<s>
Mathematical	B-Algorithm
induction	I-Algorithm
is	O
a	O
method	O
for	O
proving	O
that	O
a	O
statement	O
is	O
true	O
for	O
every	O
natural	O
number	O
,	O
that	O
is	O
,	O
that	O
the	O
infinitely	O
many	O
cases	O
all	O
hold	O
.	O
</s>
<s>
A	O
proof	B-Algorithm
by	I-Algorithm
induction	I-Algorithm
consists	O
of	O
two	O
cases	O
.	O
</s>
<s>
The	O
second	O
case	O
,	O
the	O
induction	B-Algorithm
step	I-Algorithm
,	O
proves	O
that	O
if	O
the	O
statement	O
holds	O
for	O
any	O
given	O
case	O
,	O
then	O
it	O
must	O
also	O
hold	O
for	O
the	O
next	O
case	O
.	O
</s>
<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>
Mathematical	B-Algorithm
induction	I-Algorithm
in	O
this	O
extended	O
sense	O
is	O
closely	O
related	O
to	O
recursion	O
.	O
</s>
<s>
Mathematical	B-Algorithm
induction	I-Algorithm
is	O
an	O
inference	O
rule	O
used	O
in	O
formal	O
proofs	O
,	O
and	O
is	O
the	O
foundation	O
of	O
most	O
correctness	O
proofs	O
for	O
computer	O
programs	O
.	O
</s>
<s>
Although	O
its	O
name	O
may	O
suggest	O
otherwise	O
,	O
mathematical	B-Algorithm
induction	I-Algorithm
should	O
not	O
be	O
confused	O
with	O
inductive	O
reasoning	O
as	O
used	O
in	O
philosophy	O
(	O
see	O
Problem	O
of	O
induction	B-Algorithm
)	O
.	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	B-Algorithm
proof	I-Algorithm
.	O
</s>
<s>
The	O
earliest	O
implicit	O
proof	B-Algorithm
by	I-Algorithm
mathematical	I-Algorithm
induction	I-Algorithm
is	O
in	O
the	O
al-Fakhri	O
written	O
by	O
al-Karaji	O
around	O
1000	O
AD	O
,	O
who	O
applied	O
it	O
to	O
arithmetic	O
sequences	O
to	O
prove	O
the	O
binomial	O
theorem	O
and	O
properties	O
of	O
Pascal	O
's	O
triangle	O
.	O
</s>
<s>
In	O
India	O
,	O
early	O
implicit	O
proofs	O
by	O
mathematical	B-Algorithm
induction	I-Algorithm
appear	O
in	O
Bhaskara	O
's	O
"	O
cyclic	O
method	O
"	O
.	O
</s>
<s>
None	O
of	O
these	O
ancient	O
mathematicians	O
,	O
however	O
,	O
explicitly	O
stated	O
the	O
induction	B-Algorithm
hypothesis	O
.	O
</s>
<s>
The	O
earliest	O
rigorous	O
use	O
of	O
induction	B-Algorithm
was	O
by	O
Gersonides	O
(	O
1288	O
–	O
1344	O
)	O
.	O
</s>
<s>
The	O
first	O
explicit	O
formulation	O
of	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
was	O
given	O
by	O
Pascal	O
in	O
his	O
Traité	O
du	O
triangle	O
arithmétique	O
(	O
1665	O
)	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
hypothesis	O
was	O
also	O
employed	O
by	O
the	O
Swiss	O
Jakob	O
Bernoulli	O
,	O
and	O
from	O
then	O
on	O
it	O
became	O
well	O
known	O
.	O
</s>
<s>
The	O
simplest	O
and	O
most	O
common	O
form	O
of	O
mathematical	B-Algorithm
induction	I-Algorithm
infers	O
that	O
a	O
statement	O
involving	O
a	O
natural	O
number	O
(	O
that	O
is	O
,	O
an	O
integer	O
or	O
1	O
)	O
holds	O
for	O
all	O
values	O
of	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
step	I-Algorithm
(	O
or	O
inductive	O
step	O
,	O
or	O
step	O
case	O
