<s>
In	O
mathematical	O
logic	O
,	O
the	O
Brouwer	B-Application
–	I-Application
Heyting	I-Application
–	I-Application
Kolmogorov	I-Application
interpretation	I-Application
,	O
or	O
BHK	B-Application
interpretation	I-Application
,	O
of	O
intuitionistic	O
logic	O
was	O
proposed	O
by	O
L	O
.	O
E	O
.	O
J	O
.	O
Brouwer	O
and	O
Arend	O
Heyting	O
,	O
and	O
independently	O
by	O
Andrey	O
Kolmogorov	O
.	O
</s>
<s>
This	O
is	O
specified	O
by	B-Algorithm
induction	I-Algorithm
on	O
the	O
structure	O
of	O
that	O
formula	O
:	O
</s>
<s>
The	O
BHK	B-Application
interpretation	I-Application
instead	O
takes	O
"	O
not	O
"	O
to	O
mean	O
that	O
leads	O
to	O
absurdity	O
,	O
designated	O
,	O
so	O
that	O
a	O
proof	O
of	O
is	O
a	O
function	O
converting	O
a	O
proof	O
of	O
into	O
a	O
proof	O
of	O
absurdity	O
.	O
</s>
<s>
Assume	O
that	O
0	O
=	O
1	O
,	O
and	O
proceed	O
by	O
mathematical	B-Algorithm
induction	I-Algorithm
:	O
0	O
=	O
0	O
by	O
the	O
axiom	O
of	O
equality	O
.	O
</s>
<s>
Now	O
(	O
induction	B-Algorithm
hypothesis	O
)	O
,	O
if	O
0	O
were	O
equal	O
to	O
a	O
certain	O
natural	O
number	O
n	O
,	O
then	O
1	O
would	O
be	O
equal	O
to	O
n+1	O
,	O
(	O
Peano	O
axiom	O
:	O
Sm	O
=	O
Sn	O
if	O
and	O
only	O
if	O
m	O
=	O
n	O
)	O
,	O
but	O
since	O
0	O
=	O
1	O
,	O
therefore	O
0	O
would	O
also	O
be	O
equal	O
to	O
n+1	O
.	O
</s>
<s>
By	B-Algorithm
induction	I-Algorithm
,	O
0	O
is	O
equal	O
to	O
all	O
numbers	O
,	O
and	O
therefore	O
any	O
two	O
natural	O
numbers	O
become	O
equal	O
.	O
</s>
<s>
The	O
BHK	B-Application
interpretation	I-Application
will	O
depend	O
on	O
the	O
view	O
taken	O
about	O
what	O
constitutes	O
a	O
function	O
that	O
converts	O
one	O
proof	O
to	O
another	O
,	O
or	O
that	O
converts	O
an	O
element	O
of	O
a	O
domain	O
to	O
a	O
proof	O
.	O
</s>
<s>
These	O
are	O
then	O
built	O
up	O
by	B-Algorithm
induction	I-Algorithm
into	O
more	O
complex	O
algorithms	O
.	O
</s>
<s>
If	O
one	O
takes	O
lambda	B-Language
calculus	I-Language
as	O
defining	O
the	O
notion	O
of	O
a	O
function	O
,	O
then	O
the	O
BHK	B-Application
interpretation	I-Application
describes	O
the	O
correspondence	O
between	O
natural	O
deduction	O
and	O
functions	O
.	O
</s>
