<s>
Equivalently	O
,	O
assuming	O
the	O
axiom	O
of	O
dependent	O
choice	O
,	O
a	O
relation	O
is	O
well-founded	O
when	O
it	O
contains	O
no	O
infinite	B-Algorithm
descending	I-Algorithm
chains	I-Algorithm
,	O
which	O
can	O
be	O
proved	O
when	O
there	O
is	O
no	O
infinite	O
sequence	O
of	O
elements	O
of	O
such	O
that	O
for	O
every	O
natural	O
number	O
.	O
</s>
<s>
The	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
,	O
which	O
is	O
one	O
of	O
the	O
axioms	O
of	O
Zermelo	O
–	O
Fraenkel	O
set	O
theory	O
,	O
asserts	O
that	O
all	O
sets	O
are	O
well-founded	O
.	O
</s>
<s>
Well-founded	B-Algorithm
induction	I-Algorithm
is	O
sometimes	O
called	O
Noetherian	O
induction	B-Algorithm
,	O
after	O
Emmy	O
Noether	O
.	O
</s>
<s>
On	O
par	O
with	O
induction	B-Algorithm
,	O
well-founded	O
relations	O
also	O
support	O
construction	O
of	O
objects	O
by	O
transfinite	O
recursion	O
.	O
</s>
<s>
Then	O
induction	B-Algorithm
on	O
is	O
the	O
usual	O
mathematical	B-Algorithm
induction	I-Algorithm
,	O
and	O
recursion	O
on	O
gives	O
primitive	B-Architecture
recursion	I-Architecture
.	O
</s>
<s>
If	O
we	O
consider	O
the	O
order	O
relation	O
,	O
we	O
obtain	O
complete	O
induction	B-Algorithm
,	O
and	O
course-of-values	B-Algorithm
recursion	I-Algorithm
.	O
</s>
<s>
The	O
statement	O
that	O
is	O
well-founded	O
is	O
also	O
known	O
as	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
.	O
</s>
<s>
There	O
are	O
other	O
interesting	O
special	O
cases	O
of	O
well-founded	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
When	O
the	O
well-founded	O
relation	O
is	O
the	O
usual	O
ordering	O
on	O
the	O
class	O
of	O
all	O
ordinal	B-Language
numbers	I-Language
,	O
the	O
technique	O
is	O
called	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
When	O
the	O
well-founded	O
set	O
is	O
a	O
set	O
of	O
recursively-defined	O
data	O
structures	O
,	O
the	O
technique	O
is	O
called	O
structural	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
When	O
the	O
well-founded	O
relation	O
is	O
set	O
membership	O
on	O
the	O
universal	O
class	O
,	O
the	O
technique	O
is	O
known	O
as	O
∈	B-Algorithm
-induction	I-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
.	O
</s>
<s>
The	O
set	O
of	O
strings	O
over	O
a	O
finite	O
alphabet	O
with	O
more	O
than	O
one	O
element	O
,	O
under	O
the	O
usual	O
(	O
lexicographic	O
)	O
order	O
,	O
since	O
the	O
sequence	O
is	O
an	O
infinite	B-Algorithm
descending	I-Algorithm
chain	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
the	O
union	O
of	O
the	O
positive	O
integers	O
with	O
a	O
new	O
element	O
ω	B-Language
that	O
is	O
bigger	O
than	O
any	O
integer	O
.	O
</s>
<s>
there	O
are	O
descending	O
chains	O
starting	O
at	O
ω	B-Language
of	O
arbitrary	O
great	O
(	O
finite	O
)	O
length	O
;	O
</s>
<s>
Every	O
reflexive	O
relation	O
on	O
a	O
nonempty	O
domain	O
has	O
infinite	B-Algorithm
descending	I-Algorithm
chains	I-Algorithm
,	O
because	O
any	O
constant	O
sequence	O
is	O
a	O
descending	O
chain	O
.	O
</s>
