A NNALES
SCIENTIFIQUES
DE L’U NIVERSITÉ DE
C LERMONT-F ERRAND 2
Série Mathématiques
R. O. G ANDY
General recursive functionals of finite type and hierarchies of functions
Annales scientifiques de l’Université de Clermont-Ferrand 2, tome 35, série Mathématiques, no 4 (1967), p. 5-24
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GENERAL RECURSIVE FUNCTIONALS OF FINITE TYPE
AND HIERARCHIES OF FUNCTIONS
R.O. GANDY
UNIVERSITY OF MANCHESTER, ENGLAND
INTRODUCTION
Kreisel
(in GK1)
has
that if
argued
then the number theoretic functions which
(HA) functions.
tinguish
all the
But when
we
N of natural numbers
predicatively defined
exactly
are
hyperarithmetic hierarchy
we
defined from above
(1).
On the other
Kleene’s 0
or
Spector’s W)
can
where A is arithmetic. This is not
ranges has not been
because
each
object
given
be
a
definition because the
previous definition.
in the range of the
prescribed
in this
number theoretic function which
lecture is to
can
hyperarithmetic
nevertheless, dis-
less than
over
6
form
model
a
N to be constructed
in the form
expressed
But it is
quantifier has
defined. I believe that Russell would have admitted it
of the work
basis ,
are
predicative
a
a
surely thoroughly impredicative ; a is, as it were,
hand, Spector has shown (in CS) that a complete 11:1 predicate
definitions
These
can,
the
as
the least ordinal such that
arithmetic, and let S be the first binary relation
J .
below,
the
totality
degrees of impredicativity. For example, let 7 be
constructible (sensu G6del) number-theoretic functions of order
having order-type
tifier
be
takes the
different
of classical second-order
(e, g. ,
can
beyond
go
one
as a
study
a
been
totality
over
definition,
to
speak,
from
previously (and predicatively)
non-circular definition.
and to characterise
be defined from below
so
which the quan-
more
One of the
exactly
objects
the class of
(2).
---------------
(1)
I do not hold that there is anything unsound or philosophically objectionable in the use of thoroughly impredicative definitions. But it is clear (witness the recent independence results obtained by Cohen) that the more
impredicative the definition of an object or class, the less the information which we can expect to obtain
about it from that definition.
(2)
now regard this as a very fruitful programme.
If the operation of collecting together all previously
defined functions is only repeated a finite number of times, then one obtains just those functions which are
primitive recursive in E1. If the operation is repeated any transfinite number of times then one obtains the
functions of ramified analysis. If one seeks to control the transfinite applications of the operation by requiring
that the ordinal used should be the type of a well-ordering which has already been obtained, then I believe
that the functions one obtains are just those which result from applying the operation less than ~o 0 times ,
where C. is the first ~-number as defined in section 4.
I do not
More promising, I consider, is the programme of successive diagonalisation which is considered in this lecture. But for functionals of type 3 or greater, D aax (see section 5) is thoroughly impredicative. Therefore,
if one seeks to obtain a more detailed knowledge of particular hierarchies of functions by avoiding operations
which are thoroughly impredicative, one should investigate the functionals which arise by application of Dmin.
I have found this difficult to do.
·
5
However, the line of attack
the class HA of
(in SCK1 )
function
This
beyond
at the results of
namely
rather indirect one.
expressing
if
should look
we
Let
first consider
us
of N is to per-
presupposition
by
[E] which
wish to extend
we
In other words,
diagonalising [ E ].
we
are
our
are
recur-
notion of
to consider
predicate
D(x,E) ={x} (E)
(E)
Where { x }
generally,
we
denotes the
recursion with index number
partial
It is not hard to show that the former
here and that the latter
in
defined
T)
’Equivalent’
is
predicate
means
E)
a,
and
type
2
argument E. Or more
{x} (a, E) is defined.
=
has the
predicate
that each of the three
Turing degree
as
Kleene’s 0,
E1 (introducel by
as
Tu-
equivalent
D, where
to
investigation
principles :
plication of the following three
(i) use of functionals of finite type ;
(ii) use of general recursions applied
diagonalisation
over
is recursive in either of the others .
objects considered
These considerations suggest
of
same
to x EOa and to the functional
equivalent
D(x, a) = {x} (a)
use
x
by
We remark that E itself is
(iii)
is defined,
consider
D(x,
gué
the
that HA coincides with the class of functions
suggests immediately where
HA -
a
defined
of Kleene’s functional E
use
Now Kleene has shown
the
One way of
hyperarithmetic functions.
mit curselves the
sive in E.
to be followed here is
is defined.
of the class of functions
to such
generated by repeated
ap-
functionals ;
generated by (ii).
classes
2 - RECURSIONS IN FUNCTIONALS OF FINITE TYPE
In this
section
we
cursive functional has
over a
and
a
of other p.
a
r.
particular
f. s. , by
brief resume of
0 is N and pure
of
single argument
a
numerical value and
a
type - type
pure
give
a
type
n.
A
type
partial
particular
a
n
of the definitions and results of SCK1.
finite number of
+ 1
consists
one
A
re-
arguments ; each argument ranges
of the functionals with numerical values
recursive functional
instance of
index number is allocated.
some
(p.
r.
f, )
is defined,
of the schemata S 1
-
perhaps
in terms
9 ; to each such instance
This number encodes the number and the
types
of the
ar-
guments of the p.r.f. which is being defined, the number of the schema, and the index number (s)
of the p.
r.
fs. , (if any)
S 1 - 3
result of
define
substituting
in terms of which the definition is
respectively
the value of
the successor,
a
p.
r.
f.
for
a
being made.
constant and
numerical
6
identity functionals.
argument of
an
other p.
S4
r.
defines the
f. S 5
is the
schema for
n +
the numerical value of the functional for
primitive recursion ;
1 is defined in terms of the value for
in KG. Instances of S 6
(and
each instance of the other schema
gument
as
applies
a
where
the relevant
to
type j argument
denotes
1t
string
a
exJ,
u )) for particular
for every functional
considered
as an
guments is the
in
arguments ;
of
i
are
S 7
one.
n -
the arguments of
permute
arguments of appropriate type
argument
their index
type
a
of
arguments,
and
u ,
is said to be
S 9 is
to
argument
x has been
def ined
if the
value of the p.
f. with index number
r.
c
one
being
symbols :
computation of
a , u ) is defined
The
X (1;J-2,
of
,
of the
0
type
defined for
specified
a
by G6del
dropped if in
specify freely which
always takes the first arargument. S l3’ (with j ~ 2)
0
computation
for
be
can
previously defined.
and the value of the functional
index number,
schema
type
a
reflection schema ;
a
argument
allowed to
are
we
defined functional ; in
given previously
type j-2.
