over the A-splitting tree,. I A states that if a collection of finite sequences K c A
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 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 6 K
6 A
