<s>
In	O
computability	O
theory	O
,	O
a	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
is	O
,	O
roughly	O
speaking	O
,	O
a	O
function	O
that	O
can	O
be	O
computed	O
by	O
a	O
computer	B-Application
program	I-Application
whose	O
loops	O
are	O
all	O
"	B-Language
for	I-Language
"	I-Language
loops	I-Language
(	O
that	O
is	O
,	O
an	O
upper	O
bound	O
of	O
the	O
number	O
of	O
iterations	O
of	O
every	O
loop	O
can	O
be	O
determined	O
before	O
entering	O
the	O
loop	O
)	O
.	O
</s>
<s>
Primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
form	O
a	O
strict	O
subset	O
of	O
those	O
general	O
recursive	O
functions	O
that	O
are	O
also	O
total	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
importance	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
lies	O
in	O
the	O
fact	O
that	O
most	O
computable	O
functions	O
that	O
are	O
studied	O
in	O
number	O
theory	O
(	O
and	O
more	O
generally	O
in	O
mathematics	O
)	O
are	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
addition	O
and	O
division	O
,	O
the	O
factorial	O
and	O
exponential	O
function	O
,	O
and	O
the	O
function	O
which	O
returns	O
the	O
nth	O
prime	O
are	O
all	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
showing	O
that	O
a	O
computable	O
function	O
is	O
primitive	B-Architecture
recursive	I-Architecture
,	O
it	O
suffices	O
to	O
show	O
that	O
its	O
time	O
complexity	O
is	O
bounded	O
above	O
by	O
a	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
of	O
the	O
input	O
size	O
.	O
</s>
<s>
It	O
is	O
hence	O
not	O
that	O
easy	O
to	O
devise	O
a	O
computable	O
function	O
that	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
;	O
some	O
examples	O
are	O
shown	O
in	O
section	O
below	O
.	O
</s>
<s>
The	O
set	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
is	O
known	O
as	O
PR	O
in	O
computational	O
complexity	O
theory	O
.	O
</s>
<s>
A	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
takes	O
a	O
fixed	O
number	O
of	O
arguments	O
,	O
each	O
a	O
natural	O
number	O
(	O
nonnegative	O
integer	O
:	O
{	O
0	O
,	O
1	O
,	O
2	O
,	O
...	O
}	O
)	O
,	O
and	O
returns	O
a	O
natural	O
number	O
.	O
</s>
<s>
The	O
basic	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
are	O
given	O
by	O
these	O
axioms	B-Algorithm
:	O
</s>
<s>
More	O
complex	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
can	O
be	O
obtained	O
by	O
applying	O
the	O
operations	O
given	O
by	O
these	O
axioms	B-Algorithm
:	O
</s>
<s>
The	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
are	O
the	O
basic	O
functions	O
and	O
those	O
obtained	O
from	O
the	O
basic	O
functions	O
by	O
applying	O
these	O
operations	O
a	O
finite	O
number	O
of	O
times	O
.	O
</s>
<s>
A	O
definition	O
of	O
the	O
2-ary	O
function	O
,	O
to	O
compute	O
the	O
sum	O
of	O
its	O
arguments	O
,	O
can	O
be	O
obtained	O
using	O
the	O
primitive	B-Architecture
recursion	I-Architecture
operator	O
.	O
</s>
<s>
are	O
"	O
rephrased	O
in	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
terminology	O
"	O
:	O
In	O
the	O
definition	O
of	O
,	O
the	O
first	O
equation	O
suggests	O