)	O
:	O
prove	O
that	O
for	O
every	O
,	O
if	O
the	O
statement	O
holds	O
for	O
,	O
then	O
it	O
holds	O
for	O
.	O
</s>
<s>
The	O
hypothesis	O
in	O
the	O
induction	B-Algorithm
step	I-Algorithm
,	O
that	O
the	O
statement	O
holds	O
for	O
a	O
particular	O
,	O
is	O
called	O
the	O
induction	B-Algorithm
hypothesis	O
or	O
inductive	O
hypothesis	O
.	O
</s>
<s>
To	O
prove	O
the	O
induction	B-Algorithm
step	I-Algorithm
,	O
one	O
assumes	O
the	O
induction	B-Algorithm
hypothesis	O
for	O
and	O
then	O
uses	O
this	O
assumption	O
to	O
prove	O
that	O
the	O
statement	O
holds	O
for	O
.	O
</s>
<s>
Mathematical	B-Algorithm
induction	I-Algorithm
can	O
be	O
used	O
to	O
prove	O
the	O
following	O
statement	O
P(n )	O
for	O
all	O
natural	O
numbers	O
n	O
.	O
</s>
<s>
Let	O
P(n )	O
be	O
the	O
statement	O
We	O
give	O
a	O
proof	B-Algorithm
by	I-Algorithm
induction	I-Algorithm
on	O
n	O
.	O
</s>
<s>
Induction	B-Algorithm
step	I-Algorithm
:	O
Show	O
that	O
for	O
every	O
k	O
≥	O
0	O
,	O
if	O
P(k )	O
holds	O
,	O
then	O
P( k+1	O
)	O
also	O
holds	O
.	O
</s>
<s>
Assume	O
the	O
induction	B-Algorithm
hypothesis	O
that	O
for	O
a	O
particular	O
k	O
,	O
the	O
single	O
case	O
n	O
=	O
k	O
holds	O
,	O
meaning	O
P(k )	O
is	O
true:It	O
follows	O
that	O
:	O
</s>
<s>
Equating	O
the	O
extreme	O
left	O
hand	O
and	O
right	O
hand	O
sides	O
,	O
we	O
deduce	O
that:That	O
is	O
,	O
the	O
statement	O
P( k+1	O
)	O
also	O
holds	O
true	O
,	O
establishing	O
the	O
induction	B-Algorithm
step	I-Algorithm
.	O
</s>
<s>
Induction	B-Algorithm
is	O
often	O
used	O
to	O
prove	O
inequalities	O
.	O
</s>
<s>
At	O
first	O
glance	O
,	O
it	O
may	O
appear	O
that	O
a	O
more	O
general	O
version	O
,	O
for	O
any	O
real	O
numbers	O
,	O
could	O
be	O
proven	O
without	O
induction	B-Algorithm
;	O
but	O
the	O
case	O
shows	O
it	O
may	O
be	O
false	O
for	O
non-integer	O
values	O
of	O
.	O
</s>
<s>
This	O
suggests	O
we	O
examine	O
the	O
statement	O
specifically	O
for	O
natural	O
values	O
of	O
,	O
and	O
induction	B-Algorithm
is	O
the	O
readiest	O
tool	O
.	O
</s>
<s>
Induction	B-Algorithm
step	I-Algorithm
:	O
We	O
show	O
the	O
implication	O
for	O
any	O
natural	O
number	O
.	O
</s>
<s>
Assume	O
the	O
induction	B-Algorithm
hypothesis	O
:	O
for	O
a	O
given	O
value	O
,	O
the	O
single	O
case	O
is	O
true	O
.	O
</s>
<s>
The	O
inequality	O
between	O
the	O
extreme	O
left-hand	O
and	O
right-hand	O
quantities	O
shows	O
that	O
is	O
true	O
,	O
which	O
completes	O
the	O
induction	B-Algorithm
step	I-Algorithm
.	O
</s>
<s>
In	O
practice	O
,	O
proofs	O
by	B-Algorithm
induction	I-Algorithm
are	O
often	O
structured	O
differently	O
,	O
depending	O
on	O
the	O
exact	O
nature	O
of	O
the	O
property	O
to	O
be	O
proven	O
.	O
</s>
<s>
All	O
variants	O
of	O
induction	B-Algorithm
are	O
special	O
cases	O
of	O
transfinite	B-Algorithm
induction	I-Algorithm
;	O
see	O
below	O
.	O
</s>
<s>
If	O
one	O
wishes	O
to	O
prove	O
