<s>
The	O
method	O
can	O
be	O
extended	O
to	O
prove	O
statements	O
about	O
more	O
general	O
well-founded	B-Algorithm
structures	O
,	O
such	O
as	O
trees	O
;	O
this	O
generalization	O
,	O
known	O
as	O
structural	B-Algorithm
induction	I-Algorithm
,	O
is	O
used	O
in	O
mathematical	O
logic	O
and	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
In	O
370	O
BC	O
,	O
Plato	B-Operating_System
's	O
Parmenides	O
may	O
have	O
contained	O
traces	O
of	O
an	O
early	O
example	O
of	O
an	O
implicit	O
inductive	O
proof	O
.	O
</s>
<s>
All	O
variants	O
of	O
induction	O
are	O
special	O
cases	O
of	O
transfinite	B-Algorithm
induction	I-Algorithm
;	O
see	O
below	O
.	O
</s>
<s>
Prefix	O
induction	O
can	O
simulate	O
predecessor	O
induction	O
,	O
but	O
only	O
at	O
the	O
cost	O
of	O
making	O
the	O
statement	O
more	O
syntactically	O
complex	O
(	O
adding	O
a	O
bounded	O
universal	O
quantifier	O
)	O
,	O
so	O
the	O
interesting	O
results	O
relating	O
prefix	O
induction	O
to	O
polynomial-time	O
computation	O
depend	O
on	O
excluding	O
unbounded	O
quantifiers	O
entirely	O
,	O
and	O
limiting	O
the	O
alternation	O
of	O
bounded	O
universal	O
and	O
existential	B-Algorithm
quantifiers	I-Algorithm
allowed	O
in	O
the	O
statement	O
.	O
</s>
<s>
Another	O
variant	O
,	O
called	O
complete	O
induction	O
,	O
course	B-Algorithm
of	I-Algorithm
values	I-Algorithm
induction	I-Algorithm
or	O
strong	O
induction	O
(	O
in	O
contrast	O
to	O
which	O
the	O
basic	O
form	O
of	O
induction	O
is	O
sometimes	O
known	O
as	O
weak	O
induction	O
)	O
,	O
makes	O
the	O
induction	O
step	O
easier	O
to	O
prove	O
by	O
using	O
a	O
stronger	O
hypothesis	O
:	O
one	O
proves	O
the	O
statement	O
under	O
the	O
assumption	O
that	O
holds	O
for	O
all	O
natural	O
numbers	O
less	O
than	O
;	O
by	O
contrast	O
,	O
the	O
basic	O
form	O
only	O
assumes	O
.	O
</s>
<s>
In	O
this	O
form	O
of	O
complete	O
induction	O
,	O
one	O
still	O
has	O
to	O
prove	O
the	O
base	O
case	O
,	O
,	O
and	O
it	O
may	O
even	O
be	O
necessary	O
to	O
prove	O
extra-base	O
cases	O
such	O
as	O
before	O
the	O
general	O
argument	O
applies	O
,	O
as	O
in	O
the	O
example	O
below	O
of	O
the	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
This	O
is	O
a	O
special	O
case	O
of	O
transfinite	B-Algorithm
induction	I-Algorithm
as	O
described	O
below	O
,	O
although	O
it	O
is	O
no	O
longer	O
equivalent	O
to	O
ordinary	O
induction	O
.	O
</s>
<s>
where	O
is	O
the	O
nth	O
Fibonacci	B-Algorithm
number	I-Algorithm
,	O
and	O
(	O
the	O
golden	O
ratio	O
)	O
and	O
are	O
the	O
roots	O
of	O
the	O
polynomial	O
.	O
</s>
<s>
In	O
second-order	O
logic	O
,	O
one	O
can	O
write	O
down	O
the	O
"	O
axiom	B-Algorithm
of	O
induction	O
"	O
as	O
follows	O
:	O
</s>
<s>
The	O
axiom	B-Algorithm
of	O
induction	O
asserts	O
the	O
validity	O
of	O
inferring	O
that	O
holds	O
for	O
any	O
natural	O
