manuscripta math. 38, 3 2 5 -
manuscripta
mathematica
332 (1982)
9 Springer-Verlag1982
THE
"WORLD'S
SIMPLEST
AXI~[
Fourman-A.
M.P.
OF
CHOICE"
FAILS
Scedrov
We use topos-theoretic
m e t h o d s to s h o w t h a t i n t u i tionistic
set tlleory w i t h c o u n t a b l e or ~ e p e n d e n t c h o i c e
d o e s n o t i m p l y t h a t e v e r y f a m i l y , a l l of w h o s e e l e m e n t s
are doubletons
a n d w h i c h h a s at m o s t o n e e l e m e n t , h a s a
choice function.
i.
Introduction
The
by
F.
axiom
in q u e s t i o n
Richman,
states,
(WSAC),
Vx,y
E F 9 x = y
A
VX E F " ZV,W,(V
~ w
Of
course,
for
such
function,
V x
6 F
determined
by
UF
Furthermore,
UF
is a d o u b l e t o n .
ZV
let
We
v
E =
use
ering
the
to
check
in
the
and
any
If
A
is
- x
that
the
that
the
set
which
of
CC
A
by
f
is a c h o i c e
if
~F
it
is
inhabited
{v,w}),
=
such
UF
a
of
[16]
& DC
but
set
of
A.
It
higher-order
not
arise
embedding
Joyal)
satisfies
example
can
: F §
is c o m p l e t e l y
with
[12]
simplest
examples
~f
that
A =
interpretation
[7]
formulated
UF}
E
, then
topos
F
9 v
(originating
world's
a topos
such
Also,
(v 9 w
6 A}
choice,
such
-
any
such
example
classifying
perhaps
of
be
§
{V,W})
E x.
may
universal
We
part
Zx
family,
{OF i Zv
technique
dependent
show
=
E A § Zv,w
{A i
the
a
x =
9 f(x)
F
originally
VF
of
this
this
is
easy
logic
countable
WSAC.
in t h e
consid-
This
is
technique.
well-founded
universal
example
OO25-2611/82/OO38/O325/$O1.
325
60
FOUm~N-SCEDROV
2
universally
in
2N
For d e t a i l s
[16]
or M a k k a i
intuitionistic
(as •
w
topoi
& Reyes
The
[~i].
type-theory
described
by F o u r m a n
by O s i u s
[13]
2.
[3]
on c l a s s i f y i n g
and
[2]
to T i e r n e y
interpretation
set-theory
and
and S c o t t
we r e f e r
[3]
of
in a topos
and m e r e
is
concretely,
[15].
The B a s i c M o d e l
We
introduce
inhabited
functor
is a d o u b l e t o n ,
category
non-identities
with
the c a t e g o r y
empty
would
S{
A
which
The c l a s s i f y i n g
where
{
a n d ~2 = id.
of f i n i t e l y
set and
D
if it is
topos
is the c a t e g o r y
This
be r e f l e c t e d
is
presented
a doubleton)
the e q u a l i t y
addition
a set
is the
whose
are
~ o e =~
because
universally
(equivalent
such
sets
(E
is the
and m o n o m o r p h i s m s
on such a set is d e c i d a b l e
in a n y g e o m e t r i c
of p r e d i c a t e s
to)
(monos
and
axiomatisation
or o p e r a t i o n s
forcing
this
by the
homomorphisms
to be monos).
The m o d e l s
functors
from
like K r i p k e
partial
we c o n s i d e r
9
into
models.
order
and
the c a t e g o r y
instead
maps
category.
In p a r t i c u l a r ,
restriction
by
X(B).
topos
X(~)
map
The
is g i v e n
Kripke's
X(e)
: X(E)
X(D)
definitions
[9].
lemma
The u n i v e r s a l
~ § Sets.
A
[ll]
X
together
will
w i t h an
two and a
image
is fixed
in such a p r e s h e a f
generalisation
spaces
of
are m o d e l l e d
w h i c h are e a s i l y
calculated
[I0] .
we w a n t
The
of our
a model
whose
of logic
Function
exponents
we n e e d
of o r d e r
§ X(D)
interpretation
the Y o n e d a
functor
and
§ X(D)
are
the u s u a l
to the m o r p h i s m s
by a s t r a i g h t f o r w a r d
by the c a t e g o r i c a l
using
X(E)
These
replaces
in our case,
: X(D)
are thus
of sets.