a	O
statement	O
,	O
not	O
for	O
all	O
natural	O
numbers	O
,	O
but	O
only	O
for	O
all	O
numbers	O
greater	O
than	O
or	O
equal	O
to	O
a	O
certain	O
number	O
,	O
then	O
the	O
proof	B-Algorithm
by	I-Algorithm
induction	I-Algorithm
consists	O
of	O
the	O
following	O
:	O
</s>
<s>
This	O
form	O
of	O
mathematical	B-Algorithm
induction	I-Algorithm
is	O
actually	O
a	O
special	O
case	O
of	O
the	O
previous	O
form	O
,	O
because	O
if	O
the	O
statement	O
to	O
be	O
proved	O
is	O
then	O
proving	O
it	O
with	O
these	O
two	O
rules	O
is	O
equivalent	O
with	O
proving	O
for	O
all	O
natural	O
numbers	O
with	O
an	O
induction	B-Algorithm
base	O
case	O
.	O
</s>
<s>
Induction	B-Algorithm
can	O
be	O
used	O
to	O
prove	O
that	O
any	O
whole	O
amount	O
of	O
dollars	O
greater	O
than	O
or	O
equal	O
to	O
can	O
be	O
formed	O
by	O
a	O
combination	O
of	O
such	O
coins	O
.	O
</s>
<s>
The	O
proof	O
that	O
is	O
true	O
for	O
all	O
can	O
then	O
be	O
achieved	O
by	B-Algorithm
induction	I-Algorithm
on	O
as	O
follows	O
:	O
</s>
<s>
Induction	B-Algorithm
step	I-Algorithm
:	O
Given	O
that	O
holds	O
for	O
some	O
value	O
of	O
(	O
induction	B-Algorithm
hypothesis	O
)	O
,	O
prove	O
that	O
holds	O
,	O
too	O
.	O
</s>
<s>
Therefore	O
,	O
by	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
,	O
holds	O
for	O
all	O
,	O
and	O
the	O
proof	O
is	O
complete	O
.	O
</s>
<s>
For	O
,	O
the	O
base	O
case	O
is	O
actually	O
false	O
;	O
for	O
,	O
the	O
second	O
case	O
in	O
the	O
induction	B-Algorithm
step	I-Algorithm
(	O
replacing	O
three	O
5	O
-	O
by	O
four	O
4-dollar	O
coins	O
)	O
will	O
not	O
work	O
;	O
let	O
alone	O
for	O
even	O
lower	O
.	O
</s>
<s>
It	O
is	O
sometimes	O
desirable	O
to	O
prove	O
a	O
statement	O
involving	O
two	O
natural	O
numbers	O
,	O
n	O
and	O
m	O
,	O
by	O
iterating	O
the	O
induction	B-Algorithm
process	O
.	O
</s>
<s>
That	O
is	O
,	O
one	O
proves	O
a	O
base	O
case	O
and	O
an	O
induction	B-Algorithm
step	I-Algorithm
for	O
n	O
,	O
and	O
in	O
each	O
of	O
those	O
proves	O
a	O
base	O
case	O
and	O
an	O
induction	B-Algorithm
step	I-Algorithm
for	O
m	O
.	O
See	O
,	O
for	O
example	O
,	O
the	O
proof	O
of	O
commutativity	O
accompanying	O
addition	O
of	O
natural	O
numbers	O
.	O
</s>
<s>
The	O
method	O
of	O
infinite	O
descent	O
is	O
a	O
variation	O
of	O
mathematical	B-Algorithm
induction	I-Algorithm
which	O
was	O
used	O
by	O
Pierre	O
de	O
Fermat	O
.	O
</s>
<s>
The	O
validity	O
of	O
this	O
method	O
can	O
be	O
verified	O
from	O
the	O
usual	O
principle	B-Algorithm
of	I-Algorithm
mathematical	I-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Using	O
mathematical	B-Algorithm
induction	I-Algorithm
on	O
the	O
statement	O
P(n )	O
defined	O
as	O
"	O
Q(m )	O
is	O
false	O
for	O
all	O
natural	O
numbers	O
m	O
less	O
than	O
or	O
equal	O
to	O
n	O
"	O
,	O
it	O
follows	O
that	O
P(n )	O
holds	O
for	O
all	O
n	O
,	O
which	O
means	O
that	O
Q(n )	O
is	O
false	O
for	O