given type ; this
numbers)
1
numerical
this contrasts with the schema used
the relevant ones ; whereas Kleene
applies
a
a
a
arguments
given
a
is
c
set of
ar-
selection from that set of
symbols
(u)
Naturally
such
specified
list of
a
computation
automatically
total
S5
or
finitions
By
computation
a
Functionals which
mitted)
defined
If
a
out
p.
r.
right
use
by
equivalent
are
the
description
to the
of
a
stands
an
is defined. In
the result of
of S 9
arguments.
are
a
index number for the
conjunction with S 4,S
recursiue ; they
For the definition of p.
r.
fs
(S9
per-
f. is defined for all values of all arguments it is said to be
customary
computation
{ z} (11.) both
9
subsidiary computation.
called primitive
teneral recursive functional. When the types of the arguments
a
there
are
all 0
or
a
1, these de-
ones.
we
mean
an
index number and
a
set of
arguments
of ap-
description of a computation and for its value (if
of a computation { z} ( u) can be set out on a (horizontal) tree. At each
description of a computation and, if that computation is defined, eventually
a
Suppose {y} ( f~)
as
the
on
has the form of
c
to be determined
there will also stand its value. We start
tree.
defined if
for all values of their
propriate types. We write
this is defined). The course
point
be
defined without the
be
can
be omitted.
can
only
arguments and the computation
allows the progress of
are
will
stands at
a
for the
by placing {z} ( u )
point P ;
at the vertex at the extreme left of the
the progress of the
computation beyond
this
point
is set
follows.
(i)
If y refers to
insert the
an
outright
calculation
by
schemats S1-3,
S7, then P is
a
t ip,
and at it
we
appropriate value.
(ii) If y
{ y} (6) = {Yl} ({Y2} ( ~), ~ ) say, then P is a node. Immediately to its right
there stand two points Qi , Qz ; Qi is above Q2, At Q2 we place the computation description {Y2} (6) ;
if this is defined, eventually its value - u say - will be placed at Q2. Then we place the computation description {y,) (u, 6 ) at Ql. If this is defined, then, when we eventually place a value at
Ql, we shall also place it at P.
(iii) If y refers to S9, {y} (a, b , c ) ={a} (b) then just one point Q stands immediately to the
right of P. At Q we place the computation description { a} ( ~ ), and when a value is placed at Q it
is also to be
refers to S4,
placed
at P.
7
(iv)
then P is
branch point ;
a
type j-2. At QT
in
88j,
If y refers to
at
all the
it
is
P.
at
placed
stands the
the
QT,
to its
immediately
we
can
the
use
function
Towards that and
theory.
1.
There
2.
The non-numerical
are
only
make
we
),
points Q~,
for each
one
When values have been
defined, and the value of
to that
be
played
a
1;
placed
applied
’{z} (u)
tree to obtain normal forms for
to
w’
=
derivation in recursive
number of remarks.
to the left
points (’predecessors’ )
in the
arguments
is
analogous
a
finite number of
a
&#x26;)
computation
role
a
many
(i ,
(’tJ-2,
functional h Tj-2{y}
Now
infinitely
computation description
’{z} (11.) is defined’ ; the tree plays
and
stand
right
at any
computation description
or
any
point
given point.
form
a
selection
from the list 11.’ of the non-numerical members of u .
3.
Any type can be mapped one-to-one into any higher type. Hence, if r is the highest type
occurring in 11., the different branches at a node or a branch point can each be described by an
appropriately chosen object in type r-2. Therefore a point on the tree can be uniquely specified by
a finite sequence of type r-2 objects, and hence by a single type r-2 object.
Such an object is called
,
position.
a
4.
given)
is
remark 2,
By
at
If
5.
z
Yo
y of the tree
position
any
will be denoted
Thus (3 and 11
is the vertex then
be
can
will be
point
at
a
given
tree -
that u
so
(if defined)
the value
at
y
type r-1.
given primitively recursively
part of u . Examination of the
of fi
case
(i) - (iv)
at the immediate
node, the value of p
=
where 8 is
at the
in terms of
shows that else-
predecessor
point immediately
of y,
below y .
the
points immediately
the
position
is
y. Hence,
give
can
9(z,
y)
1t 0’ 11,
primitive recursive.
6. The value of Y) at
y
j3(y). Also
number
a
have
we
we
single
a
functionals of
are
defined in terms of the value
simply
y itself, and, if y is the upper
So
by
computation description (for
evidently
and u. 0’ where u o is the numerical
where P(y)
the
specify
can
we
a
a
to the
position
right
is
recursive
easily
defined,
we
predicate L(z ,
related to its values at the
correctly
from P(yL
determined
There may be branches of
of ~y ,
even
primitive
y is
cannot
1t"
y,
give
T))
arbitrarily great length through
primitive
a
and the values of p at
w ,
recursion for
T)(y).
But
which expres ses that the value of n at
to the
points immediately
right
of y. Y) is then said to
be l ocal l y correct at y.
7. We
The
at
interested
are
nodes,
or
at branch
predicate which
lowest
points where j
point
at
is
a
primitive
node is
can
be
r,
Tl(y)
values
is
when y describes
on
the
represented by
expressed by quantifying
recursive in those
represented by
choice has been made will
we
in the
precise expression of this latter predicate depends
will show that the condition
a
only
the
variables, #
zero
obviously require
a
universal
type
particular way
r-2
type
over
quantifier).
obtain :
y is
a
position
=
(QI )P(z , 1t, ~, y , ... )
8
then
on
the tree.
in which the choice
But
(For example,
type r-2,
r-3
object.
point
a
certain variables of
and ’y .
functional of
a
if
little
type r-3,
choosing
expressing
Thus
thought
using
a
the
that such
remark 5
denotes the
where ....
versal
on T)
quantifier suffices. )
shall refer to
we
To
8.
lowing
clarify
then
Y2
In
of
extention
hence if T) is
p-positions
We
locally
which
are
y 2.
a
tree.
with the
In
points
on
particular, they
below
comes
that
between
is
1~’
points
required
to
a
the tree,
and J
are
the
and every
tip,
locally correct,
are
w
=
and the
=
There may be
an
the lower
all
at
In the latter
point,
points
at
a
(TI) [(y) (y
is
locally
an
r
is defined.
of the tree there will be
Y) -positions
the
just
are
711-positions.
(Eil) [(y) (y
is
p is
correct at
locally
defined,
below
let
case
a
an
assigned
points
on
the
Observe that the relation
us
p,
2, then p will be
positions .
Y) - Y)(Yo) = w]
on
y)
we
= w]
&#x26;
see
that if
the tree,
branch,
or
there may be
an
point,
p say,
so
pn ,
a
the case {z}
point
(M)
at which the
a
schema
indefinitely by simply
that in either
case
there is
an
such that any branch which passes
point, does
reach
a
tip.
Hence
correctly given by
computation descriptions along p will
for any such Tll. Here the branch p
p. of positions along it.
of type 1.