~
of r e s t r i c t i o n s
corresponding
of d o m a i n s
automorphism
section
The c a t e g o r y
transition
be a p a i r
in this
is g i v e n by the f o r g e t f u l
family
326
F = {A I Z x,
x 6 Ak
is
FOU~iAN-SCEDROV
then
represented
Since
A :
In w o r d s ,
OF
AF(E)
representable
singleton
and
A
3.
--~ [F • D, A]
is r e p r e s e n t e d
from
F x E
A(D)
valid.
firstly
AF(E)
for
show that
(assuming
the corresponding
natural
constant
numbers
and
that
Y
is
n
we
have
6 N
some
~(D)
D
We d e f i n e
a natural
f
by
f(a,n)
(i.e.
for
By Y o n e d a
=
the non-trivial
and
~
transforma-
E~Zf
"if
: F §
A
the
which
presheaf
hypotheses
E Y
such
S~
DC
hold
are
suppose
given
~(n,y)
Then
by
CC
for
that
transformation
: D • N § Y
for
and
f
(~,n)
n
6
(D • I~)
(B)
E N ~I{(B)).
represents
327
by
D C i~i
I[- ~ ( n , Y n ) -
(as b e f o r e )
of
in o u r m e t a t h e o r y ) .
topos
Now
S {.
topos
choice
principles
in
6 Y(D)
: D + B
OF.
is i n h a b i t e d
Models
6 ~ Zy
(Y(~)) (yn)
~
= ~
= N.
functor
Yn
is t h e
is a
so no n a t u r a l
in a n y p r e s h e a f
D [[-Vn
where
(D)
with
and dependent
functor
E
(F • E) (D) § A(D)
in a n y
choice,
the
A F.
set of n a t u r a l
(where
(F • E)
since
Presheaf
CC
space
I# W A S C .
countable
The
A
(by c o n s t r u c t i o n ) ,
Thus
: {A}.
similarly.
is a d o u b l e t o n
However,
F(D)
function
by t h e
is no m a p
and
= ~,
~-- [F • E, A]
to
Since
Thus
OF.
Generalities
We
[E, A F]
the automorphism
are
the
AF(D)
is a d o u b l e t o n "
WSAC
calculate
~
there
: F +
F(E)
AF(E)
F • E § A.
Zf
So
functor
functor).
automorphism,
tion
the
we now
transformations
respects
by
3
an
element
of
each
4
FOU~4AN-SCEDROV
Y~](D)
D~ V n
a n d by the p e r s i s t e n c e
6 ~.~
We n o w
(n,f(n)).
consider
sequences.
over
a cover
of
is a set,
tree,
finite
over
the
IA
Ve
6 K
9
Ve
6 AN
9
Ve
[ Va
6 A
DC
trees
of
principle
6A
is s i m i l a r .
finite
of
that
K c A <N
Va
forcing
for
states
sequences
of
induction
if a
is p e r s i s t e n t
e ~ < a > 6 K,
1
(or bar)
Ze s
K
e ca
2
inductive
then
< > 6 K.
hypotheses
The
fail
of
fan
special
We c a l l
IA
9 e~a>
2)
and
FT a n d
bar
3)
+
e
6 K]
(conjoined)
induction,
(A : 2 a n d A = N).
[5],
3
the
topoi,
BI
are
Such principles
may
[14].
finite
and
pointwise:
AN
6 K
K.
in s h e a f m o d e l s
computed
i),
for
theorem,
cases
in p r e s h e a f
are
argument
induction
A
the A - s p l i t t i n g
collection
and
If
The
property
countable
sequences
t
(D) ~ A(D) N
and
A<~
If
K
(D)
is a s u b p r e s h e a f
~ A(D) < ~
of
A <N
and
D li- " h y p o t h e s e s
then
K(D)
In g e n e r a l ,
However,
every
c A(D)
<N
there
if e a c h
extension
is a p e r s i s t e n t
is no r e a s o n
restriction
of
of
e
in
why
map
K"
cover.
it s h o u l d
from
A<N(D)
I A for
be
A(D)
belongs
inductive.
is o n t o
to
K(D)
from
D I~
~<a>
6 K
we o b t a i n
328
for each
a
6 A(D)
and
then
FOU~N-SCEDROV
D li- V a
Thus
D If- e
and
- C<a>
5
6 K
.
K(D)
is i n d u c t i v e .
Dii - <
>
9
]~
are
E K
By a p p e a l
to
IK(D)
Since
this
2
shows
There
take
and
that
FT
are probably
the
analysis
model
of
every
restriction
onto),
w
easier
from
This
suffices
appeal
to
see
for a n y
A(D)
this
We
models.
b u t we w a n t
show that
presheaf
A.