every	O
natural	O
number	O
n	O
.	O
</s>
<s>
whereupon	O
the	O
induction	B-Algorithm
principle	O
"	O
automates	O
"	O
n	O
applications	O
of	O
this	O
step	O
in	O
getting	O
from	O
P(0 )	O
to	O
P(n )	O
.	O
</s>
<s>
This	O
could	O
be	O
called	O
"	O
predecessor	O
induction	B-Algorithm
"	O
because	O
each	O
step	O
proves	O
something	O
about	O
a	O
number	O
from	O
something	O
about	O
that	O
number	O
's	O
predecessor	O
.	O
</s>
<s>
A	O
variant	O
of	O
interest	O
in	O
computational	O
complexity	O
is	O
"	O
prefix	O
induction	B-Algorithm
"	O
,	O
in	O
which	O
one	O
proves	O
the	O
following	O
statement	O
in	O
the	O
induction	B-Algorithm
step	I-Algorithm
:	O
</s>
<s>
The	O
induction	B-Algorithm
principle	O
then	O
"	O
automates	O
"	O
log2n	O
applications	O
of	O
this	O
inference	O
in	O
getting	O
from	O
P(0 )	O
to	O
P(n )	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
is	O
called	O
"	O
prefix	O
induction	B-Algorithm
"	O
because	O
each	O
step	O
proves	O
something	O
about	O
a	O
number	O
from	O
something	O
about	O
the	O
"	O
prefix	O
"	O
of	O
that	O
number	O
—	O
as	O
formed	O
by	O
truncating	O
the	O
low	O
bit	O
of	O
its	O
binary	O
representation	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
viewed	O
as	O
an	O
application	O
of	O
traditional	O
induction	B-Algorithm
on	O
the	O
length	O
of	O
that	O
binary	O
representation	O
.	O
</s>
<s>
If	O
traditional	O
predecessor	O
induction	B-Algorithm
is	O
interpreted	O
computationally	O
as	O
an	O
n-step	O
loop	O
,	O
then	O
prefix	O
induction	B-Algorithm
would	O
correspond	O
to	O
a	O
log-n-step	O
loop	O
.	O
</s>
<s>
Because	O
of	O
that	O
,	O
proofs	O
using	O
prefix	O
induction	B-Algorithm
are	O
"	O
more	O
feasibly	O
constructive	O
"	O
than	O
proofs	O
using	O
predecessor	O
induction	B-Algorithm
.	O
</s>
<s>
Predecessor	O
induction	B-Algorithm
can	O
trivially	O
simulate	O
prefix	O
induction	B-Algorithm
on	O
the	O
same	O
statement	O
.	O
</s>
<s>
Prefix	O
induction	B-Algorithm
can	O
simulate	O
predecessor	O
induction	B-Algorithm
,	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	B-Algorithm
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>
whereupon	O
the	O
induction	B-Algorithm
principle	O
"	O
automates	O
"	O
log	O
log	O
n	O
applications	O
of	O
this	O
inference	O
in	O
getting	O
from	O
P(0 )	O
to	O
P(n )	O
.	O
</s>
<s>
This	O
form	O
of	O
induction	B-Algorithm
has	O
been	O
used	O
,	O
analogously	O
,	O
to	O
study	O
log-time	O
parallel	O
computation	O
.	O
</s>
<s>
Another	O
variant	O
,	O
called	O
complete	O
induction	B-Algorithm
,	O
course	O
of	O
values	O
induction	B-Algorithm
or	O
strong	O
induction	B-Algorithm
(	O
in	O
contrast	O
to	O
which	O
the	O
basic	O
form	O
of	O
induction	B-Algorithm
is	O
sometimes	O
known	O
as	O
weak	O
induction	B-Algorithm
)	O
,	O
makes	O
the	O
induction	B-Algorithm
step	I-Algorithm
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>