9
now
index number for
continue the branch
and will be
And the
{z} (n) is defined, then
arguments. Consider
conveniently
lowest such
at all these
same
20132013&#x3E;
but y has not the form of
below p.
the function Xn.
the
T)-position
at each successive
(points of)
are
T)-position
correct at
tified with the infinite sequence
r-2 ; if
{z} ( u)
Hence
node where p takes the upper
p’-positions lying
rectly specified by
=
its values at all the
the tree is well-founded ; since also ’below’ is well-founded
computation description
is defined for all
correct
points beyond y ;
at
locally
infinite branch
is
infinite branch. Now there must be
through
sequence)
a
on
recursive in the indicated
primitive
arguments b .
the
point y
correct at all
at the vertex .
same
computation description {y} ( 6 )
repeating
Y1 (as
if
Y2
First suppose that
normal forms.
Thus, by remarks 6, 7 and manipulation of quantifiers,
with the
r-2
lie below y.
T)-positions which
at
T)
uniquel y determined by
and that the values of n and ’~’
-
is not defined.
type
is
T) is
I
finite sequences of
beyond Y ,
or
{z} ( U)
where
dependance
T)’" it follows by transfinite induction from remark 8 that the Tj -positions coincide
and
both for Y)
the
introduce the fol-
we
and the values of T)
by fi (y)
is determined
Now
able to obtain the
beyond’
or
emphasise
well-founded relation. We say that Y1 is beyond
Suppose, conversely,
’below
of
the values
on
correct at y then its value there is
are
now
n-position depend only
an
71(y). Evidently this n
value
To
omitted.
are
uni-
single
a
Y1 if there is an T)-position 5 which is a node and Y1 (resp. Y2)
(resp. lower) r¡-position immediately to the right of 5. Then the value
Then every branch of the tree
a
quantifiers
(In fact
them.
is said to be beloW
and whether y is
an
2 the
putative positions (i. e. represents
two
are
Yl , Y2
Furthermore ’below’ is
is
r
over
n-positions.
is an extension of the upper
of
case
=
the relation of the definitions in remarks 6 and 7 to 1~,
definition : if
objects)
variables, and (Q1;) quantifiers
r-3
type
any r¡’ which is
If r&#x3E;2, then p
can
a
value
locally
be
cor-
is iden-
be taken to be of
type
Hence
{z} (n) is defined
11 is
Now it
n-position
an
recursive R.
primitive
Thus for all r ~ 2
y)
correct at
we
below p
is
an
n-position below p-&#x3E;
specifies
(En)
-
(En) (Tr-3) R(z,M.o,
=
Also ’x specifies
a
tip)].
1; r-3,
Y.
tip’ is primitive recursive.
a
have
{z} en) is defined
with
(11) (p) [(y) (y
be shown that :
can
y is
with
locally
=
=
(a’-’ ) (Eyr-2) M(z, u ,
(2, 3)
recursive M.
primitive
Alternatively
{z} (11.) is defined
(ET)) [(y) (y
&#x26;
If
r &#x3E;
If
r
2, then this
2 it
=
a
is
Ei
primitive
is
an
( p ) (En) (P(p~) specifies
in
correct at
y)
tip] .
be reduced to the form
can
primitive recursive.
are
primitive recursive in u ,
recursion in z ,
T)l,
11.,
and
In
general
quantifiers (x) is of
numerical
it cannot be further
then the second term of the
so
we
course
reduced, but
conjunction
can
we
remark
be reduced to
have
{ z} ( u ) is defined
(the
a
locally
be reduced to the form
can
where A and B
that if
is
=
=
(E T) 1) C (z , w , T) 1) ;
to
equivalent
primitive
a
(2. 6 )
recursion in E and hence also
E 1 ).
3 - FURTHER RESULTS ABOUT RECURSIVE FUNCTIONALS
In this
greater
of the
section
than 2.
type
There is then
tree
R is
for { z}
shall suppose that
(The results
in
objects
we
(u)
a
can
r
=
be extended to the
r-2 must be
that
case r &#x3E;
2, but to do
no
so
a
list
n
has
type
specific well-ordering
assumed).
natural extension of the relation below.
is beneath the
variable in the
2 - i. e.
point S,
if R is below S
or
Namely,
we
if there is
a
say that
branch
a
point
point
R
on
the
P such that
branch
through Q1" S is on a branch through Qj, and i j. Then the relation ’beneath
or beyond’
gives a total ordering of points on the tree. If {z} (11.) is defined then this is a wellIf
not, then there is a ’furthest down’ branch p , such that every branch which (eventually)
ordering.
on
a
goes beneath p
all the
points
on
comes
to
a
tip.
case
the relation ’beneath
the tree which lie beneath p ; and
Thus whether {z} (M-) is defined
the well-ordered
In this
part of
the tree.
or
not, there is
an
We define
z
10
a
or
beyond’
is
unique value is assigned
to all these
1 {z} (B)] which gives
|{z} (u )1. This is a rational
ordinal
well-ordering
a
of
points .
the order type of
notation, because
it
be shown that if
can
Kleene’s system of ordinal notations,
adapt
we
Spector’s
or
notion of
recur-
well-orderings, by substituting ’recursive in u ’ for ’recursive’, then the supremum of the ordinals represented by such notation is wIn as defined above. Note that we could replace u by u ’ ,
sive
"
for it is
the non-numerical
only
part
The notion of the ordinal of
also
Kleene’s E.
uses
of the
least one of which is
at
the ordinals of the
And from this
3.1, There is
Of
the
course
a
p.
theory
is well known.
of the
type
E is recursive in
(The
following
as
an
we are
going
We
by
shall also
(N.B. A =
constant
could be
x
n
E).
can
also be
and E will be
but then
proved,
replaced by
ROG1 ) .
a
well-ordering
functional E r which
a
form if E is recursive in u ;; and this will
rest of this section
C is the
range)
therefore
we
assume
that
example Spector’s
quantifiers
after the
are
binding
limited
occurence
notation of Addison-Mostowski-
this notation to the
apply
For
symbols by
done
by giving
types except
tions -
though
El 2 (2) ~
the
a
each
highest.
in many contexts
2). Spector’s
colon,
type
For
we
in
theorem mentioned in the introduction
Finally,
we
write
recursive in the
[ u]
example,
theorem
to
numeral
can
can
be
mean
distinguish
are
list u may be
we
as
above.