Since
is an a u t o m o r p h i s m
we
to
our
(hence
have
IA
that
restricting
D if- e
to
presheaves,
to p r e s h e a f
further.
above
Eil-
whence
by c o n s t a n t
pass
ways
IA
Dil-
Then
BI
a little
satisfies
suppose
given
and
by t h e r e m a r k s
Now
E K
"hypotheses
to
D
E K
to
for
show
IK(E)
we
see
every
that
of
that
e
for
K"
D II- < > 6 K
6 A<N(D)
K(E)
we obtain
IA
is i n d u c t i v e .
Eli-
<
>
By
Thus
E K.
I = IA
A more
general
presheaf
categories.
4.
requires
We
shall
A Well-Founded
Here
we work
show that
suitable
adding
Again
the
A
(Cantor
space)
induction
of
this
the m o d e l
may
principles
induction
in
over
here.
for
we t a k e
in
2~
CC
solution
family
329
space
space)
and
DC
or
family
or c l a s s i f y i n g
of e l e m e n t s
on t h e
on a
w
(in a
or
the corresponding
sheaves
(sheaves
in
destroying
the universal
an A - i n d e x e d
t y p e of m o d e l
constructed
be e m b e d d e d
without
function
we consider
To a d d
such
not pursue
inside
set
a choice
of
a discussion
Example
extension)
topos.
This
treatment
topoi
of
2IN
(2~T)A
is d i s c u s s e d
F.
6
FOURMAN-SCEDROV
in d e t a i l
[5].
in F o u r m a n
In a n y
& Scott
such model
the
represented
as the
sheaf
functions.
In o u r
example,
Cantor
space
(2~) A
to
The
internal
sheaf
isomorphic
to
the
A.
show
They
family
~
that
some
if t h e r e
inhabited
e 6 [[ f(x)
To
see
the
the
F
satisfy
WSAC.
that
CC
every
disjoint
we have
I(2A)
of
w
the
use
required
and
the
2A
if
This
hold
cover
is
sections
~
has
for e a c h
§ 2~
corresponding
shows
2).
x
6 F
such
a
this
that our new
a refinement
we n o t e
Thus
in
by
in the m o d e l
of
w
any
by a g l o b a l
principles
of g l o b a l
there
zero-aimensional
sets because
realized
§
of
in t h i s m o d e l
is d e c i d a b l e .
sequences
internal
section
in t h e m o d e l
sections
which
define
functions.
}
WSAC.
Remarks
The choice
tion
A.
DC
c a n be
ZF + DC
5.
as the
then
determines
(2A) ~
open
clopen
statement
to c h o o s e
~
(2~) A
: (2~)A
a
~a
§
and
(2~) A
mutually
and we can
internally
it is a d o u b l e t o n .
then
~
As
a function
that
existence
from
a subsheaf
satifies
to t h e r e s u l t s
§ ~]]
projection
= 72]] .
space
sense
of
(2~)A
is a n y m o r p h i s m
: ~
that
of
of g l o b a l
o p e n of
~ 6 [[f
that
is a u n i q u e
cannot
open
is r e p r e s e n t e d
Suppose
yield
elements
generate
by a family
(contrary
model
is
F.
over
would
space
projections
space which
F § A
that
required
inhabited
is a f u n c t i o n
there
& Hyland
2 ~-valued
if it is i n h a b i t e d
generated
to
We now
defined
Cantor
continuous
b y the v a r i o u s
Cantor
that
corresponding
trivial
internal
of
from any
isomorphic
the
the condition
The
given
and Feurman
.
projections
from a set
of
are
2~
[6]
of
principle
separable
X2 + 1
has
closures
a root
WSAC
arose
from
in c o n s t r u c t i v e
it h a s
two a n d
330
they
the c o n s i d e r a mathematics:
are distinct.
FOU~4AN-~EDROV
In the
field,
R
which
lives
given
by the
of
X2 + 1
extension
since
in
~,
inclusion
are
in w h i c h
the
adding
w
with
and c o m p l e x
set
a choice
the t r a n s i t i o n
conjugation,
A.
There
set of r o o t s
yield
covering.
on the
of
just our
this w o u l d
S{
§
in the m o d e l
However,
open
7
the roots
can be no field
becomes
inhabited
function.
a root universally
This corresponds
internal
maps
gives
to t a k i n g
us
an
sheaves
locale
o
o
with
the t r a n s i t i o n
generated
~f(*)
or
= a2 ]
which
DC
(2A) ~
I
pass
if we use the
space
maps
o
indicated.
by the e l e m e n t s
The use of
CC
o
~f(*)
are p e r m u t e d
(2 A)
to our
formal
These
This
= aI ]
by
in the m o d e l
locale
and
the a c t i o n
of
w
is
of
B.
to see that
final m o d e l m a y be c i r c u m v e n t e d
locale
(2A) ~
coincide
if
in p l a c e
I
(2A)
of the
is v a l i d
[4]~
Acknowledgements.
