<s>
Transfinite	B-Algorithm
induction	I-Algorithm
is	O
an	O
extension	O
of	O
mathematical	B-Algorithm
induction	I-Algorithm
to	O
well-ordered	O
sets	O
,	O
for	O
example	O
to	O
sets	O
of	O
ordinal	B-Language
numbers	I-Language
or	O
cardinal	O
numbers	O
.	O
</s>
<s>
Let	O
be	O
a	O
property	O
defined	O
for	O
all	O
ordinals	B-Language
.	O
</s>
<s>
Then	O
transfinite	B-Algorithm
induction	I-Algorithm
tells	O
us	O
that	O
is	O
true	O
for	O
all	O
ordinals	B-Language
.	O
</s>
<s>
Successor	O
case	O
:	O
Prove	O
that	O
for	O
any	O
successor	O
ordinal	B-Language
,	O
follows	O
from	O
(	O
and	O
,	O
if	O
necessary	O
,	O
for	O
all	O
)	O
.	O
</s>
<s>
Limit	O
case	O
:	O
Prove	O
that	O
for	O
any	O
limit	O
ordinal	B-Language
,	O
follows	O
from	O
for	O
all	O
.	O
</s>
<s>
All	O
three	O
cases	O
are	O
identical	O
except	O
for	O
the	O
type	O
of	O
ordinal	B-Language
considered	O
.	O
</s>
<s>
Zero	O
is	O
sometimes	O
considered	O
a	O
limit	O
ordinal	B-Language
and	O
then	O
may	O
sometimes	O
be	O
treated	O
in	O
proofs	O
in	O
the	O
same	O
case	O
as	O
limit	O
ordinals	B-Language
.	O
</s>
<s>
Transfinite	O
recursion	O
is	O
similar	O
to	O
transfinite	B-Algorithm
induction	I-Algorithm
;	O
however	O
,	O
instead	O
of	O
proving	O
that	O
something	O
holds	O
for	O
all	O
ordinal	B-Language
numbers	I-Language
,	O
we	O
construct	O
a	O
sequence	O
of	O
objects	O
,	O
one	O
for	O
each	O
ordinal	B-Language
.	O
</s>
<s>
As	O
an	O
example	O
,	O
a	O
basis	O
for	O
a	O
(	O
possibly	O
infinite-dimensional	O
)	O
vector	O
space	O
can	O
be	O
created	O
by	O
starting	O
with	O
the	O
empty	O
set	O
and	O
for	O
each	O
ordinal	B-Language
α	O
>	O
0	O
choosing	O
a	O
vector	O
that	O
is	O
not	O
in	O
the	O
span	O
of	O
the	O
vectors	O
.	O
</s>
<s>
for	O
all	O
ordinals	B-Language
α	O
,	O
where	O
denotes	O
the	O
restriction	O
of	O
Fs	O
domain	O
to	O
ordinals	B-Language
<	O
α	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
of	O
induction	B-Algorithm
,	O
we	O
may	O
treat	O
different	O
types	O
of	O
ordinals	B-Language
separately	O
:	O
another	O
formulation	O
of	O
transfinite	O
recursion	O
is	O
the	O
following:Transfinite	O
Recursion	O
Theorem	O
(	O
version	O
2	O
)	O
'''	O
.	O
</s>
<s>
The	O
uniqueness	O
of	O
the	O
sequence	O
satisfying	O
these	O
properties	O
can	O
be	O
proved	O
using	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
one	O
can	O
define	O
objects	O
by	O
transfinite	O
recursion	O
on	O
any	O
well-founded	B-Algorithm
relation	I-Algorithm
R	O
.	O
(	O
R	O
need	O
not	O
even	O
be	O
a	O
set	O
;	O
it	O
can	O
be	O
a	O
proper	O
class	O
,	O
provided	O
it	O
is	O
a	O
set-like	O
relation	O
;	O
i.e.	O
</s>
<s>
Proofs	O
or	O
constructions	O
using	O
induction	B-Algorithm
and	O
recursion	O
often	O
use	O
the	O
axiom	O
of	O
choice	O
to	O
produce	O
a	O
well-ordered	O
relation	O
that	O
can	O
be	O
treated	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
However	O
,	O
if	O
the	O
relation	O
in	O
question	O
is	O
already	O
well-ordered	O
,	O
one	O
can	O
often	O
use	O
transfinite	B-Algorithm
induction	I-Algorithm
without	O
invoking	O
the	O
axiom	O
of	O
choice.In	O
fact	O
,	O
the	O
domain	O
of	O
the	O
relation	O
does	O
not	O
even	O
need	O
to	O
be	O
a	O
set	O
.	O
</s>
<s>
For	O
example	O
,	O
many	O
results	O
about	O
Borel	O
sets	O
are	O
proved	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
on	O
the	O
ordinal	B-Language
rank	O
of	O
the	O
set	O
;	O
these	O
ranks	O
are	O
already	O
well-ordered	O
,	O
so	O
the	O
axiom	O
of	O
choice	O
is	O
not	O
needed	O
to	O
well-order	O
them	O
.	O
</s>
<s>
The	O
following	O
construction	O
of	O
the	O
Vitali	O
set	O
shows	O
one	O
way	O
that	O
the	O
axiom	O
of	O
choice	O
can	O
be	O
used	O
in	O
a	O
proof	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
:	O
</s>
<s>
First	O
,	O
well-order	O
the	O
real	O
numbers	O
(	O
this	O
is	O
where	O
the	O
axiom	O
of	O
choice	O
enters	O
via	O
the	O
well-ordering	O
theorem	O
)	O
,	O
giving	O
a	O
sequence	O
,	O
where	O
β	O
is	O
an	O
ordinal	B-Language
with	O
the	O
cardinality	O
of	O
the	O
continuum	O
.	O
</s>
<s>
For	O
inductions	B-Algorithm
and	O
recursions	O
of	O
countable	O
length	O
,	O
the	O
weaker	O
axiom	O
of	O
dependent	O
choice	O
is	O
sufficient	O
.	O
</s>