number	O
from	O
the	O
base	O
case	O
and	O
the	O
induction	O
step	O
.	O
</s>
<s>
The	O
first	O
quantifier	O
in	O
the	O
axiom	B-Algorithm
ranges	O
over	O
predicates	O
rather	O
than	O
over	O
individual	O
numbers	O
.	O
</s>
<s>
This	O
is	O
a	O
second-order	O
quantifier	O
,	O
which	O
means	O
that	O
this	O
axiom	B-Algorithm
is	O
stated	O
in	O
second-order	O
logic	O
.	O
</s>
<s>
Axiomatizing	O
arithmetic	O
induction	O
in	O
first-order	O
logic	O
requires	O
an	O
axiom	B-Algorithm
schema	O
containing	O
a	O
separate	O
axiom	B-Algorithm
for	O
each	O
possible	O
predicate	O
.	O
</s>
<s>
The	O
article	O
Peano	O
axioms	B-Algorithm
contains	O
further	O
discussion	O
of	O
this	O
issue	O
.	O
</s>
<s>
The	O
axiom	B-Algorithm
of	O
structural	B-Algorithm
induction	I-Algorithm
for	O
the	O
natural	O
numbers	O
was	O
first	O
formulated	O
by	O
Peano	O
,	O
who	O
used	O
it	O
to	O
specify	O
the	O
natural	O
numbers	O
together	O
with	O
the	O
following	O
four	O
other	O
axioms	B-Algorithm
:	O
</s>
<s>
0	O
is	O
not	O
in	O
the	O
range	B-Algorithm
of	O
.	O
</s>
<s>
This	O
is	O
not	O
an	O
axiom	B-Algorithm
,	O
but	O
a	O
theorem	O
,	O
given	O
that	O
natural	O
numbers	O
are	O
defined	O
in	O
the	O
language	O
of	O
ZFC	O
set	O
theory	O
by	O
axioms	B-Algorithm
,	O
analogous	O
to	O
Peano	O
's	O
.	O
</s>
<s>
One	O
variation	O
of	O
the	O
principle	O
of	O
complete	O
induction	O
can	O
be	O
generalized	O
for	O
statements	O
about	O
elements	O
of	O
any	O
well-founded	B-Algorithm
set	I-Algorithm
,	O
that	O
is	O
,	O
a	O
set	O
with	O
an	O
irreflexive	O
relation	O
<	O
that	O
contains	O
no	O
infinite	B-Algorithm
descending	I-Algorithm
chains	I-Algorithm
.	O
</s>
<s>
Every	O
set	O
representing	O
an	O
ordinal	B-Language
number	I-Language
is	O
well-founded	B-Algorithm
,	O
the	O
set	O
of	O
natural	O
numbers	O
is	O
one	O
of	O
them	O
.	O
</s>
<s>
Applied	O
to	O
a	O
well-founded	B-Algorithm
set	I-Algorithm
,	O
transfinite	B-Algorithm
induction	I-Algorithm
can	O
be	O
formulated	O
as	O
a	O
single	O
step	O
.	O
</s>
<s>
To	O
prove	O
that	O
a	O
statement	O
P(n )	O
holds	O
for	O
each	O
ordinal	B-Language
number	I-Language
:	O
</s>
<s>
Show	O
,	O
for	O
each	O
ordinal	B-Language
number	I-Language
n	O
,	O
that	O
if	O
P(m )	O
holds	O
for	O
all	O
,	O
then	O
P(n )	O
also	O
holds	O
.	O
</s>
<s>
This	O
form	O
of	O
induction	O
,	O
when	O
applied	O
to	O
a	O
set	O
of	O
ordinal	B-Language
numbers	I-Language
(	O
which	O
form	O
a	O
well-ordered	O
and	O
hence	O
well-founded	B-Algorithm
class	O
)	O
,	O
is	O
called	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
It	O
is	O
an	O
important	O
proof	O
technique	O
in	O
set	O
theory	O
,	O
topology	B-Architecture
and	O
other	O
fields	O
.	O
</s>
<s>
Proofs	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
typically	O
distinguish	O
three	O
cases	O
:	O
</s>
<s>
