<s>
In	O
mathematics	O
,	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
(	O
also	O
known	O
as	O
the	O
axiom	B-General_Concept
of	I-General_Concept
foundation	I-General_Concept
)	O
is	O
an	O
axiom	O
of	O
Zermelo	O
–	O
Fraenkel	O
set	O
theory	O
that	O
states	O
that	O
every	O
non-empty	O
set	O
A	O
contains	O
an	O
element	O
that	O
is	O
disjoint	B-Algorithm
from	O
A	O
.	O
</s>
<s>
The	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
together	O
with	O
the	O
axiom	O
of	O
pairing	O
implies	O
that	O
no	O
set	O
is	O
an	O
element	O
of	O
itself	O
,	O
and	O
that	O
there	O
is	O
no	O
infinite	O
sequence	O
(	O
an	O
)	O
such	O
that	O
ai+1	O
is	O
an	O
element	O
of	O
ai	O
for	O
all	O
i	O
.	O
</s>
<s>
With	O
the	O
axiom	O
of	O
dependent	O
choice	O
(	O
which	O
is	O
a	O
weakened	O
form	O
of	O
the	O
axiom	O
of	O
choice	O
)	O
,	O
this	O
result	O
can	O
be	O
reversed	O
:	O
if	O
there	O
are	O
no	O
such	O
infinite	O
sequences	O
,	O
then	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
is	O
true	O
.	O
</s>
<s>
Hence	O
,	O
in	O
this	O
context	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
is	O
equivalent	O
to	O
the	O
sentence	O
that	O
there	O
are	O
no	O
downward	O
infinite	O
membership	O
chains	O
.	O
</s>
<s>
Given	O
the	O
other	O
axioms	O
of	O
Zermelo	O
–	O
Fraenkel	O
set	O
theory	O
,	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
is	O
equivalent	O
to	O
the	O
axiom	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
The	O
axiom	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
tends	O
to	O
be	O
used	O
in	O
place	O
of	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
in	O
intuitionistic	O
theories	O
(	O
ones	O
that	O
do	O
not	O
accept	O
the	O
law	O
of	O
the	O
excluded	O
middle	O
)	O
,	O
where	O
the	O
two	O
axioms	O
are	O
not	O
equivalent	O
.	O
</s>
<s>
In	O
addition	O
to	O
omitting	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
,	O
non-standard	O
set	O
theories	O
have	O
indeed	O
postulated	O
the	O
existence	O
of	O
sets	O
that	O
are	O
elements	O
of	O
themselves	O
.	O
</s>
<s>
Let	O
A	O
be	O
a	O
set	O
,	O
and	O
apply	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
to	O
{A},	O
which	O
is	O
a	O
set	O
by	O
the	O
axiom	O
of	O
pairing	O
.	O
</s>
<s>
We	O
see	O
that	O
there	O
must	O
be	O
an	O
element	O
of	O
 { A } 	O
which	O
is	O
disjoint	B-Algorithm
from	O
 { A } 	O
.	O
</s>
<s>
Since	O
the	O
only	O
element	O
of	O
 { A } 	O
is	O
A	O
,	O
it	O
must	O
be	O
that	O
A	O
is	O
disjoint	B-Algorithm
from	O
 { A } 	O
.	O
</s>
<s>
So	O
,	O
since	O
,	O
we	O
cannot	O
have	O
A	O
∈	O
A	O
(	O
by	O
the	O
definition	O
of	O
disjoint	B-Algorithm
)	O
.	O
</s>
<s>
Applying	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
to	O
S	O
,	O
let	O
B	O
be	O
an	O
element	O
of	O
S	O
which	O
is	O
disjoint	B-Algorithm
from	O
S	O
.	O
By	O
the	O
definition	O
of	O
S	O
,	O
B	O
must	O
be	O
f(k )	O
for	O
some	O
natural	O
number	O
k	O
.	O
However	O
,	O
we	O
are	O
given	O
that	O
f(k )	O
contains	O
f( k+1	O
)	O
which	O
is	O
also	O
an	O
element	O
of	O
S	O
.	O
So	O
f( k+1	O
)	O
is	O
in	O
the	O
intersection	O
of	O
f(k )	O
and	O
S	O
.	O
This	O
contradicts	O
the	O
fact	O
that	O
they	O
are	O
disjoint	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
The	O
hereditarily	O
finite	O
sets	O
,	O
Vω	O
,	O
satisfy	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
(	O
and	O
all	O
other	O
axioms	O
of	O
ZFC	O
except	O
the	O
axiom	O
of	O
infinity	O
)	O
.	O
</s>
<s>
So	O
if	O
one	O
forms	O
a	O
non-trivial	O
ultrapower	O
of	O
Vω	O
,	O
then	O
it	O
will	O
also	O
satisfy	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
.	O
</s>
<s>
The	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
enables	O
defining	O
the	O
ordered	O
pair	O
(	O
a	O
,	O
b	O
)	O
as	O
{a,{a,b}};	O
see	O
ordered	O
pair	O
for	O
specifics	O
.	O
</s>
<s>
Otherwise	O
,	O
apply	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
to	O
u	O
to	O
get	O
an	O
element	O
w	O
of	O
u	O
which	O
is	O
disjoint	B-Algorithm
from	O
u	O
.	O
</s>
<s>
Since	O
w	O
is	O