Thus
recursive in u .
empty,
can
(1 )’;
7t}(l)
some
of
We shall
for
a
normally
predicate
11:~ (0) = (El 1t
HA) (0),
types
if this
type -
a
and
parentheses,
separated
of the argu-
were
required
omit the numerals
of numbers and func-
E2 (1 ) E1 1 (E
it is obvious that
the set of all
objects - possibly
course
=
but ’0’ cannot be
1),
while
replaced
of
the numerical
a
prescribed type -
which
part of is irrelevant. It will
predicates, but, as in the previous pa(highest) type (s) of the arguments ; we use exactly
between functionals and
(0)
means
Theorem 2.1
the set . of all number-theoretic
can
be
expressed by [u] (r)
but must not include functionals of
state
same
to omit the ’0’.
it may be necessary to exhibit the
notation
write
we
be written
misleading
arguments
subscript).
a
In the
numerals to indicate the
place
may
list of functionals u ;; of
be necessary to
tions which
parentheses.
could omit the ’
we
and it would therefore be
Now
2
shall not need to indicate the number of
for all
same
comparing
details will appear in
HA. If the recursive predicate which is the scope of the quantifiers involves
ments ; (we
the
r &#x3E;
placing ’IC’ (where
functionals, these will be exhibited
ragraph,
E
from u ,
computations
to be concerned with formulae in which the
constant
never
using
operator for predicates with type r-2 arguments.
as
are
Given two
then
operators for partial predicates in recursive function
argument of v
written
by 1’,
point P ;
u -
of the variable.
it
branch
a
which satisfies
of 3. 1 for
analogue
ordinal with the ordinals
given
a
fairly elaborate ;
obviously has a particularly simple
applications we shall make. For the
In what follows
from the
compared
one
theorem :
of such selection
in range ; this will be indicated
Shoenfield
has
if
especially
method which is recursive in u and E for
a
method is
v (z, n , Ea)
f.
An
existence
in the
case
r.
r-2 will appear
Theorem 3. 1
be the
defined,
obtain the
existence
represents the
there is
computation.
can
one
one
suppose
very useful tool,
a
points Q, immediately to the right of
the ordinal of the computation at P.
also compare it with
one can
proves to be
computation
a
particular,
In
at the
computations
of u which is relevant.
corollaries to 3,1 :
11
type greater
C
than
p~-1
r.
predicates Qr func( u : r), where the
3.11 (Axiom of choice for [11.]). I f R is recursive, then
That the
hand side is included in the left hand side follows from 3.11 ; the
right
by examining
the
3.12 is
an
proof of 2.1,
analogue of
and
establishing
the familiar
that all the bound variables may be limited to
A§
Recursive =
shall
we
continually
find that when
type
2
objects
to functions recursive in those
parameters is
tifier in
recursive function
particular case
GK3). He proposes
has been
logy
investigated by
number should be taken to be
set is the jntersection of
rithmetic functions
a
Kreisel
a
and
set
hyperarithmetic
(or notations for them)
(or
(1),
n
=
the
(or numbers)
function
a
numerical quan-
a
E, the basis of this
that the
of
analogue
analogue
of
a
ana-
natural
recursive
a
with the set of all
hypera-
will take for the
of
For
of
analogue
it) ;
index for
an
set of functions
hyperarithmetic
the
For the
theory.
(in GK2
parameters, then
as
appear
quantifier limited
ordinary
[1].
equality
(*)
Indeed,
is shown
converse
ease
expression we
analogue
than
this
is
of
no im(rather
sets)
;
hyperarithmetic functions
change
evidently
Kreisel’s analogue for (*) can be written :
of N the set of all
Then
portance.
(1 ~t
where
C)
that the
means
predicates
must still be defined for all
predicates
are
to be restricted to
type
1
in
arguments
[C] -
but the
arguments in type 1. Now the methods of proof of 3.12 also
suffice to prove
which is
just the generalisation
to
arbitrary
an
in which E is recursive of Kreisel’s
u.
analogue
of (*).
A further
(jj~.
corollary
This may be
well-ordered
theorem
(where,
which
compared
in the
were
with the
so
an
undefined
fact that there is
And
segment has ordinal
(see ROG2 ),
as
of 3.1 is that there is
just
does the former lead to
the
as
a
a
proof
computation {z} (n) whose
recursive linear
ordering
latter fact leads to
a
proof
ordinal is
whose initial
of
Spector’s
of :
introduction, D(x ; 11. ) == {x} (11.) is defined). This theorem shows that the remarks
made in the introduction about
obtaining
a
diagonal predicate
for E
by
means
of quan-
tification from below also apply to any type 2 functional in which E is recursive. Hierarchies based
this idea will be discussed in section 5.
on
By using 3.
Let
us
same
denote
1 it is easy to show that
by D[ 11.]
conventions
as
(1) Kreisel’s analogy
is
the set of all
before for
indicating
incorrectly stated.
In
the
our
our
purpose the
complete predicate
E [E]
analogy described
x
(highest) type (s)
notation his
&#x26;
p.r.f. {z} (a, E) need only be defined
w-models, whose hard core consists
However for
a
predicates
a
where the
deration of
Àx. D(x ; 11.) is
{z} (a,E)
of the
for
analogue
=
ranges
a
for
over
N ;
arguments.
[ u ] ) (11. : 0) ,
we
adopt
the
Then :
recursive set is
0 ,
I
for hyperarithmetic a. This arises
of the hyperarithmetic functions.
in the text is easier to
(Z § ?
naturally
in the consi-
handle, and is sufficiently suggestive .
3. 21
and also :
3. 22
The class of
predicates denoted by
of Kreisel’s
analogy,
of the class of
4 - THE HIERARCHIES
We first
the theorems
in E’
some
of the
4.1 Primitive recursive in E
This is true both for
in
our
generalisation
predicates.
’hyperarithmetic’.
can
explored
hyperarithmetic hierarchy
We do this to
and also to
single
provide illustrations
for
out those theorems which
in the final section.
Arithmetic.
=
of numbers and of functions.
predicates
forward inductive one ; it
to be
of the
important properties
of the last two sections,
partial analogy
a
more
the definition of
as
and notations
form the basis of
enumerable
recursively
analogue,
[E] AND [E1]
give proofs of
using ’recursive
either side of this equation is the
The
proof
either way is
a
straight-
be found in SCK1.
2.2 and 4.1
This is
an
immediate consequence of 2.1"
This is
an
immediate consequence of 4.1 and 2, 3.
4. 4. For any type 2 functional F and any function ex,
Àx. D(x ; a , F) is complete for
For
simplicity
shall omit all reference to
we
and let
S~
cursion theorem
we
be
given,
be the set of
can
Recall that there is
a
find
an
primitive
a
in
our
non-past-secured
index number
e
proof.
’~i (a ;:
Let
a
0) .
11:~ predicate ( ~ ) (Ey) R(J3(yLa)
sequence numbers for
R(u , a).
Now
by
the
re-
which satisfies :
recursive substitution function
S2 satisfying
I claim that
computation {e} (1 a, F). Corresponding to each u E S~ there will
tree, and conversely. Further the computation at any point beyond which
Consider the tree for the
be
branch
a
there
are no
to the
are
point
on
branch
the
points
will
surely
be
defined, since
top clause in the definition of {e}.
it must be
a
part of
a
computation according
Thus the tree has infinite branches if and
only
if there
unsecured infinite sequences ; this proves the theorem.
D[E] (I)
4, 5
This follows
immediately
from 4. 3 and 4. 4.
=
7~(1) .