n	O
is	O
a	O
so-called	O
limit	O
ordinal	B-Language
.	O
</s>
<s>
Strictly	O
speaking	O
,	O
it	O
is	O
not	O
necessary	O
in	O
transfinite	B-Algorithm
induction	I-Algorithm
to	O
prove	O
a	O
base	O
case	O
,	O
because	O
it	O
is	O
a	O
vacuous	O
special	O
case	O
of	O
the	O
proposition	O
that	O
if	O
P	O
is	O
true	O
of	O
all	O
,	O
then	O
P	O
is	O
true	O
of	O
m	O
.	O
It	O
is	O
vacuously	O
true	O
precisely	O
because	O
there	O
are	O
no	O
values	O
of	O
that	O
could	O
serve	O
as	O
counterexamples	O
.	O
</s>
<s>
The	O
principle	O
of	O
mathematical	O
induction	O
is	O
usually	O
stated	O
as	O
an	O
axiom	B-Algorithm
of	O
the	O
natural	O
numbers	O
;	O
see	O
Peano	O
axioms	B-Algorithm
.	O
</s>
<s>
It	O
is	O
strictly	O
stronger	O
than	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
in	O
the	O
context	O
of	O
the	O
other	O
Peano	O
axioms	B-Algorithm
.	O
</s>
<s>
The	O
trichotomy	O
axiom	B-Algorithm
:	O
For	O
any	O
natural	O
numbers	O
n	O
and	O
m	O
,	O
n	O
is	O
less	O
than	O
or	O
equal	O
to	O
m	O
if	O
and	O
only	O
if	O
m	O
is	O
not	O
less	O
than	O
n	O
.	O
</s>
<s>
It	O
can	O
then	O
be	O
proved	O
that	O
induction	O
,	O
given	O
the	O
above-listed	O
axioms	B-Algorithm
,	O
implies	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
.	O
</s>
<s>
The	O
following	O
proof	O
uses	O
complete	O
induction	O
and	O
the	O
first	O
and	O
fourth	O
axioms	B-Algorithm
.	O
</s>
<s>
Suppose	O
there	B-Algorithm
exists	I-Algorithm
a	O
non-empty	O
set	O
,	O
S	O
,	O
of	O
natural	O
numbers	O
that	O
has	O
no	O
least	O
element	O
.	O
</s>
<s>
Moreover	O
,	O
except	O
for	O
the	O
induction	O
axiom	B-Algorithm
,	O
it	O
satisfies	O
all	O
Peano	O
axioms	B-Algorithm
,	O
where	O
Peano	O
's	O
constant	O
0	O
is	O
interpreted	O
as	O
the	O
pair	O
(	O
0	O
,	O
0	O
)	O
,	O
and	O
Peano	O
's	O
successor	O
function	O
is	O
defined	O
on	O
pairs	O
by	O
for	O
all	O
and	O
.	O
</s>
<s>
As	O
an	O
example	O
for	O
the	O
violation	O
of	O
the	O
induction	O
axiom	B-Algorithm
,	O
define	O
the	O
predicate	O
as	O
or	O
for	O
some	O
and	O
.	O
</s>
<s>
Peano	O
's	O
axioms	B-Algorithm
with	O
the	O
induction	O
principle	O
uniquely	O
model	O
the	O
natural	O
numbers	O
.	O
</s>
<s>
Replacing	O
the	O
induction	O
principle	O
with	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
allows	O
for	O
more	O
exotic	O
models	O
that	O
fulfill	O
all	O
the	O
axioms	B-Algorithm
.	O
</s>
<s>
It	O
is	O
mistakenly	O
printed	O
in	O
several	O
books	O
and	O
sources	O
that	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
is	O
equivalent	O
to	O
the	O
induction	O
axiom	B-Algorithm
.	O
</s>
<s>
A	O
common	O
mistake	O
in	O
many	O
erroneous	O
proofs	O
is	O
to	O
assume	O
that	O
is	O
a	O
unique	O
and	O
well-defined	O
natural	O
number	O
,	O
a	O
property	O
which	O
is	O
not	O
implied	O
by	O
the	O
other	O
Peano	O
axioms	B-Algorithm
.	O
</s>