disjoint	B-Algorithm
from	O
u	O
,	O
every	O
element	O
of	O
w	O
is	O
ranked	O
.	O
</s>
<s>
Applying	O
the	O
axioms	O
of	O
replacement	O
and	O
union	O
to	O
combine	O
the	O
ranks	O
of	O
the	O
elements	O
of	O
w	O
,	O
we	O
get	O
an	O
ordinal	B-Language
rank	O
for	O
w	O
,	O
to	O
wit	O
.	O
</s>
<s>
Then	O
apply	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
to	O
the	O
set	O
 { X , Y } 	O
(	O
which	O
exists	O
by	O
the	O
axiom	O
of	O
pairing	O
)	O
.	O
</s>
<s>
We	O
see	O
there	O
must	O
be	O
an	O
element	O
of	O
 { X , Y } 	O
which	O
is	O
also	O
disjoint	B-Algorithm
from	O
it	O
.	O
</s>
<s>
By	O
the	O
definition	O
of	O
disjoint	B-Algorithm
then	O
,	O
we	O
must	O
have	O
either	O
Y	O
is	O
not	O
an	O
element	O
of	O
X	O
or	O
vice	O
versa	O
.	O
</s>
<s>
Let	O
the	O
non-empty	O
set	O
S	O
be	O
a	O
counter-example	O
to	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
;	O
that	O
is	O
,	O
every	O
element	O
of	O
S	O
has	O
a	O
non-empty	O
intersection	O
with	O
S	O
.	O
We	O
define	O
a	O
binary	O
relation	O
R	O
on	O
S	O
by	O
,	O
which	O
is	O
entire	O
by	O
assumption	O
.	O
</s>
<s>
The	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
was	O
also	O
shown	O
to	O
be	O
independent	O
from	O
the	O
other	O
axioms	O
of	O
ZF(C )	O
,	O
assuming	O
they	O
are	O
consistent	O
.	O
</s>
<s>
The	O
proof	O
involves	O
(	O
and	O
led	O
to	O
the	O
study	O
of	O
)	O
Rieger-Bernays	O
permutation	O
models	O
(	O
or	O
method	O
)	O
,	O
which	O
were	O
used	O
for	O
other	O
proofs	O
of	O
independence	O
for	O
non-well-founded	O
systems	O
(	O
and	O
)	O
.	O
</s>
<s>
Subsequently	O
,	O
the	O
axiom	O
of	O
choice	O
and	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
were	O
added	O
to	O
exclude	O
models	O
with	O
some	O
undesirable	O
properties	O
.	O
</s>
<s>
The	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
together	O
with	O
the	O
axiom	O
of	O
pairing	O
also	O
prohibit	O
such	O
a	O
universal	O
set	O
.	O
</s>
<s>
In	O
particular	O
,	O
ZF	O
without	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
already	O
prohibits	O
such	O
a	O
universal	O
set	O
.	O
</s>
<s>
have	O
themselves	O
as	O
their	O
only	O
elements	O
)	O
is	O
consistent	O
with	O
the	O
theory	O
obtained	O
by	O
removing	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
from	O
ZFC	O
.	O
</s>
<s>
Various	O
non-wellfounded	O
set	O
theories	O
allow	O
"	O
safe	O
"	O
circular	O
sets	O
,	O
such	O
as	O
Quine	O
atoms	O
,	O
without	O
becoming	O
inconsistent	O
by	O
means	O
of	O
Russell	O
's	O
paradox	O
.	O
</s>
<s>
This	O
statement	O
is	O
even	O
equivalent	O
to	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
(	O
if	O
we	O
work	O
in	O
ZF	O
with	O
this	O
axiom	O
omitted	O
)	O
.	O
</s>
<s>
From	O
any	O
model	O
which	O
does	O
not	O
satisfy	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
,	O
a	O
model	O
which	O
satisfies	O
it	O
can	O
be	O
constructed	O
by	O
taking	O
only	O
sets	O
in	O
.	O
</s>
<s>
Comparing	O
ZF	O
with	O
type	O
theory	O
,	O
Alasdair	O
Urquhart	O
wrote	O
that	O
"	O
Zermelo	O
's	O
system	O
has	O
the	O
notational	O
advantage	O
of	O
not	O
containing	O
any	O
explicitly	O
typed	O
variables	O
,	O
although	O
in	O
fact	O
it	O
can	O
be	O
seen	O
as	O
having	O
an	O
implicit	O
type	O
structure	O
built	O
into	O
it	O
,	O
at	O
least	O
if	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
is	O
included	O
.	O
</s>
<s>
The	O
concept	O
of	O
well-foundedness	B-Algorithm
and	O
rank	O
of	O
a	O
set	O
were	O
both	O
introduced	O
by	O
Dmitry	O
Mirimanoff	O
(	O
1917	O
)	O
cf	O
.	O
</s>
<s>
Mirimanoff	O
however	O
did	O
not	O
consider	O
his	O
notion	O
of	O
regularity	O
(	O
and	O
well-foundedness	B-Algorithm
)	O
as	O
an	O
axiom	O
to	O
be	O
observed	O
by	O
all	O
sets	O
;	O
in	O
later	O
papers	O
Mirimanoff	O
also	O
explored	O
what	O
are	O
now	O
called	O
non-well-founded	O
sets	O
(	O
"	O
extraordinaire	O
"	O
in	O
Mirimanoff	O
's	O
terminology	O
)	O
.	O
</s>
<s>
In	O
these	O
theories	O
,	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
must	O
be	O
modified	O
.	O
</s>