[E])
Since
13
is Kreisel’s definition of the
analogue
of
’recursively enumerable’,
we
in
justified
were
] (1 t [ u ] ) the generalisation of his
calling
analogue.
This is
consequence of 3. 21.
a
This follows from 4.5" 4. 61.
This is
Evidently
we
and’Z’.
interchange’1t’
may
Therefore
instance of 3.12 ; it shows that any class of functions which is
an
hyparithmetic operations
to
finite set of functions
a
provides
a
generated by applying
ð~ comprehension
model for the
axiom with free function variables.
4. 9
(1) .
This is
immediate consequence of 4. 7 and 4. 8.
an
We
now
Let R be
follows.
define what it
whereR~
in
turn to the
means
is
apply the hyperjump
an
limit
a
W1l+1.
point
Hence
w’1
for
an
ascending
call any solution of the
An
so
W:l.
w’1
equation C
(indeed, they
(March ’64) that
me
are
So
we can
=
w~,
Wt
probably
he has
Addison second number class.
for
by c~
are
construct
just
the ordinals of
recursively
in
then its limit
We
now
JRS).
We
We
investigate
give
the
a new
denote
an
This
E1
then
index.
former
To
as
type
0 and of
distinguish
that W2 is the least ordinal not
a
proof
of this
arbitrary
fact,
of
predicates
based
on
which
point
will be
Hence
see
is the
are
that
if
we
C-number.
certainly
all less
represented
in the Kleeneare
in
type
1
list of variables of
arguments
can
or
(c)
all be
for the
[E1].
It is known that
[E1]ç
types
0 and 1. We suppose that
ð~ (1)
so
a
p.
r.
f.
we
a
standard
that the two formulae and the
primitively recursively
predicate
determined from the
shall if necessary refer to the
whose index is
n.
We shall also
functional, meaning thereby the index of its graph, and
14
(see TT
ideas which will be used in the next section .
between this index and that for
function
w~
well-orderings
or
then
suggests the conjecture that the Kleene-Addison ordinals
quantifier form
a A-index. We write Q n]
of the index of
less than
are
sequence of notations
a
so
va-
[For ’probably’ read ’not’. W. H. Richter
system of indexing has been adopted for all predicates of
number of
but
predicate A,
And it is not hard to
all less than
proved
~R~,
initial ordinal. If
are
c~,~, , . , ,
what ordinals
number-theoretic
some
the
C-number, then if T)
a
which is recursive
]
those less than
precisely
and
wi
E N ; indeed these
Further if
E1.
denote
can
predicate
on1 We may remark that the ordinals defined in Addison-Kleene A-K
than
tells
we
We
predicate Bx.x=x~
approciated by considering
which is
one
a
E~.
as
with the
starting
of invariance of this process for constant
sequence of initial ordinals each
then also
r~
in
n
times
hierarchy
which is recursive in
result of this process is
be
can
for
idea of the extent of this
some
operation
degree
of such ordinals ;
give
can
well-ordering relation,
Evidently the
initial ordinal to be
primitive recursive
are
if
to
do not here discuss the
We define
so
number-theoretic
Thus the extent of
rying R).
We
a
is the ordinal of R.
E1. (We
which is
hierarchy [E1],
we
speak
shall write
Enl
(c)
for the value of the function.
Evidently
there
are
primitive
recursive substitution functions
(analogous to Kleene’s S functions) which determine the index of the predicate which results from
a given predicate by substituting particular numbers, or functions with particular index numbers ,
for some of the arguments. It makes perfectly good sense to speak of an index number for a predicate of
no
variables.
5.1. There is
a
primitive recursive function (D(z , f) such that, if F is
a
functional with A -index
number f, then
We first prove this when
induction
by
considering
on
is
z
the construction of
index for
an
(considered
z
The construction of ~ is
primitive recursion.
a
an
as
index).
We illustrate the definition of ~
by
three cases.
S. 7 : We must define ~(z ~ f)
this is
so
that
easily done.
S.4 : We require
We have
merely to bring the
E~
form, thus defining ~(z , f) in
El 2 and 14
form
LHS to
terms of
f),
O(z2f).
S.8 : We require
right hand sides can
4D(z, , f). Evidently, in
The
be
of
each case, the
brought
to
respectively, thus defining ~(z , f) in terms
defining equations for ~(z , f) can be expressed as a pri-
mitive recursion.
Thus
in
we can
obtain A-indices for the
2.1, 2. 2, 2. 3, and 2. 5, for the
). Then, using
required index, ~(z , f) for
in which
2,1 and 2. 3 for
object
the
case
primitive
the
recursive
we are
the II12
predicates I, J, M and
2 and F
interested (viz. r
form,
å2,
then
[F]
El
E
and
w.
Q, E, D,
D [F]
D(z ; F) - (Ew) [ (D (z , f)] ( c, w) .
For
Since
the E12
only type 2
form, we obtain
the
predicate :
{z} ( ~ , F) is defined &#x26; {z} ( ~i, F) =
5. 2, If F E
=
2. 2 and 2. 5 for
A which appear
A,2
it follows that
[El]
We close this section with
a
is included in aid does not exhaust
rather
interesting
15
theorem about
A’ 2,
[E1] .
as was
to be shown.
be
let
For,
an
(see e, g.
It is well known
The RHS is recursive in
arithmetic formula whose
SCK
2)
free function variables
only
and .
a
are
that
hence
E1 and ;
for suitable recursive R.
Therefore
and, since
we
can
As
which is
in
types
not
closed with
ROG1.
greater
matter of
a
replace II by E,
than 2 in which
respect
to
respect
to the
could
any list of para-
replace E, by
predicates
hyperjump and satisfies (A’ 2 t C) (0) C ; this will be proved
[E1] (1) provides a minimum B-model (i, e, a model which is
=
of the natural numbers -
well-orderings
order arithmetic whose
E1
we
is recursive.
is the least class C of functions and number-theoretic
fact,
From this it follows that
standard with
have
we
5.4 follows by 3. 12. Notice that in 5.4
From this
meters with
then
strongest comprehension
axiom is
AMI)
see
that
for
A’ 2
for
(with
a
system
or
of second
without
function
parameters).
6 - THE SUPERJUMP OPERATION
In the
objects)
introduction it
to the
both be
thought
tional.
They
of
as
the result of
reason
for
superjump
are
good
reasons
admitting
to the
It therefore
are
a
Two candidates
the
seems
type
and
stopping
is
type
a
1
can
2 func-
operation :
defined) .
jump operation ;
but there
hyperhyperjump (i.
seems
no
the result of
e.
particular
applying
the
on.
straight
proceed
to the
to
3 functional which is of the
suggest
the
at
hyperjump operation E1,
class of functions recursive in
a
shall call the super jump
but not the
so
We
on
({x} (a , F)
natural to pass
superjump.
we
--~
for
hyperjump,
hyperjump) ;
recursive in the
We need
diagonalising
both instances of what
are
from recursive operations (on type
and the passage from there to the
jump operation E,
F
Now there
observed that the passage
was
an
same
themselves :
- 1 otherwise
16
of
hierarchy
investigation
recursive
type
of this
degree
1 and
type
2
objects
which
hierarchy.
as
the
superjump operation .