to	O
choose	O
to	O
obtain	O
;	O
the	O
second	O
equation	O
suggests	O
to	O
choose	O
to	O
obtain	O
.	O
</s>
<s>
A	O
primitive	B-Architecture
recursive	I-Architecture
definition	O
is	O
.	O
</s>
<s>
Since	O
the	O
recursion	O
runs	O
over	O
the	O
second	O
argument	O
,	O
we	O
begin	O
with	O
a	O
primitive	B-Architecture
recursive	I-Architecture
definition	O
of	O
the	O
reversed	O
subtraction	O
,	O
.	O
</s>
<s>
Its	O
recursion	O
then	O
runs	O
over	O
the	O
first	O
argument	O
,	O
so	O
its	O
primitive	B-Architecture
recursive	I-Architecture
definition	O
can	O
be	O
obtained	O
,	O
similar	O
to	O
addition	O
,	O
as	O
.	O
</s>
<s>
In	O
some	O
settings	O
it	O
is	O
natural	O
to	O
consider	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
that	O
take	O
as	O
inputs	O
tuples	O
that	O
mix	O
numbers	O
with	O
truth	O
values	O
(	O
that	O
is	O
t	O
for	O
true	O
and	O
f	O
for	O
false	O
)	O
,	O
or	O
that	O
produce	O
truth	O
values	O
as	O
outputs	O
.	O
</s>
<s>
Exponentiation	O
and	O
primality	B-Algorithm
testing	I-Algorithm
are	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
Given	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
e	O
,	O
f	O
,	O
g	O
,	O
and	O
h	O
,	O
a	O
function	O
that	O
returns	O
the	O
value	O
of	O
g	O
when	O
e≤f	O
and	O
the	O
value	O
of	O
h	O
otherwise	O
is	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
By	O
using	O
Gödel	O
numberings	O
,	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
can	O
be	O
extended	O
to	O
operate	O
on	O
other	O
objects	O
such	O
as	O
integers	O
and	O
rational	O
numbers	O
.	O
</s>
<s>
If	O
integers	O
are	O
encoded	O
by	O
Gödel	O
numbers	O
in	O
a	O
standard	O
way	O
,	O
the	O
arithmetic	O
operations	O
including	O
addition	O
,	O
subtraction	O
,	O
and	O
multiplication	O
are	O
all	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
Similarly	O
,	O
if	O
the	O
rationals	O
are	O
represented	O
by	O
Gödel	O
numbers	O
then	O
the	O
field	O
operations	O
are	O
all	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
In	O
the	O
following	O
we	O
observe	O
that	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
can	O
be	O
of	O
four	O
types	O
:	O
</s>
<s>
The	O
functions	O
16-20	O
and	O
#G	O
are	O
of	O
particular	O
interest	O
with	O
respect	O
to	O
converting	O
primitive	B-Architecture
recursive	I-Architecture
predicates	O
to	O
,	O
and	O
extracting	O
them	O
from	O
,	O
their	O
"	O
arithmetical	O
"	O
form	O
expressed	O
as	O
Gödel	O
numbers	O
.	O
</s>
<s>
#A	O
:	O
A	O
function	O
φ	O
definable	O
explicitly	O
from	O
functions	O
Ψ	O
and	O
constants	O
q1	O
,	O
...	O
qn	O
is	O
primitive	B-Architecture
recursive	I-Architecture
in	O
Ψ	O
.	O
</s>
<s>
#B	O
:	O
The	O
finite	O
sum	O
Σy	O
<	O
z	O
ψ(x, y )	O
and	O
product	O
Πy	O
<	O
zψ(x, y )	O
are	O
primitive	B-Architecture
recursive	I-Architecture
in	O
ψ	O
.	O
</s>
<s>
#C	O
:	O
A	O
predicate	O
P	O
obtained	O
by	O
substituting	O
functions	O
χ1	O
,...,	O
χm	O
for	O
the	O
respective	O
variables	O
of	O
a	O
predicate	O
Q	O
is	O
primitive	B-Architecture
recursive	I-Architecture
in	O
χ1	O
,...,	O
χm	O
,	O
Q	O
.	O
