<s>
In	O
mathematics	O
,	O
primitive	B-Algorithm
recursive	I-Algorithm
set	I-Algorithm
functions	I-Algorithm
or	O
primitive	B-Algorithm
recursive	I-Algorithm
ordinal	I-Algorithm
functions	I-Algorithm
are	O
analogs	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
,	O
defined	O
for	O
sets	O
or	O
ordinals	B-Language
rather	O
than	O
natural	O
numbers	O
.	O
</s>
<s>
A	O
primitive	B-Algorithm
recursive	I-Algorithm
set	I-Algorithm
function	I-Algorithm
is	O
a	O
function	O
from	O
sets	O
to	O
sets	O
that	O
can	O
be	O
obtained	O
from	O
the	O
following	O
basic	O
functions	O
by	O
repeatedly	O
applying	O
the	O
following	O
rules	O
of	O
substitution	O
and	O
recursion	O
:	O
</s>
<s>
A	O
primitive	B-Algorithm
recursive	I-Algorithm
ordinal	I-Algorithm
function	I-Algorithm
is	O
defined	O
in	O
the	O
same	O
way	O
,	O
except	O
that	O
the	O
initial	O
function	O
F(x,y )	O
=	O
x∪{y}	O
is	O
replaced	O
by	O
F(x )	O
=	O
x∪{x}	O
(	O
the	O
successor	O
of	O
x	O
)	O
.	O
</s>
<s>
The	O
primitive	B-Algorithm
recursive	I-Algorithm
ordinal	I-Algorithm
functions	I-Algorithm
are	O
the	O
same	O
as	O
the	O
primitive	B-Algorithm
recursive	I-Algorithm
set	I-Algorithm
functions	I-Algorithm
that	O
map	O
ordinals	B-Language
to	O
ordinals	B-Language
.	O
</s>
<s>
Examples	O
of	O
primitive	B-Algorithm
recursive	I-Algorithm
set	I-Algorithm
functions	I-Algorithm
:	O
</s>
<s>
For	O
example	O
,	O
the	O
ordinal	B-Language
function	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
,	O
because	O
the	O
constant	O
function	O
with	O
value	O
ω	B-Language
(	O
or	O
any	O
other	O
infinite	O
set	O
)	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
,	O
so	O
one	O
might	O
want	O
to	O
add	O
this	O
constant	O
function	O
to	O
the	O
initial	O
functions	O
.	O
</s>
<s>
The	O
notion	O
of	O
a	O
set	O
function	O
being	O
primitive	B-Architecture
recursive	I-Architecture
in	O
ω	B-Language
has	O
the	O
same	O
definition	O
as	O
that	O
of	O
primitive	B-Architecture
recursion	I-Architecture
,	O
except	O
with	O
ω	B-Language
as	O
a	O
parameter	O
kept	O
fixed	O
,	O
not	O
altered	O
by	O
the	O
primitive	B-Architecture
recursion	I-Architecture
schemata	O
.	O
</s>
<s>
Lα	O
is	O
closed	O
under	O
primitive	B-Algorithm
recursive	I-Algorithm
set	I-Algorithm
functions	I-Algorithm
iff	O
α	O
is	O
closed	O
under	O
each	O
for	O
all	O
.	O
</s>