T h e m e t h o d s u s e d in this p a p e r are not
new.
T h e y are p a r t of the f o l k - l o r e of t o p o s - t h e o r y and
are u l t i m a t e l y due to the i n s i g h t s of Bill L a w v e r e and
A n d r ~ Joyal.
T h e i r i n f l u e n c e is g r a t e f u l l y a c k n o w l e d g e d .
References
1
DUi~.~ETT, M . A . E . f
U n i v e r s i t y Press,
E l e m e n t s of i n t u i t i o n i s m
Oxford
(1977)
2
F O U ~ i A N , M.P. : The logic of topoi.
In: H a n d b o o k of
Mathematical Logic
(ed. J. B a r w i s e ) , N o r t h Holland,
Amsterdam
(1977), 1 0 5 3 - 1 0 9 0
3
F O U R M A N , M.P. : Sheaf m o d e l s for set theory.
and A p p l i e d A l g e b r a 19 (1980), 91-101
4
F O U R M A N , M.P. and GRAYSON, R.J.: F o r m a l spaces induct i o n p r i n c i p l e s and c o m p l e t e n e s s t h e o r e m s .
In
preparation
F O U R M A N , M.P. and HYLAIqD, J.M.E.: Sheaf m o d e l s for
analysis.
In: A p p l i c a t i o n s of S h e a v e s
(Proc. L.M.S.
D u r h a m S y m p o s i u m , 1977), S p r i n g e r L e c t u r e N o t e s in
Math. 753 (1979)
331
.
Oxford
J. Pure
8
FOURMAN-SCEDROV
FOURMAN, M.P. and SCOTT, D.S.: Sheaves and logic.
A p p l i c a t i o n s of Sheaves
(Proc. L.M.S. D u r h a m
Symposium, 1977), S p r i n g e r Lecture Notes in Math.
753 (1979)
In:
FREYD, P.: The a x i o m of choice.
A l g e b r a 19 (1981), 103-125
J. Pure
JOYAL, A. and REYES,
ical logic.
J. Pure
m o d e l s in c a t e g o r A l g e b r a (to appear)
G.: Generic
and A p p l i e d
and A p p l i e d
KRIPKE, S . : S e m a n t i c a l a n a l y s i s of i n t u i t i o n i s t i c
logic I.
Z. math. logik u Grundl. Math. 9 (1963),
67-96
i0
LAWVERE, F.W.: I n t r o d u c t i o n .
G e o m e t r y and Logic.
Springer
274 (1972), 1-12
In: Poposes, A l g e b r a i c
L e c t u r e Notes in Math.
ii
MACLANE, S. : C a t e g o r i e s
cian
Springer-Verlag,
12
MAKKAI,
logic.
13
OSIUS, G.:
L o g i c a l and S e t - T h e o r e t i c a l Tools in
E l e m e n t a r y Topoi.
In: M o d e l T h e o r y and Topoi .
S p r i n g e r L e c t u r e Notes in Math. 445 (1975), 297-346
14
SCEDROV, A . : C o n s i s t e n c y and I n d e p e n d e n c e Proofs in
i n t u i t i o n i s t i c set theory.
P r o c e e d i n g s of the N e w
M e x i c o C o n f e r e n c e on C o n s t r u c t i v e M a t h e m a t i c s
(F.
Richman, ed.), S p r i n g e r Lecture Notes (to appea r )
for the W o r k i n g
Heidelberg
Mathemati-
M. and REYES, G. : F i r s t - o r d e r c a t e g o r i c a l
S p r i n g e r Lecture Notes in Math. 611 (1977)
15
SCOTT,
16
TIERNEY, M. : F o r c i n g T o p o l o g i e s and C l a s s i f y i n g Topoi~
in Algebra, T o p o l o g y and C a t e g o r y Theory; a
c o l l e c t i o n of p a p e r s in honor of Samuel E i l e n b e r g
(A. Heller, ed.), A c a d e m i c Press, New York (1976),
211-219
D.S.:
Sheaf
models
for set theory.
D e p a r t m e n t of M a t h e m a t i c s
Columbia University
N e w York, N Y
10027
U.S.A.
Department
University
Ann Arbor,
U.S.A.
of M a t h e m a t i c s
of M i c h i g a n
MI
48109
(Received October 12, 1981)
332
To a p p e a r