</s>
<s>
#D	O
:	O
The	O
following	O
predicates	O
are	O
primitive	B-Architecture
recursive	I-Architecture
in	O
Q	O
and	O
R	O
:	O
</s>
<s>
#E	O
:	O
The	O
following	O
predicates	O
are	O
primitive	B-Architecture
recursive	I-Architecture
in	O
the	O
predicate	O
R	O
:	O
</s>
<s>
#F	O
:	O
Definition	O
by	O
cases	O
:	O
The	O
function	O
defined	O
thus	O
,	O
where	O
Q1	O
,	O
...	O
,	O
Qm	O
are	O
mutually	O
exclusive	O
predicates	O
(	O
or	O
"	O
ψ(x )	O
shall	O
have	O
the	O
value	O
given	O
by	O
the	O
first	O
clause	O
that	O
applies	O
)	O
,	O
is	O
primitive	B-Architecture
recursive	I-Architecture
in	O
φ1	O
,	O
...	O
,	O
Q1	O
,	O
...	O
Qm	O
:	O
</s>
<s>
φ(y,x )	O
=	O
χ( y	O
,	O
COURSE-φ	O
( y	O
;	O
x2	O
,	O
...	O
xn	O
)	O
,	O
x2	O
,	O
...	O
xn	O
then	O
φ	O
is	O
primitive	B-Architecture
recursive	I-Architecture
in	O
χ	O
.	O
</s>
<s>
Thus	O
in	O
order	O
to	O
define	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
one	O
has	O
to	O
use	O
the	O
following	O
trick	O
by	O
Gödel	O
.	O
</s>
<s>
Such	O
a	O
number	O
can	O
therefore	O
represent	O
the	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
until	O
a	O
given	O
n	O
.	O
</s>
<s>
Let	O
h	O
be	O
a	O
1-ary	O
primitive	B-Architecture
recursion	I-Architecture
function	O
defined	O
by	O
:	O
</s>
<s>
The	O
generalization	O
to	O
any	O
k-ary	O
primitive	B-Architecture
recursion	I-Architecture
function	O
is	O
trivial	O
.	O
</s>
<s>
The	O
use	O
of	O
this	O
operator	O
may	O
result	O
in	O
a	O
partial	B-Algorithm
function	I-Algorithm
,	O
that	O
is	O
,	O
a	O
relation	O
with	O
at	O
most	O
one	O
value	O
for	O
each	O
argument	O
,	O
but	O
does	O
not	O
necessarily	O
have	O
any	O
value	O
for	O
any	O
argument	O
(	O
see	O
domain	B-Algorithm
)	O
.	O
</s>
<s>
An	O
equivalent	O
definition	O
states	O
that	O
a	O
partial	O
recursive	O
function	O
is	O
one	O
that	O
can	O
be	O
computed	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
A	O
total	B-Algorithm
recursive	O
function	O
is	O
a	O
partial	O
recursive	O
function	O
that	O
is	O
defined	O
for	O
every	O
input	O
.	O
</s>
<s>
Every	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
is	O
total	B-Algorithm
recursive	O
,	O
but	O
not	O
all	O
total	B-Algorithm
recursive	O
functions	O
are	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
The	O
Ackermann	O
function	O
A(m,n )	O
is	O
a	O
well-known	O
example	O
of	O
a	O
total	B-Algorithm
recursive	O
function	O
(	O
in	O
fact	O
,	O
provable	O
total	B-Algorithm
)	O
,	O
that	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
There	O
is	O
a	O
characterization	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
as	O
a	O
subset	O
of	O
the	O
total	B-Algorithm
recursive	O
functions	O
using	O
the	O
Ackermann	O
function	O
.	O
</s>
<s>
This	O
characterization	O
states	O
that	O
a	O
function	O
is	O
primitive	B-Architecture
recursive	I-Architecture
if	O
and	O
only	O
if	O
there	O
is	O
a	O
natural	O
number	O
m	O
such	O
that	O
the	O
function	O
can	O
be	O
computed	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
that	O
always	O
halts	O
within	O
A(m,n )	O
or	O
fewer	O