The	O
name	O
"	O
strong	O
induction	B-Algorithm
"	O
does	O
not	O
mean	O
that	O
this	O
method	O
can	O
prove	O
more	O
than	O
"	O
weak	O
induction	B-Algorithm
"	O
,	O
but	O
merely	O
refers	O
to	O
the	O
stronger	O
hypothesis	O
used	O
in	O
the	O
induction	B-Algorithm
step	I-Algorithm
.	O
</s>
<s>
In	O
this	O
form	O
of	O
complete	O
induction	B-Algorithm
,	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	B-Algorithm
.	O
</s>
<s>
Complete	O
induction	B-Algorithm
is	O
equivalent	O
to	O
ordinary	O
mathematical	B-Algorithm
induction	I-Algorithm
as	O
described	O
above	O
,	O
in	O
the	O
sense	O
that	O
a	O
proof	O
by	O
one	O
method	O
can	O
be	O
transformed	O
into	O
a	O
proof	O
by	O
the	O
other	O
.	O
</s>
<s>
Suppose	O
there	O
is	O
a	O
proof	O
of	O
by	O
complete	O
induction	B-Algorithm
.	O
</s>
<s>
Then	O
holds	O
for	O
all	O
if	O
and	O
only	O
if	O
holds	O
for	O
all	O
,	O
and	O
our	O
proof	O
of	O
is	O
easily	O
transformed	O
into	O
a	O
proof	O
of	O
by	O
(	O
ordinary	O
)	O
induction	B-Algorithm
.	O
</s>
<s>
If	O
,	O
on	O
the	O
other	O
hand	O
,	O
had	O
been	O
proven	O
by	O
ordinary	O
induction	B-Algorithm
,	O
the	O
proof	O
would	O
already	O
effectively	O
be	O
one	O
by	O
complete	O
induction	B-Algorithm
:	O
is	O
proved	O
in	O
the	O
base	O
case	O
,	O
using	O
no	O
assumptions	O
,	O
and	O
is	O
proved	O
in	O
the	O
induction	B-Algorithm
step	I-Algorithm
,	O
in	O
which	O
one	O
may	O
assume	O
all	O
earlier	O
cases	O
but	O
need	O
only	O
use	O
the	O
case	O
.	O
</s>
<s>
Complete	O
induction	B-Algorithm
is	O
most	O
useful	O
when	O
several	O
instances	O
of	O
the	O
inductive	O
hypothesis	O
are	O
required	O
for	O
each	O
induction	B-Algorithm
step	I-Algorithm
.	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>
Another	O
proof	O
by	O
complete	O
induction	B-Algorithm
uses	O
the	O
hypothesis	O
that	O
the	O
statement	O
holds	O
for	O
all	O
smaller	O
more	O
thoroughly	O
.	O
</s>
<s>
For	O
proving	O
the	O
induction	B-Algorithm
step	I-Algorithm
,	O
the	O
induction	B-Algorithm
hypothesis	O
is	O
that	O
for	O
a	O
given	O
the	O
statement	O
holds	O
for	O
all	O
smaller	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
hypothesis	O
now	O
applies	O
to	O
and	O
,	O
so	O
each	O
one	O
is	O
a	O
product	O
of	O
primes	O
.	O
</s>
<s>
We	O
shall	O
look	O
to	O
prove	O
the	O
same	O
example	O
as	O
above	O
,	O
this	O
time	O
with	O
strong	O
induction	B-Algorithm
.	O
</s>
<s>
Induction	B-Algorithm
step	I-Algorithm
:	O
Given	O
some	O
,	O
assume	O
holds	O
for	O
all	O
with	O
.	O
</s>
<s>
inequality	O
of	O
arithmetic	O
and	O
geometric	O
means	O
for	O
all	O
powers	O
of	O
2	O
,	O
and	O
then	O
used	O
backwards	O
induction	B-Algorithm
to	O
show	O
it	O
for	O
all	O
natural	O
numbers	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
step	I-Algorithm
must	O
be	O
proved	O
for	O
all	O
values	O
of	O
n	O
.	O
To	O
illustrate	O
this	O
,	O
Joel	O
E	O
.	O