6D is
obviously
recursive in the
and since E is also,
superjump operation ;
we
see
that
is ; thus £ is recursive in the superjump operation. Conversely that operation is recursive in 6D
and
using 3. 2,
The
raise the
new
types of all variables by
should
is
certainly
E ) by E3 ,
E(=
imperfect
of Kreisel’s
generalisation
theorems
analogy
it is not hard to show that it is also recursive in S
the
more
a
as
which
and
one,
reason
fruitful in
was
replace
to be
suggesting
E
discussed in section 3
this is not
always
theorems than it would be if
(E a) (F(a)
suggests
that if
we
in the theorems of section 4, then
by 8,
discussed,
function for XF .
representing
may be exhibited
where V is
For
result.
analogy
(3).
0). The
=
reason
the case ; but the
we
simply replaced
why
our
analogy
is
Consider
follows.
partial recursive. According
to Kleene’s
definition, this computation is only defined if
~( a ,8 )
is defined for all functions a ;; but the value of the
founded
models)
computation depends only on the values
of 4D for those functions which are recursive in ~.a , ~ (a , 8). Kleene’s definition is thoroughly impredicative ; for example, using Mostowski’s undecidable sentence (sec AM2) one can construct a
(1) recursive in E which will be defined for all a in some w-models (and indeed in some welltained
that
of set
by ajoining
but will not be defined for all
theory,
of
Spector’s
theorem
(4. 26)
having arbitrary type
gument 8.
It is therefore natural to seek
can
assign unambiguously
Let It., be
naturally,
a
a
3
definition ’from below’ ; and
a
should hold. But whereas Kleene
functionals
we
arguments,
a
value to
we
are
(u , F)
We cannot
so we
was
here concerned
looser definition of is
theory
ob-
expect therefore
cannot
expect
that the
concerned with recursive
only
with the
particular
ar-
defined’, which will apply whenever
(*).
partial functional of type 2, and let
that {z}
in any w-model of the
the axiom of the existence of inaccessible ordinals.
Xz. D(z ; S) should be capable of
analogue
a
it
be
a
list of
is defined if it is defined in Kleene’s sense,
objects of types 5 2 ;
and whenever the
we
say,
computation
---------------
(3)
To show that the superjump is recursive in
in F , 8. Let G be defined by :
Evidently
G E
[F] and F E [G] ,
so
that [F]
9,
=
it will be sufficient to show that
[G] ,
Then
which
gives
the
required recursion.
17
(Eal [F] ) R(a , F)
is recursive
requires
an
(by S8),
of F
application
the value of F is defined for the
argument
in
question. We
say that F is recursively satisfactory if
(z) [(x) ({z} (x , w , F)
Now
say that {z}
we
defined)2013-~F(Bx.{z} (x, F))
is
(n , 8 ) is 14inímally de f ined,
of the inductive definition
given by
Kleene when
and write
refers to
z
n,
a
is
defined] .
8)
if it
satisfies the clauses
scheme other than
S83, while for that
schema the clause is :
(zil (a , u , 1&#x26;»
is
recursively satisfactory.
S) for
We write
But the
converse
6.1. There is
a
Kleene’s definition ;
is false.
implication
by
F)
8)
such that :
crucial
computation tree ; the
a
[F]) (F(a)
point
=
0) ==
(E a f
is that
we
can
define
[F] R(a , F) ,
whenever F is
here. 6.1 shows that the extent of the class
we
n ,
which satisfies
(E ex í
and which is defined for all
(a,
if
However :
induction up the
recursive relation
only
evidently
partial recursive functional
The definition of X is
a
is defined if and
recursively satisfactory. We
[8] (2) does not depend on which
shall not
give
the
proof
definition of ’is defined’
adopt.
Before
between
[8]
investigating
and
A§ ;
the
analogues
5. 2 leads
us
to
of the theorems of section 4,
expect
that
[8]~
C ,02,
and
we
shall examine the relation
shall show that the inclusion is
we
proper.
6, 2. Let c be
(Kz)
will be
a
list of
defined
Sl-3, S7, then
where the
RHS to
If
~~
z
means
outright.
recursive function
So in this
refers to
1. There is
S9,
we
a
case w(z)
a
=
If
z
refers to S4"
is determined
6(z, e).
write
18
by
recursive
primitive
of the recursion theorem ; let
is determined
primitive
form.
by
objects of
the
e
be
we
an
function o such that
index for
(~.
If
z
refers to
write :
manipulations
necessary to
bring
the
easily determined primitive recursive T ; in this case 4)(z) i (e),
mined outright - a fact which explains the possibility of giving a primitive recursive
Towards S8 : let F be the functional whose graph is Xaq. [f] (a , c , q) ; so
with
say,
=
an
the
parameters
The
proof of this
to S8,
Then there is
c .
a
recursive function
primitive
is very similar to the
proof
e(f)
and
definition
that F
primitive
Thus
is
recursive p,
determined
(Kz)
and set
from
primitive recursively
and
and e,
z
(~ itself is primitive
so
Q. E. D.
We shall not
only needs
to
the
give
consider
proof.
condition. Thus
A’ 2
can
point is
tree with
that in
countably
deciding
if
D,in (z ;
c ,
9) holds,
many branches at each branch
one
point ;
proposed evaluative function T) at a branch point is described by
imitate the analysis described in section 2 for the case r
2, using
a
=
formulae.
The
For if X
inclusion is
c
Using
number
one
The crucial
computation
a
further the local correctness of
only
refers
z
p (e, z).
=
recursive.
A’ 2
on
such that
and is left to the reader. Now, when
of 5,1,
for ~ .
depends
write
we
say, with
a
is deter-
so
(z ;
z ,
an
c,
remark 5
immediate consequence of 6. 2, while 6. 3 shows that it must be proper.
8)
E
[ .9 1 ,
we
should have
in section 2 and
a
contradiction
the
by
diagonal argument.
6.2, it is easy to prove the following
B, such that for all putative positions ’~ which lie below
a
lowest
a
position)
Lemma :
There is
a
non-terminatinf branch of the
computation {z} ( c , t6) ,
~B ~ ( c , y ,
z ,
b)
ae
[(b
= 0 and y
is not
v
(b = ~ (Y ))J .
particular, if the computation is defined, then the above predicate determines the computation
description number b correctly at every position.
Using the above predicate we obtain a 11:~ formula for D... through the equivalence :
In
D,sx (z ;
where
p
is the
8 ) _ (p) [(m)
is
c
c ,
(Ep) A(a , P,, c ) is recursive in E ,
primitive recursive function 4S(a), (where
Since
a
(p~
a
position)
--j
member of the infinite sequence p of
X
6.6
fine
mth
c ,
8)
is
(En)
( po
a
tip)] ,
putative positions. This
complete for
proves 6, 5.