steps	O
,	O
where	O
n	O
is	O
the	O
sum	O
of	O
the	O
arguments	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
.	O
</s>
<s>
An	O
important	O
property	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
is	O
that	O
they	O
are	O
a	O
recursively	O
enumerable	O
subset	O
of	O
the	O
set	O
of	O
all	O
total	B-Algorithm
recursive	O
functions	O
(	O
which	O
is	O
not	O
itself	O
recursively	O
enumerable	O
)	O
.	O
</s>
<s>
This	O
means	O
that	O
there	O
is	O
a	O
single	O
computable	O
function	O
f(m,n )	O
that	O
enumerates	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
,	O
namely	O
:	O
</s>
<s>
For	O
every	O
m	O
,	O
the	O
function	O
h(n )	O
=	O
f(m,n )	O
is	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
f	O
can	O
be	O
explicitly	O
constructed	O
by	O
iteratively	O
repeating	O
all	O
possible	O
ways	O
of	O
creating	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
provably	O
total	B-Algorithm
.	O
</s>
<s>
But	O
if	O
this	O
equals	O
some	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
,	O
there	O
is	O
an	O
m	O
such	O
that	O
h(n )	O
=	O
f(m,n )	O
for	O
all	O
n	O
,	O
and	O
then	O
h(m )	O
=	O
f(m,m )	O
,	O
leading	O
to	O
contradiction	O
.	O
</s>
<s>
However	O
,	O
the	O
set	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
is	O
not	O
the	O
largest	O
recursively	O
enumerable	O
subset	O
of	O
the	O
set	O
of	O
all	O
total	B-Algorithm
recursive	O
functions	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
set	O
of	O
provably	O
total	B-Algorithm
functions	I-Algorithm
(	O
in	O
Peano	O
arithmetic	O
)	O
is	O
also	O
recursively	O
enumerable	O
,	O
as	O
one	O
can	O
enumerate	O
all	O
the	O
proofs	O
of	O
the	O
theory	O
.	O
</s>
<s>
While	O
all	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
are	O
provably	O
total	B-Algorithm
,	O
the	O
converse	O
is	O
not	O
true	O
.	O
</s>
<s>
Primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
tend	O
to	O
correspond	O
very	O
closely	O
with	O
our	O
intuition	O
of	O
what	O
a	O
computable	O
function	O
must	O
be	O
.	O
</s>
<s>
Certainly	O
the	O
initial	O
functions	O
are	O
intuitively	O
computable	O
(	O
in	O
their	O
very	O
simplicity	O
)	O
,	O
and	O
the	O
two	O
operations	O
by	O
which	O
one	O
can	O
create	O
new	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
are	O
also	O
very	O
straightforward	O
.	O
</s>
<s>
However	O
,	O
the	O
set	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
does	O
not	O
include	O
every	O
possible	O
total	B-Algorithm
computable	O
function	O
—	O
this	O
can	O
be	O
seen	O
with	O
a	O
variant	O
of	O
Cantor	O
's	O
diagonal	O
argument	O
.	O
</s>
<s>
This	O
argument	O
provides	O
a	O
total	B-Algorithm
computable	O
function	O
that	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
This	O
argument	O
can	O
be	O
applied	O
to	O
any	O
class	O
of	O
computable	O
(	O
total	B-Algorithm
)	O
functions	O
that	O
can	O
be	O
enumerated	O
in	O
this	O
way	O
,	O
as	O
explained	O
in	O
the	O
article	O
Machine	O
that	O
always	O
halts	O
.	O
</s>
<s>
Note	O
however	O
that	O
the	O
partial	O
computable	O
functions	O
(	O
those	O
