<s>
In	O
set	O
theory	O
,	O
-induction	O
,	O
also	O
called	O
epsilon-induction	B-Algorithm
or	O
set-induction	O
,	O
is	O
a	O
principle	O
that	O
can	O
be	O
used	O
to	O
prove	O
that	O
all	O
sets	O
satisfy	O
a	O
given	O
property	O
.	O
</s>
<s>
Considered	O
as	O
an	O
axiomatic	O
principle	O
,	O
it	O
is	O
called	O
the	O
axiom	B-Algorithm
schema	I-Algorithm
of	I-Algorithm
set	I-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
The	O
principle	O
implies	O
transfinite	B-Algorithm
induction	I-Algorithm
and	O
recursion	O
.	O
</s>
<s>
It	O
may	O
also	O
be	O
studied	O
in	O
a	O
general	O
context	O
of	O
induction	B-Algorithm
on	O
well-founded	B-Algorithm
relations	I-Algorithm
.	O
</s>
<s>
There	O
are	O
many	O
transitive	O
sets	O
-	O
in	O
particular	O
the	O
set	O
theoretical	O
ordinals	B-Language
.	O
</s>
<s>
But	O
indeed	O
if	O
is	O
any	O
-transitive	O
class	O
,	O
then	O
still	O
and	O
a	O
version	O
of	O
set	O
induction	B-Algorithm
for	O
holds	O
inside	O
of	O
.	O
</s>
<s>
Ordinals	B-Language
may	O
be	O
defined	O
as	O
transitive	O
sets	O
of	O
transitive	O
sets	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
situation	O
in	O
the	O
first	B-Language
infinite	I-Language
ordinal	I-Language
,	O
the	O
set	O
of	O
natural	O
numbers	O
,	O
is	O
discussed	O
in	O
more	O
detail	O
below	O
.	O
</s>
<s>
As	O
set	O
induction	B-Algorithm
allows	O
for	O
induction	B-Algorithm
in	O
transitive	O
sets	O
containing	O
,	O
this	O
gives	O
what	O
is	O
called	O
transfinite	B-Algorithm
induction	I-Algorithm
and	O
definition	O
by	O
transfinite	O
recursion	O
using	O
,	O
indeed	O
,	O
the	O
whole	O
proper	O
class	O
of	O
ordinals	B-Language
.	O
</s>
<s>
With	O
ordinals	B-Language
,	O
induction	B-Algorithm
proves	O
that	O
all	O
sets	O
have	O
ordinal	B-Language
rank	O
and	O
the	O
rank	O
of	O
an	O
ordinal	B-Language
is	O
itself	O
.	O
</s>
<s>
The	O
theory	O
of	O
Von	O
Neumann	O
ordinals	B-Language
describes	O
such	O
sets	O
and	O
,	O
there	O
,	O
models	O
the	O
order	O
relation	O
,	O
which	O
classically	O
is	O
provably	O
trichotomous	O
and	O
total	O
.	O
</s>
<s>
Of	O
interest	O
there	O
is	O
the	O
successor	O
operation	O
that	O
maps	O
ordinals	B-Language
to	O
ordinals	B-Language
.	O
</s>
<s>
In	O
the	O
classical	O
case	O
,	O
the	O
induction	B-Algorithm
step	I-Algorithm
for	O
successor	O
ordinals	B-Language
can	O
be	O
simplified	O
so	O
that	O
a	O
property	O
must	O
merely	O
be	O
preserved	O
between	O
successive	O
ordinals	B-Language
(	O
this	O
is	O
the	O
formulation	O
that	O
is	O
typically	O
understood	O
as	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
)	O
</s>
<s>
The	O
sets	O
are	O
-well-founded	O
.	O
</s>
<s>
For	O
a	O
binary	O
relation	O
on	O
a	O
set	O
,	O
well-foundedness	B-Algorithm
can	O
be	O
defined	O
by	O
requiring	O
a	O
tailored	O
induction	B-Algorithm
property	O
:	O
in	O
the	O
condition	O
is	O
abstracted	O
to	O
,	O
i.e.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
for	O
a	O
well-founded	B-Algorithm
relation	I-Algorithm
,	O
there	O
are	O
no	O
infinite	O
descending	O
-sequences	O
and	O
so	O
also	O
.	O
</s>
<s>
Classically	O
,	O
well-foundedness	B-Algorithm
of	O
a	O
relation	O
on	O
a	O
set	O
can	O
also	O
be	O
characterized	O
by	O
the	O
strong	O
property	O
of	O
minimal	O
element	O
existence	O
for	O
every	O
subset	O
.	O
</s>
<s>
Consider	O
set	O
induction	B-Algorithm
for	O
a	O
negated	O
predicate	O
.	O
</s>
<s>
Given	O
the	O
extra	O
assumptions	O
,	O
the	O
postulate	O
of	O
mere	O
non-existence	O
of	O
infinite	O
descending	O
chains	O
is	O
relatively	O
weak	O
compared	O
to	O
set	O
induction	B-Algorithm
.	O
</s>
<s>
So	O
assuming	O
the	O
induction	B-Algorithm
principle	O
makes	O
the	O
chain	O
existence	O
contradictory	O
also	O
.	O
</s>
<s>
One	O
studies	O
set	O
induction	B-Algorithm
for	O
the	O
class	O
of	O
sets	O
that	O
are	O
not	O
equal	O
to	O
.	O
</s>
<s>
In	O
a	O
theory	O
with	O
set	O
