutive occurrences o f a giv en one o f 8, V, V,
in o n e operat ion (")• (4) Hi s does n o t contract pairs ( o r n-t uples )
of nat ural numbers int o s ingle nat ural numbers (").
In §§ 5 and 6 we hav e seen t he way t o bridge t he differences (1)
and (2), respectively. We arrange o u r t reat ment t o al l ow t he choice
(A o r B) o f f ollowing [16] o r [17], and als o t he choice ( a or b) o f
assuming Markov's princ iple or not. Shanin calls a f ormula normal,
if i t contains no V o r 3 ; we c all a f ormula s eminormal, i f i t contains n o 3 , except i n parts o f t he f o r m 3x P(x ) (P(x ) prime), a n d
(") He requires a n ex t ra s tep because
property E i
ready
built into the representing function
n
( z ,( x
ily
i 1not ion i n [ 3] (c f. P I pp. 503, 508).
2 x Is B x
, (")
)
n 0
F
...x
, 162
n
y o
) rA ( x
p
= 0
0 r,
x
& e
n
x n
i )e
, xo
vr
'E x
he does n't assume f o r h is e „ t h e
x
of no u r predic ate T
into
n , the handling of the realiz abil( zw
,x
l )(x )
, =nx
n 0)
, ys
y =i )
wm
,p
wl
he
ir
c?
h
i
.
no V. A l l o wi n g s eminormal instead of only normal components, we
obtain completely semiregular (semi regular) fo rmu la s instead o f
completely regular (regular) ones.
Under Choice b, t o each f ormula E we n o w correlate a lis t b o f
variables d
e
i r E expresses v e realizes E(Y)» a n d r E expresses «E(Y) is realiz ,able»,
d we can read b s E as ob solve E(Y)» and s E (below) as , ,
is
solvable».
iE ( Y )
( 1.1I f E is s eminormal, b is empt y (/ = 0) and b s E is E.
> In t he remaining cases E s hall not be s eminormal. W e wr i t e a
0for t he lis t of variables a
13
)l
1, 2. aI f E is A
a
,conjunction,
r e b l isaa
n„ I &c o r t hen
blt 3.
e
I
d
f
E
is
A
&
d
t hen b is u, a
»disjunction,
a,t lA a V ko
lkA A
,[u
0
D
a
t
s dA
,
w
h
a
n
s ae
a r n
cl,b
a
k
e
m 4.k iI f E is A D dB, t hen b is b and, i f n 0 , b s E is
n
d
olksb .e& a
n so
b
rVa[a
fr& w
E
k ms hA oD 3 YIT,,(13
A
r,sir[I Te
> lu rs
a
for
d
m = 0 reduces t o ' , iot a s A D s Bj); but i f n 0 , b s E is
n
E
e1,
t=
e
1 aD, b syBhi. )
fa
s A
.
(
b
i ka
s
l&
ole
ki 5. I f E is —7A, t hen b is empt y and b s E is v a [ a s A] .
a&
lr a
i
s
n
D
I f E is yY xAmwhere x consists o f k > 0 v ariables a n d A does
, 6.
t3
s
n
t
A
d
m
a
k wi t h y , t hen b is a and b s E is y x[3 y
begin
ad
e
enot
1
n
ui_
,3 YnTk(an
dr&
on ( b m ,
lT, In
s , A l l ( wh i c h f or n 0 reduces to V X[s A] )zIgyn)
)
ita
Yo
&
aky n
7.
I f) E is 3 l A where x consists o f k 0 v ariables and A does
t
t
ios
y n
,
,
e begin
ba
not
wi t h 3 , t hen b is t , a and b s E is a s A.