that	O
need	O
not	O
be	O
defined	O
for	O
all	O
arguments	O
)	O
can	O
be	O
explicitly	O
enumerated	O
,	O
for	O
instance	O
by	O
enumerating	O
Turing	B-Architecture
machine	I-Architecture
encodings	O
.	O
</s>
<s>
Other	O
examples	O
of	O
total	B-Algorithm
recursive	O
but	O
not	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
are	O
known	O
:	O
</s>
<s>
The	O
function	O
that	O
takes	O
m	O
to	O
Ackermann(m,m )	O
is	O
a	O
unary	O
total	B-Algorithm
recursive	O
function	O
that	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
The	O
Paris	O
–	O
Harrington	O
theorem	O
involves	O
a	O
total	B-Algorithm
recursive	O
function	O
that	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
The	O
1-place	O
predecessor	O
function	O
is	O
primitive	B-Architecture
recursive	I-Architecture
,	O
see	O
section	O
#Predecessor	O
.	O
</s>
<s>
They	O
proved	O
that	O
the	O
predecessor	O
function	O
still	O
could	O
be	O
defined	O
,	O
and	O
hence	O
that	O
"	O
weak	O
"	O
primitive	B-Architecture
recursion	I-Architecture
also	O
defines	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
The	O
class	O
of	O
iterative	O
functions	O
is	O
defined	O
the	O
same	O
way	O
as	O
the	O
class	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
except	O
with	O
this	O
weaker	O
rule	O
.	O
</s>
<s>
These	O
are	O
conjectured	O
to	O
be	O
a	O
proper	O
subset	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
<s>
Course-of-values	B-Algorithm
recursion	I-Algorithm
defines	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
Some	O
forms	O
of	O
mutual	B-Algorithm
recursion	I-Algorithm
also	O
define	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
The	O
functions	O
that	O
can	O
be	O
programmed	O
in	O
the	O
LOOP	O
programming	O
language	O
are	O
exactly	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
The	O
main	O
limitation	O
of	O
the	O
LOOP	O
language	O
,	O
compared	O
to	O
a	O
Turing-complete	B-Algorithm
language	O
,	O
is	O
that	O
in	O
the	O
LOOP	O
language	O
the	O
number	O
of	O
times	O
that	O
each	O
loop	O
will	O
run	O
is	O
specified	O
before	O
the	O
loop	O
begins	O
to	O
run	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
primitive	B-Architecture
recursive	I-Architecture
programming	O
language	O
is	O
one	O
that	O
contains	O
basic	O
arithmetic	O
operators	O
(	O
e.g.	O
</s>
<s>
+	O
and	O
−	O
,	O
or	O
ADD	O
and	O
SUBTRACT	O
)	O
,	O
conditionals	O
and	O
comparison	O
(	O
IF-THEN	O
,	O
EQUALS	O
,	O
LESS-THAN	O
)	O
,	O
and	O
bounded	B-Language
loops	I-Language
,	O
such	O
as	O
the	O
basic	O
for	B-Language
loop	I-Language
,	O
where	O
there	O
is	O
a	O
known	O
or	O
calculable	O
upper	O
bound	O
to	O
all	O
loops	O
(	O
FOR	O
i	O
FROM	O
1	O
TO	O
n	O
,	O
with	O
neither	O
i	O
nor	O
n	O
modifiable	O
by	O
the	O
loop	O
body	O
)	O
.	O
</s>
<s>
No	O
control	O
structures	O
of	O
greater	O
generality	O
,	O
such	O
as	O
while	O
loops	O
or	O
IF-THEN	O
plus	O
GOTO	B-Application
,	O
are	O
admitted	O
in	O
a	O
primitive	B-Architecture
recursive	I-Architecture
language	O
.	O
</s>
<s>
Its	O
computing	O
power	O
coincides	O