induction	B-Algorithm
,	O
a	O
with	O
the	O
described	O
recursive	O
property	O
is	O
not	O
actually	O
a	O
set	O
in	O
the	O
first	O
place	O
.	O
</s>
<s>
In	O
first-order	O
set	O
theories	O
,	O
the	O
common	O
framework	O
,	O
the	O
set	O
induction	B-Algorithm
principle	O
is	O
an	O
axiom	O
schema	O
,	O
granting	O
an	O
axiom	O
for	O
any	O
predicate	O
(	O
i.e.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
is	O
a	O
single	O
axiom	O
,	O
formulated	O
with	O
a	O
universal	O
quantifier	O
only	O
over	O
elements	O
of	O
the	O
domain	O
of	O
discourse	O
,	O
i.e.	O
</s>
<s>
If	O
is	O
a	O
set	O
and	O
the	O
induction	B-Algorithm
schema	O
is	O
assumed	O
,	O
the	O
above	O
is	O
the	O
instance	O
of	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
for	O
.	O
</s>
<s>
Hence	O
,	O
assuming	O
set	O
induction	B-Algorithm
over	O
a	O
classical	O
logic	O
(	O
i.e.	O
</s>
<s>
Meanwhile	O
,	O
the	O
set	O
induction	B-Algorithm
schema	O
does	O
not	O
imply	O
excluded	O
middle	O
,	O
while	O
still	O
being	O
strong	O
enough	O
to	O
imply	O
strong	O
induction	B-Algorithm
principles	O
,	O
as	O
discussed	O
above	O
.	O
</s>
<s>
So	O
within	O
such	O
a	O
set	O
theory	O
framework	O
,	O
set	O
induction	B-Algorithm
is	O
a	O
strong	O
principle	O
strictly	O
weaker	O
than	O
regularity	O
.	O
</s>
<s>
When	O
adopting	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
and	O
full	O
Separation	O
,	O
CZF	O
equals	O
standard	O
ZF	O
.	O
</s>
<s>
Because	O
of	O
its	O
use	O
in	O
the	O
set	O
theoretical	O
treatment	O
of	O
ordinals	B-Language
,	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
was	O
formulated	O
by	O
von	O
Neumann	O
in	O
1925	O
.	O
</s>
<s>
The	O
theory	O
does	O
not	O
prove	O
all	O
set	O
induction	B-Algorithm
instances	O
.	O
</s>
<s>
Regularity	O
is	O
classically	O
equivalent	O
to	O
the	O
contrapositive	O
of	O
set	O
induction	B-Algorithm
for	O
negated	O
statements	O
,	O
as	O
demonstrated	O
.	O
</s>
<s>
Moreover	O
,	O
an	O
induction	B-Algorithm
schema	O
stated	O
in	O
terms	O
of	O
a	O
negated	O
predicate	O
is	O
then	O
just	O
as	O
strong	O
as	O
one	O
in	O
terms	O
of	O
a	O
predicate	O
variable	O
,	O
as	O
the	O
latter	O
simply	O
equals	O
.	O
</s>
<s>
As	O
the	O
equivalences	O
with	O
the	O
contrapositive	O
of	O
set	O
induction	B-Algorithm
have	O
been	O
discussed	O
,	O
the	O
task	O
is	O
to	O
translate	O
regularity	O
back	O
to	O
a	O
statement	O
about	O
a	O
general	O
class	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
principle	O
is	O
a	O
stronger	O
tool	O
for	O
the	O
task	O
of	O
proving	O
by	O
ruling	O
out	O
the	O
existence	O
of	O
counter-examples	O
,	O
and	O
it	O
is	O
one	O
often	O
also	O
adopted	O
in	O
constructive	O
frameworks	O
.	O
</s>
<s>
The	O
transitive	O
von	O
Neumann	O
model	O
of	O
the	O
standard	O
natural	O
numbers	O
is	O
the	O
first	B-Language
infinite	I-Language
ordinal	I-Language
.	O
</s>
<s>
Then	O
,	O
the	O
principle	O
derived	O
from	O
set	O
induction	B-Algorithm
complete	O
induction	B-Algorithm
.	O
</s>
<s>
complete	O
induction	B-Algorithm
is	O
already	O
implied	O
by	O
standard	B-Algorithm
form	I-Algorithm
of	I-Algorithm
the	I-Algorithm
mathematical	I-Algorithm
induction	I-Algorithm
schema	I-Algorithm
.	O
</s>
<s>
Abstracting	O
away	O
the	O
signature	O
symbols	O
,	O
this	O
can	O
be	O
demonstrated	O
to	O
still	O
be	O
close	O
to	O
set	O
induction	B-Algorithm
:	O
Using	O
induction	B-Algorithm
,	O
proves	O
that	O
every	O
is	O
either	O
zero	O
or	O
has	O
a	O
computable	O
unique	O
predecessor	O
with	O
.	O
</s>
<s>
By	O
design	O
this	O
trivially	O
holds	O
for	O
zero	O
,	O
and	O
so	O
,	O
akin	O
to	O
the	O
bottom	O
case	O
in	O
set	O
induction	B-Algorithm
,	O
the	O
implication	O
is	O
equivalent	O
to	O
just	O
.	O
</s>
<s>
As	O
in	O
the	O
set	O
theory	O
case	O
,	O
one	O
may	O
consider	O
induction	B-Algorithm
for	O
negated	O
predicates	O
and	O
take	O
the	O
contrapositive	O
.	O
</s>