~a (1 )
it is also recursive in
a
is
8,
can
therefore de-
is the Godel number of the arithmetic
predicate A) ,
such that :
19
and
we
evidently the computation
Then
(maximally)
c ).
Q. E. D,
only if ( a) (Ep)
particular, therefore, there is a ð~ index for the type 2 functional D~ which represents
a(t + 1 ) , 8), Further
C [Di] , Now we can repeat the proofs of 6.2-6.6,
using D), 8, in place of lb, and 11:~ in place of ~2, etc. And this process can then be repeated.
is defined
if and
In
In
general,
let
Df
= Xa . 0, and let
Then, for n &#x3E; 0,
These two theorems
were
announced in ROG3
(4).
7 - THE ANALOGY BETWEEN E AND S
In this section
at the
beginning
we
of the
work out the details of the
last
section.
In
analogy
particular,
4. 2-4. ?. One might suppose that the analogue of
7~
we
between E and 8 which
seek to prove the
was
analogues
would be the set C of all
discussed
of theorems
predicates :
(4) While revising this paper I have noticed that the results of section 6 can be extended to higher types ; the
only important modification is that the two-function-quantifier forms are replaced by forms with a single
quantifier of appropriate type.
Let
be a recursive one-to-one mapping of type n into (type n - {0°}). Let a:+1
~,i° . 0; let
be defined recursively from a°+1 so as to satisfy a~+i(i°)
a°+1(f(i°)). We define a generalised superjump
operator ÖD+3 by :
=
And
we
define
Then, for n &#x3E; 1,
The results of section 7
can
also be extended ; in
particular
20
However this will not do
with
with
a
(5). Using
recursive S
primitive
Thus if the
two
in the normal form for
quantifier
The correct
i
n)
are
a
restricts those quantifiers,
It is in fact
We shall
opposite kind.
generally permissible.
7. 1. There is
a
types
of quan-
of 4.2 would lead
analogue
we
which
to
a
see
of
one
11:~.)
shall call
permute
restricted function
a
us
to
that
~ti* ~
quantifier
single numerical
a
quantifiers .
predicates :
2) ,
[F , u which
note that in the term
are
relevant.
quantifier and
But the reduction to
restricted function
of all
(types
alternating quantifiers ;
shown in 7. 3.
arrives at
to restricted function
applied
the non-numerical members of u
this will be
quantifier ;
require the permutation of
the reduction of the
while the
cannot all be
(which
is the set
string
only
permissible
tional
combining quantifiers,
on
[F, u 1) R(u, 0:1’...’ aQ, F)
(F)
where the
ç 11:~,
manipulations by
11:~ predicate
a
1t~
of
analogue
on
obviously incompatible.
are
for this failure is that the
reason
such that
z
Therefore,
1, then C
arse 5
11
a
(SCK1 XXXIV)
Kleene’s theorem
of
types
expect [919 C ; these
The
by
find
readily
we
and 2. 3.
primitive recursive P, by 2.1
tifiers.
the
3. 2
n
=
unrestricted func-
an
1 in the above form would
with restricted functional
quantifier of
C ; it follows that the second sort of permutation is not
We first prove two lemmas.
primitive recursive
P such that :
For,
(T) is locally
The
t,
’each
phrase
z ;
tricted to
correct for
S. 8 point .. , ’
can
&#x26; each S.8
be
point
expressed by
a
of this
computation
predicate
is
which is
an
application
primitive
of
H)].
recursive in
=
Let II21+
as an
I
by 3.2" { z} (b, C , H, E) a (t) can be expressed using an existential function quantifier res[G, H, E] ; and then 3. 11 allows the function quantifiers to be pulled to the front, leaving
the numerical
E
{z} (t, G, H, E)
quantifiers
be defined
to be
by
expressed
replacing ’f
[F ,
as
u
recursions in E.
This completes the
proof.
[F" 11. , E]’ and allowing
J
I ’ in the definition of
argument of R. Repeated application of 7. 1 shows that
n2* 11:~+.
But the
converse
is also
true ; for
(5) At the
time
?.1 - ?, 3
are
of giving this paper I wrongly
therefore later additions.
supposed
21
that it would do.
The definition of
and theorems
The
dition
on
[F , u I
we see
the
ordinal of the
hand side
implication
be
can
right follows. Finally,
restricted function
replaced by
computation for
con-
each
a
in
must agree with E for all relevant argu-
=
from left to
And if F satisfies the
(F)o .
E for
proved by substituting
right, then by transfinite induction on the
that [F’ , u~1 (1)
[(F)1, u , E] (1) ; thus (F)o
the
right
to left is
right
from which the
ments,
on
from
implication
the numerical
quantifiers
quantifiers.
This proves
7,3.
(a)
Let
A(u)
all type 2
functions
be
a
predicate of arguments of type 5 2 which has
quantifiers
recursive in u and those type 2 variables whose
tifiers of types &#x3E;2.
A(.u)
Then
Xt .
is, say,
2 a( t ) 3 f3( t ) ).
In this way all
7L1and
2
type
By 7. 1
of 4.2,
S
belong
induction
By
a
obtain
we can
in the
in
Similarly
Now for
the
obtain
general
further
application
This is the
quantifier
the inner
along
11:
can
a
which the
in 7. 5.
Z2*
operations
depends
type
2
which
on
soon
can
we
where
a:&#x3E;
a
predicate
of
be
of F
only
the
we
case
show that the
that
z
graphs
refers to
an
of
ap-
have
which the function
quantifiers
(and
we
[ u , F] .
restricted to
are
do not need to exhibit
we
obtain the
required
n 2*
form.
can
be restricted
proof
(b)
for the
of ’l. 3.
here, also,
not terminate.
So it is not to be
The scope of the initial
a
type
1
in
fact,
expected
an
Then
the inner
that E could
branch
replace
quantifier (F) contains only objects
arguments.
the formula therefore
Hence
we
quantifier. That this is indeed the
22
a
equality.
arguments u . The truth of
denumerable set of
a
case
argument of the previous paragraph applies. However,
restricted, since its range must include, if possible,
n2* .
replaced by
in condition
as
of 2. 3 shows that
that the inclusion in 7. 5 is,
for
proofs
completes the proof of 7. 4.
that the
so
Examination of their
apply 2,1, 2, 2,
we
of 4. 3. Examination of the
see
the index x,
on
restricted). So, by 7. 3
recursive in F and the
the values
quantifier
quan-
from [ ~I to [b, G , E] .
and the result will be
8 ) belongs
of 7. 3 and its dual
formula of
are
front,
We shall consider
(type 1) quantifier
computation does
a
no
to
form.
in 2, 4 cannot be
We shall
Consider
~i* .
to
graph of
restricted ;
quantifier
to the
First, by induction
recursions in 8
analogue
be
there are
G( a ~ &#x3E;)) ;
[6]) B « ,
to alter the restriction
pulled
since it is itself
in hand shows that the inner
a
be
equivalent formula in
an
hypothesis
can
can
the definition of ~,
restrictions,
we
apply ?.1
can
analogue
S83. By
of
we
satisfies :
11:~..
primitive recursions
plication
Now
which
11:~..