with	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
A	O
variant	O
of	O
the	O
LOOP	O
language	O
is	O
Douglas	O
Hofstadter	O
's	O
BlooP	B-Language
in	O
Gödel	O
,	O
Escher	O
,	O
Bach	O
.	O
</s>
<s>
Adding	O
unbounded	O
loops	O
(	O
WHILE	O
,	O
GOTO	B-Application
)	O
makes	O
the	O
language	O
general	O
recursive	O
and	O
Turing-complete	B-Algorithm
,	O
as	O
are	O
all	O
real-world	O
computer	O
programming	O
languages	O
.	O
</s>
<s>
The	O
definition	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
implies	O
that	O
their	O
computation	O
halts	O
on	O
every	O
input	O
(	O
after	O
a	O
finite	O
number	O
of	O
steps	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
halting	O
problem	O
is	O
undecidable	O
for	O
general	O
recursive	O
functions	O
,	O
even	O
if	O
they	O
are	O
total	B-Algorithm
.	O
</s>
<s>
The	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
are	O
closely	O
related	O
to	O
mathematical	O
finitism	O
,	O
and	O
are	O
used	O
in	O
several	O
contexts	O
in	O
mathematical	O
logic	O
where	O
a	O
particularly	O
constructive	O
system	O
is	O
desired	O
.	O
</s>
<s>
Primitive	B-Architecture
recursive	I-Architecture
arithmetic	O
(	O
PRA	O
)	O
,	O
a	O
formal	O
axiom	B-Algorithm
system	O
for	O
the	O
natural	O
numbers	O
and	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
on	O
them	O
,	O
is	O
often	O
used	O
for	O
this	O
purpose	O
.	O
</s>
<s>
Similarly	O
,	O
many	O
of	O
the	O
syntactic	O
results	O
in	O
proof	O
theory	O
can	O
be	O
proved	O
in	O
PRA	O
,	O
which	O
implies	O
that	O
there	O
are	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
that	O
carry	O
out	O
the	O
corresponding	O
syntactic	O
transformations	O
of	O
proofs	O
.	O
</s>
<s>
Such	O
a	O
proof	O
establishes	O
that	O
the	O
consistency	O
of	O
a	O
theory	O
T	O
implies	O
the	O
consistency	O
of	O
a	O
theory	O
S	O
by	O
producing	O
a	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
that	O
can	O
transform	O
any	O
proof	O
of	O
an	O
inconsistency	O
from	O
S	O
into	O
a	O
proof	O
of	O
an	O
inconsistency	O
from	O
T	O
.	O
One	O
sufficient	O
condition	O
for	O
a	O
consistency	O
proof	O
to	O
be	O
finitistic	O
is	O
the	O
ability	O
to	O
formalize	O
it	O
in	O
PRA	O
.	O
</s>
<s>
Recursive	O
definitions	O
had	O
been	O
used	O
more	O
or	O
less	O
formally	O
in	O
mathematics	O
before	O
,	O
but	O
the	O
construction	O
of	O
primitive	B-Architecture
recursion	I-Architecture
is	O
traced	O
back	O
to	O
Richard	O
Dedekind	O
's	O
theorem	O
126	O
of	O
his	O
Was	O
sind	O
und	O
was	O
sollen	O
die	O
Zahlen	O
?	O
</s>
<s>
Primitive	B-Architecture
recursive	I-Architecture
arithmetic	O
was	O
first	O
proposed	O
by	O
Thoralf	O
Skolem	O
in	O
1923	O
.	O
</s>
<s>
The	O
current	O
terminology	O
was	O
coined	O
by	O
Rózsa	O
Péter	O
(	O
1934	O
)	O
after	O
Ackermann	O
had	O
proved	O
in	O
1928	O
that	O
the	O
function	O
which	O
today	O
is	O
named	O
after	O
him	O
was	O
not	O
primitive	B-Architecture
recursive	I-Architecture
,	O
an	O
event	O
which	O
prompted	O
the	O
need	O
to	O
rename	O
what	O
until	O
then	O
were	O
simply	O
called	O
recursive	O
functions	O
.	O
</s>