(A
,k)&ae
)k
B
o
slY i . •
.-&
s& o
fLEMMA 3.1b. T o each fo rmu la E(Y) containing o n ly t h e va rE
,Y
. xfy, y i
&
• are partial recursive functions YE and b
i.A
iables
A Y,
] there
1
„ (y)
[ T I• f b solve
T
•kE
Y
A
l
s u c h
tE(Y),
h athen
t :y
J&
)»n
kI
.u> (vi)
,
E
I)f&e realizes E(Y),& then bE(e, Y) (is defined and) = • • •
( a
where
,( b , dY )
( i s
skl ,A
itb PROOF,
A
kd 1 e) by
f iind.
n o en t he
d number of logic al symbols i n E, wi t h cases
-,corresponding
t o those i n t he def init ion o f b s E.
,a kiad
n
d
)
2
Th e n b i s empt y a n d b s E is E. ( v )
isx» isrCASEe1: Ea is sl eminormal.
i
z
, s YE(Y) = E
Let
a
s eo s
E
Y
Q
and
av b y ( i i ) b E
lE
Y By )Lemma 2.1b (i), E(Y) is true, i.e. c i
i(,EY( )
realizes
E(Y).
e
solve
t()y( Y.L) E(Y);
e m and
m (f or / 0 ) ( c l
E
i,a&
ir ed a l i z e
(,i2•n
163
/
0
s c . / f E1 o b r
Y
/).)( .Y ) .
)I.U
( . v =
1f&
(i
).
(byL
„e
sa
o t
l„)b
v
e,sE
x1
E
(
e
,
(,3
Y
Y
CASE 3: E is A
b is
l u.V a A
lYE(b
h
,if
, ,u 1 (CASE B) , ( 1 ,
Y ) )
i f u > 1 (CASE C) . Supapose
Ye
2) , b tsolve
c E(Y). I n Case A, t hen e
A
al((y),
.s ,yoAl v e
l S t E(Y).a Th e ot her t wo cases are s imilar. ( v i ) Say e.g. n
el0izes
(1
( ayY. =) , 2 , s
o
c,n
2
b
(if
(l e(e) y.
=
h
, g B),
y . ( 2p , 0, .0, ( 8 A
v0
0(CASE
realizes
3
0
i,Y E) nE(Y). Idn Case. A, ( e )
)=
1
a
(,y(r (eiee )
U
solve
(bAl((e)i,
i, Y))0, (6A1((e)1,1
,s2
1
r e a so
l i 0,
z e
s
)n
a sIl i AI(Y),
A
Under
Choice
a,
we
modif
of b s E t o mak e the
,3
Y
)
)
1
,
0
s
oy the
l vconstruction
e
)i1
A
Y
t
)
(
)
Y
)
lz (. e
result
alway
s
n
o
r
ma
l
:
f
i
r
s
t
replac
e
eac
h
p
a
r
t
o f E o f t he f o r m
E
(
Y
)
.
snw
0
o
=
0
(s A
3xP(x)
(P(x f) prime)
by - 7 V X
b
h
)
i
\
1
g=
aA l
under
Choice b, and f inally replace each resulting 3 yT by 1
.l(l V e )
0
[e
To
7
1
distinguish
this
f rom the f ormer b s E, we may writ e t hem b s
(5r(L
>
( A(Y e
,0
and
b s
a
3
E
l]e
0
C
) 2,
Y
h
ted
, t the
y
( X )Ef,rom
hA functions
e Sn
.Ta
t)(s ) C
A
r eWe
E
ap
s giv
n pde f our
8 definitions
of s E. Say E is E
a
l
y
1
,
b
o
hlS
)E
and
pV
E
le none
cV h oEf eE
t
Y
) )
(Ba,
e,E
)C
A
3 sb[br s„e eE
ct,t. i volr bBb,w
a us Ehutis e
=
a
(i0
oA
],M
e[b,3>l sE
ya, E r k o v
1
s
(r2
u init
a ion,
p
l p b is empty, b s E as defined earlier and s E as
)e
,,ii . 3 def
fS
each
b [ b when
s
'.u s
s
e
,es,uo
(i] ,T s s n o wec oinc ide; s o t he not at ion i s consistent. Respectively,
defined
Id
p
r
i
n
c
i
o
Y
=
m
0e
hr is
s0
is E wh e n E is ) c omplet ely regular, c omplet ely s emi,ao E
n
p
l E ( and
e
l()(regular,
,]d3 e i s regular,
s emiregular.
