<s>
In	O
mathematics	O
,	O
a	O
binary	O
relation	O
is	O
called	O
well-founded	B-Algorithm
(	O
or	O
wellfounded	B-Algorithm
or	O
foundational	O
)	O
on	O
a	O
class	O
if	O
every	O
non-empty	O
subset	O
has	O
a	O
minimal	O
element	O
with	O
respect	O
to	O
,	O
that	O
is	O
,	O
an	O
element	O
not	O
related	O
by	O
(	O
for	O
instance	O
,	O
"	O
is	O
not	O
smaller	O
than	O
"	O
)	O
for	O
any	O
.	O
</s>
<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	B-Algorithm
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>
In	O
order	O
theory	O
,	O
a	O
partial	O
order	O
is	O
called	O
well-founded	B-Algorithm
if	O
the	O
corresponding	O
strict	O
order	O
is	O
a	O
well-founded	B-Algorithm
relation	I-Algorithm
.	O
</s>
<s>
In	O
set	O
theory	O
,	O
a	O
set	O
is	O
called	O
a	O
well-founded	B-Algorithm
set	I-Algorithm
if	O
the	O
set	O
membership	O
relation	O
is	O
well-founded	B-Algorithm
on	O
the	O
transitive	O
closure	O
of	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	B-Algorithm
.	O
</s>
<s>
A	O
relation	O
is	O
converse	O
well-founded	B-Algorithm
,	O
upwards	O
well-founded	B-Algorithm
or	O
Noetherian	O
on	O
,	O
if	O
the	O
converse	O
relation	O
is	O
well-founded	B-Algorithm
on	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
rewriting	O
systems	O
,	O
a	O
Noetherian	B-Algorithm
relation	I-Algorithm
is	O
also	O
called	O
terminating	O
.	O
</s>
<s>
Well-founded	B-Algorithm
induction	B-Algorithm
is	O
sometimes	O
called	O
Noetherian	B-Algorithm
induction	I-Algorithm
,	O
after	O
Emmy	O
Noether	O
.	O
</s>
<s>
On	O
par	O
with	O
induction	B-Algorithm
,	O
well-founded	B-Algorithm
relations	I-Algorithm
also	O
support	O
construction	O
of	O
objects	O
by	O
transfinite	O
recursion	O
.	O
</s>
<s>
Let	O
be	O
a	O
set-like	O
well-founded	B-Algorithm
relation	I-Algorithm
and	O
a	O
function	O
that	O
assigns	O
an	O
object	O
to	O
each	O
pair	O
of	O
an	O
element	O
and	O
a	O
function	O
on	O
the	O
initial	O
segment	O
of	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
the	O
well-founded	B-Algorithm
relation	I-Algorithm
,	O
where	O
is	O
the	O
set	O
of	O
all	O
natural	O
numbers	O
,	O
and	O
is	O
the	O
graph	O
of	O
the	O
successor	O
function	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	B-Algorithm
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	B-Algorithm
.	O
</s>
<s>
When	O
the	O
well-founded	B-Algorithm
relation	I-Algorithm
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	B-Algorithm
set	I-Algorithm
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	B-Algorithm
relation	I-Algorithm
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>
Well-founded	B-Algorithm
relations	I-Algorithm
that	O
are	O
not	O
totally	O
ordered	O
include	O
:	O
</s>
<s>
This	O
is	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
.	O
</s>
<s>
Examples	O
of	O
relations	O
that	O
are	O
not	O
well-founded	B-Algorithm
include	O
:	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>
This	O
relation	O
fails	O
to	O
be	O
well-founded	B-Algorithm
even	O
though	O
the	O
entire	O
set	O
has	O
a	O
minimum	O
element	O
,	O
namely	O
the	O
empty	O
string	O
.	O
</s>
<s>
If	O
is	O
a	O
well-founded	B-Algorithm
relation	I-Algorithm
and	O
is	O
an	O
element	O
of	O
,	O
then	O
the	O
descending	O
chains	O
starting	O
at	O
are	O
all	O
finite	O
,	O
but	O
this	O
does	O
not	O
mean	O
that	O
their	O
lengths	O
are	O
necessarily	O
bounded	O
.	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>
The	O
Mostowski	O
collapse	O
lemma	O
implies	O
that	O
set	O
membership	O
is	O
a	O
universal	O
among	O
the	O
extensional	O
well-founded	B-Algorithm
relations	I-Algorithm
:	O
for	O
any	O
set-like	O
well-founded	B-Algorithm
relation	I-Algorithm
on	O
a	O
class	O
that	O
is	O
extensional	O
,	O
there	O
exists	O
a	O
class	O
such	O
that	O
is	O
isomorphic	O
to	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>
<s>
In	O
the	O
context	O
of	O
the	O
natural	O
numbers	O
,	O
this	O
means	O
that	O
the	O
relation	O
<	O
,	O
which	O
is	O
well-founded	B-Algorithm
,	O
is	O
used	O
instead	O
of	O
the	O
relation	O
≤	O
,	O
which	O
is	O
not	O
.	O
</s>
<s>
In	O
some	O
texts	O
,	O
the	O
definition	O
of	O
a	O
well-founded	B-Algorithm
relation	I-Algorithm
is	O
changed	O
from	O
the	O
definition	O
above	O
to	O
include	O
these	O
conventions	O
.	O
</s>