E
quantifiers
hence also of
This is the
quantifiers precede it ; (c)
6]) (G) B(a , G) _ (G)
For, classically,
form
p renex normal
unrestricted ; (b) each type 1 quantifier is restricted
universal and
are
a
and
only
may expect that the
case
is shown
by :
Let
proof.
be
it
Let
a
be
a
finite sequence of
number which encodes
a
selection b - perhaps arranged in
{z} (a ......
to denote
Let
sions W n ,
G)
duction
we
If
there is
an
that if y
z
If
where t*
and
the
analyse
a
refers to
z
If
z
an
is
,
a
string
to
recur-
of numerical
some
functional’ ; the analysis will be by
These clauses
application
then there
clauses which relate
are
ar-
an
in-
y(2Z 3a)
to
all recursive in y .
are
of S 8 in which,
a
say,
(from
functional F
the
list u)
some
we
can
be
t to the list of numerical
is to be taken
of G
application
an
is ap-
as
arguments
encoded
the above-mentioned F.
by
refers to
an
by S8,
(t)
of G &#x26;
application
G whenever that
particular
determine
[y(2z 38)
recursively
=
0
==
on
or
(Ep) (p
expressed recursively
computations.
recursive
This
Here
a.
Thus this clause
then the value
of y
be
can
freely
chosen
subject
condition :
{z} (a : u , G) which lies
(z) (a)
prefixing
(y(2z’,~*") = y(2Yl 3t.&#x26;)) :-~ y(2Z3&#x26;) = y(2Y 3 a))
If Y satisfies the above clauses for all z, a, then it will determine the value of {z}
for
7.7
(with respect
and E.
refers to
consistency
(d)
a
B( 11. 0’ 11. y) (where u .
predicate ’y represents
determine which member of u
(y) (y
can
say that the function y represents G
we
arithmetic formula
encodes the result of
a
(c)
a
given ;
refers to schemata Sl -3, S4, S7, S9,
is recursive in y , u ,
to
{z} (a : u ))
We write
u .
then the clause is :
plied,
z
different order - from the sequence
a
a
represents G, then
certain other values of ’~ .
(b)
numbers, and also indicates
an of
ao,...,
computations.
on
(a)
multiplet
a
2 ; this will remain fixed throughout the
1 and
if :
Evidently
guments), such
Now
types
).
an , n
functional G be
a
of
objects
Thus
in
is
in Y ,
’y represents
predicate Q(’y ,
completes
the
u ,
proof
computation
z , a , ’y,
below
a
the
branch &#x26;
as
computation description
(n) (p.
is not
And this clause does
some
Hence,
non-terminating branch.
lowest
a
Ei .
Ei ).
is defined.
G with
respect
And then the
a
tip
at any
point
on
8)
is
to recursions
typical predicate
complete
for
For, since E1 E [8] , given any recursive R(a , u ,
23
7~(E~ :
E1)
we
(a :
u ,
G)))]
that y vanishes for all undefined
ensure
of
in u , G’
is
can
be
equivalent
of 7. 6.
n ,
the tree for
Thus :
the tree for {z}
on
(a : y , G)
in remark 5 of section 2 ,
2) .
can
find z,
so
that :
expressed by
to
(a , ii
Then
,
This theorem
gives
is
maximally
an
alternative
~i (E1 :
7.7 shows that
For
of inclusions.
This theorem
7. 8 ; but this has only been
It
is
was
pointed
certainly not
In
with it.
out
2)
C
defined if and
proof
only
if
(a) R(a ,
u ,
,E1).
Q, E, D,
of 6.6.
D,_ [8] (2) :
together with 7.5
and
and 7.6 this
gives
expected analogue of 4. 5. Actually we could add
proved (in 6.6) when the types of the arguments are 1.
in
gives
the
section 6 that
coextensive with
seeking analogues
(E2*1l[E]).
we
cannot
expect
On the other hand,
to further theorems about the
to obtain the
analogue
I believe that
’=
circle
a
7~(2)’
to
of 4.6.
Dmin [@] is coextensive
hyperarithmetic hierarchy,
one
must
distinguish between those occurrences (explicit of implicit) of 7t} which depend on 4. 62. ,
those which depend on 4. 5. The analogue for the former will be D,in [@] ; for the latter D.ax [8]
equivalently, 7~* .
therefore
and
or,"
REFERENCES
A-K
J. W. ADDISON and S. C. KLEENE - A note
Soc. 8 (1957) pp. 1002-1006.
KG
K.
12
GÖDEL - Uber eine bisher
(1958) pp. 210-287.
function quantification, Proc. Amer. Math.
on
noch nicht benutzte Erweiterung des
fini ten Standpunktes,
ROG1
R.O. GANDY - Selection operators for recursive functionals,
ROG2
R. O. GANDY - Proof of Mostowski’s con jecture,
ROG3
R.O. GANDY - The Analytic hierarchy and recursive functionals,
Amer. Math. Soc. June 1962.
in
Bull. Acad. Pol.
GK1 G. KREISEL - La Prédicativité, Bull. Soc. Math.
France 88
Dialectica
preparation.
Sci.
8
(1960)
pp. 571-574.
(Abstract), Monthly
notices
(1960) pp. 371-391.
of potential total ity, Infinitistic
GK2
G. KREISEL - Set theoretic problems suggested by the notion
Methods, Warsaw 1961 pp. 103-140.
GK3
G. KREISEL - Modell theoretic invariants : applications to recursive and hyperarithmetic operations, Proc. Theory of Models, Symposium held at Berkeley, July 1963.
SCK
S. C. KLEENE - Recursive functionals and Quantifiers of finite types I, Trans. Amer. Math.
Soc. 91 (1959) pp. 1-52.
AMI
A. MOSTOWSKI - Formal systems of analysis based
Methods, Warsaw 1961, pp. 141-166.
AM2
A. MOSTOWSKI - An undecidable arithmetical statement, Fund. Math. 36 (1949) pp. 143-164.
CS
C.
JRS
J. R. SHOENFIELD - The form of the negation of
Symp. Pure. Math. 5 (1962) pp. 131-134.
TT
T. TUGUE - Predicates recursive in
Univ. St. Paul 8 (1959) pp. 97-117.
SPECTOR - Hyperarithmetical quantifiers,
a
on an
rule
of proof, Infinitistic
Fund. Math. 48 (1961) pp. 113-120.
a
predicate, Recursive Function Theory Proc.
type 2 object
24
infinitistic
and Kleene
hierarchies,
Comment. Math.