X
j u
iL
v1
(C
y,nb n c t i o
X
3 As s umingb Mark ov 's princ iple, i f Def init ion A a o r Ba
sE THEOREM 3.
,0
A
(e
A
. s
in
i chosen,
M
a
sr eachu f ormula E:
is
f
o
,S
0
l[U
cA
[ b
b ens E is
M
m
(vii)
d equiv
. alent t o r E.
(idi
)I,(E
e U ns d e r Def init ion A a (Ab), the t heorem is immediat e f rom
A PROOF.
b
n
p(0
A
„r3S
E
Lemma
3.1a (3.1b).
Under Def init ion Ba (Bb), i t f ollows thence b y
,k)Lemma
ilD
, E 2.2.
.Y
.b
i,Y
3)0
I
]FE
1
3,)(()a
N
I
V
l
BIBLIOGRAPHY
7i(0),V
Tl
I
)3
,.fe
,PI
V K uNr t G6DEL, Ober eine bis her noc h nic ht beniitz te Erweiterung des f initO
()((A
ten Standpunktes, i n Dialec tic a, v o l. 12, nos . 3/4 (47/48), 1958, p p . 280le
bV 287.
h
1
e.e
1=
,a
3 S . C. KLEENE, O n t h e int erpret at ion o f int uit ionis t ic n u mb e r theory , i n
e
r21
t().,y,pb
J ournal o f Sy mbolic Logic , v ol. 10 (1945), pp. 109-124.
b
1
0
-E
A
ai S . C. KLEENE, O n the int uit ion istic logic , i n Proceedings o f the Tent h I n[3]
A
-,y(b
r[
l-Y
b
A
,ab
164
(-)2,
,
m
(-)(Y
eS
e
-0
0)tb
)0
,2)
eE
t&
(,r
ri
,(6
e]
Y
s
te
\)a
,
Y
,)1
)lar
Y
()1
re
)
ternational Congress o f Philosophy (Ams terdam, Aug. 11-18, 1948), A msterdam (Nort h-Holland), 1949, pp. 741-743 (fasc. 2).
[4] S . C. KLEENE, Recursive functions and int uit ionis t ic mathematics, i n Proceedings o f t he I nt ernat ional Congress o f Mathematic ians (Cambridge,
Mass., Aug. 30-Sept. 6, 1950), 1952, v ol. 1, pp. 679-685.
[51 S . C. KLEENE, Introduc tion t o Metamathematics, Ams terdam (No rt h -Ho lland), G roningen (Noordhoff), Ne w Y ork and Toront o (V a n Nos trand),
1952, X-550 pp.
[6] S . C. KLEENE, Realiz ability , i n Summaries o f talk s presented at the Summer Ins titute o f Sy mbolic Logic i n 1957 a t Co rn e ll Univ ers ity , v o l. 1,
pp. 100-104, reprint ed i n Cons t ruc t iv it y i n Mat hemat ic s , A ms t e rd a m
(North-Holland) 1959, pp. 285-289.
[7] S . C. KLEENE, Th e Foundations o f I nt uit ionis t ic Mathematic s , i n preparation.
[8] G e o r g KREISEL, Interpretation o f analysis by means o f constructive func tionals o f f in it e ty pes , i n Cons truc tiv ity i n Mathematic s , A ms t erdam
(North-Holland), 1959, p p . 101-128.
[9] G e o r g KREISEL, Elementary completeness properties o f intuitonis tic logic
wit h a not e o n negations o f prenex f ormulae, i n J ournal o f Sy mbolic
Logic, v ol. 23 (1958), pp. 317-330.
[101 G e o rg KREISEL, T h e non-deriv abilit y o f ( x ) i l ( x ) _ _ _ , ( E x ) 7
primit iv e recursive, i n int uit ionis t ic f o rma l systems (abstract), i n J ourA ( . 1 nal
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The Univ ers it y o f Wisconsin.
Received April 28, 1960.
S. C. KLEENE
165