<s>
In	O
mathematics	O
,	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
states	O
that	O
every	O
non-empty	O
set	O
of	O
positive	O
integers	O
contains	O
a	O
least	O
element	O
.	O
</s>
<s>
The	O
phrase	O
"	O
well-ordering	B-Algorithm
principle	I-Algorithm
"	O
is	O
sometimes	O
taken	O
to	O
be	O
synonymous	O
with	O
the	O
"	O
well-ordering	O
theorem	O
"	O
.	O
</s>
<s>
Depending	O
on	O
the	O
framework	O
in	O
which	O
the	O
natural	O
numbers	O
are	O
introduced	O
,	O
this	O
(	O
second	O
order	O
)	O
property	O
of	O
the	O
set	O
of	O
natural	O
numbers	O
is	O
either	O
an	O
axiom	B-Algorithm
or	O
a	O
provable	O
theorem	O
.	O
</s>
<s>
In	O
Peano	O
arithmetic	O
,	O
second-order	O
arithmetic	O
and	O
related	O
systems	O
,	O
and	O
indeed	O
in	O
most	O
(	O
not	O
necessarily	O
formal	O
)	O
mathematical	O
treatments	O
of	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
,	O
the	O
principle	O
is	O
derived	O
from	O
the	O
principle	B-Algorithm
of	I-Algorithm
mathematical	I-Algorithm
induction	I-Algorithm
,	O
which	O
is	O
itself	O
taken	O
as	O
basic	O
.	O
</s>
<s>
Considering	O
the	O
natural	O
numbers	O
as	O
a	O
subset	O
of	O
the	O
real	O
numbers	O
,	O
and	O
assuming	O
that	O
we	O
know	O
already	O
that	O
the	O
real	O
numbers	O
are	O
complete	O
(	O
again	O
,	O
either	O
as	O
an	O
axiom	B-Algorithm
or	O
a	O
theorem	O
about	O
the	O
real	O
number	O
system	O
)	O
,	O
i.e.	O
,	O
every	O
bounded	O
(	O
from	O
below	O
)	O
set	O
has	O
an	O
infimum	O
,	O
then	O
also	O
every	O
set	O
of	O
natural	O
numbers	O
has	O
an	O
infimum	O
,	O
say	O
.	O
</s>
<s>
One	O
can	O
(	O
even	O
without	O
invoking	O
the	O
regularity	B-General_Concept
axiom	I-General_Concept
)	O
show	O
that	O
the	O
set	O
of	O
all	O
natural	O
numbers	O
such	O
that	O
"	O
is	O
well-ordered	O
"	O
is	O
inductive	O
,	O
and	O
must	O
therefore	O
contain	O
all	O
natural	O
numbers	O
;	O
from	O
this	O
property	O
one	O
can	O
conclude	O
that	O
the	O
set	O
of	O
all	O
natural	O
numbers	O
is	O
also	O
well-ordered	O
.	O
</s>
<s>
This	O
mode	O
of	O
argument	O
is	O
the	O
contrapositive	O
of	O
proof	O
by	O
complete	O
induction	B-Algorithm
.	O
</s>
<s>
It	O
is	O
known	O
light-heartedly	O
as	O
the	O
"	O
minimal	O
criminal	O
"	O
method	O
and	O
is	O
similar	O
in	O
its	O
nature	O
to	O
Fermat	B-Language
's	I-Language
method	O
of	O
"	O
infinite	O
descent	O
"	O
.	O
</s>
<s>
Garrett	O
Birkhoff	O
and	O
Saunders	O
Mac	O
Lane	O
wrote	O
in	O
A	O
Survey	O
of	O
Modern	O
Algebra	O
that	O
this	O
property	O
,	O
like	O
the	O
least	O
upper	O
bound	O
axiom	B-Algorithm
for	O
real	O
numbers	O
,	O
is	O
non-algebraic	O
;	O
i.e.	O
,	O
it	O
cannot	O
be	O
deduced	O
from	O
the	O
algebraic	O
properties	O
of	O
the	O
integers	O
(	O
which	O
form	O
an	O
ordered	O
integral	O
domain	O
)	O
.	O
</s>