Cohen	O
proposed	O
the	O
following	O
argument	O
,	O
which	O
purports	O
to	O
prove	O
by	O
mathematical	B-Algorithm
induction	I-Algorithm
that	O
all	O
horses	O
are	O
of	O
the	O
same	O
color	O
:	O
</s>
<s>
Induction	B-Algorithm
step	I-Algorithm
:	O
assume	O
as	O
induction	B-Algorithm
hypothesis	O
that	O
within	O
any	O
set	O
of	O
horses	O
,	O
there	O
is	O
only	O
one	O
color	O
.	O
</s>
<s>
The	O
base	O
case	O
is	O
trivial	O
,	O
and	O
the	O
induction	B-Algorithm
step	I-Algorithm
is	O
correct	O
in	O
all	O
cases	O
.	O
</s>
<s>
However	O
,	O
the	O
argument	O
used	O
in	O
the	O
induction	B-Algorithm
step	I-Algorithm
is	O
incorrect	O
for	O
,	O
because	O
the	O
statement	O
that	O
"	O
the	O
two	O
sets	O
overlap	O
"	O
is	O
false	O
for	O
and	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	B-Algorithm
"	O
as	O
follows	O
:	O
</s>
<s>
In	O
words	O
,	O
the	O
base	O
case	O
and	O
the	O
induction	B-Algorithm
step	I-Algorithm
(	O
namely	O
,	O
that	O
the	O
induction	B-Algorithm
hypothesis	O
implies	O
)	O
together	O
imply	O
that	O
for	O
any	O
natural	O
number	O
.	O
</s>
<s>
The	O
axiom	B-Algorithm
of	O
induction	B-Algorithm
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	B-Algorithm
step	I-Algorithm
.	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	B-Algorithm
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>
In	O
first-order	O
ZFC	O
set	O
theory	O
,	O
quantification	O
over	O
predicates	O
is	O
not	O
allowed	O
,	O
but	O
one	O
can	O
still	O
express	O
induction	B-Algorithm
by	O
quantification	O
over	O
sets	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	B-Algorithm
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	B-Algorithm
,	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	B-Algorithm
of	I-Algorithm
mathematical	I-Algorithm
induction	I-Algorithm
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	B-Algorithm
,	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	B-Algorithm
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>
Therefore	O
,	O
by	O
the	O
complete	O
induction	B-Algorithm
principle	O
,	O
P(n )	O
holds	O
for	O
all	O
natural	O
numbers	O
n	O
;	O
so	O
S	O
is	O
empty	O
,	O
a	O
contradiction	O
.	O
</s>
<s>
Moreover	O
,	O
except	O
for	O
the	O
induction	B-Algorithm
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	B-Algorithm
axiom	B-Algorithm
,	O
define	O
the	O
predicate	O
as	O
or	O
for	O
some	O
and	O
.	O
</s>
<s>
Then	O
the	O
base	O
case	O
P(0,0 )	O
is	O
trivially	O
true	O
,	O
and	O
so	O
is	O
the	O
induction	B-Algorithm
step	I-Algorithm
:	O
if	O
,	O
then	O
.	O
</s>
<s>
Peano	O
's	O
axioms	B-Algorithm
with	O
the	O
induction	B-Algorithm
principle	O
uniquely	O
model	O
the	O
natural	O
numbers	O
.	O
</s>
<s>
Replacing	O
the	O
induction	B-Algorithm
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	B-Algorithm
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>
