The Consistency of the Axiom of Choice, Filozofia, Filozofia - Artykuły
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The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis
Author(s): Kurt Godel
Source: Proceedings of the National Academy of Sciences of the United States of America,
Vol. 24, No. 12 (Dec. 15, 1938), pp. 556-557
Published by: National Academy of Sciences
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556
MA THEMA TICS: K.
GODEL
PROC.N. A. S.
THE CONSISTENCY OF THE AXIOM OF CHOICE AND OF THE
GENERALIZED CONTINUUM-HYPOTHESIS
BY KURT G6DEL
THE
INSTITUTE
FOR ADVANCED
STUDY
Communicated November 9, 1938
THEOREM.
Let T be the
system of
axioms
for set-theory
obtained
from
v. Neumann's
system
S*
1
by leaving
out theaxiom
of
choice
(Ax.
III
3*); then,
if
T is
consistent,
it remains
so,
if
the
following
propositions
1-4 are
adjoined
simultaneously
as new axioms:
1.
The axiom of choice
(i.e.,
v. Neumann's Ax. III
3*)
2.
The
generalized
Continuum-Hypothesis (i.e.,
the statement that
2t
=
N,
+
1
holds for
any
ordinal
a)
3. The
existence of linear non-measurable sets such that both
they
and
their
complements
are one-to-one
projections
of two-dimensional
comple-
ments of
analytic
sets
(and
which therefore are
B2-sets
in Lusin's termi-
nology2)
4. The
existence of linear
complements
of
analytic
sets,
which are of
the
power
of the continuum and contain no
perfect
subset.
A
corresponding
theorem
holds,
if T denotes the
system
of Princ.
Math.3 or Fraenkel's
system
of axioms for set
theory,4 leaving
out in both
cases the
axiom of choice but
including
the axiom of
infinity.
The
proof
of the "abovetheorems is constructive in the sense
that,
if
a
contradiction
were obtained in the
enlarged system,
a contradiction in
T
could
actually
be exhibited.
The method
of
proof
consists in
constructing
on the basis of the
axioms
of T5 a model
for which the
propositions
1-4 are true. This
model,
roughly speaking,
consists of all
"mathematically
constructible"
sets,
where
the
term
"constructible" is to be understood in
the
semiintuitionistic
sense which
excludes
impredicative procedures.
This means
"construc-
tible" sets
are defined to be those sets which can be obtained
by
Russell's
ramified
hierarchy
of
types,
if extended to include transfinite
orders.
The extension
to transfinite orders has the
consequence
that the model
satisfies
the
impredicative
axioms of set
theory,
because an axiom of re-
ducibility
can be
proved
for
sufficiently
high
orders. Furthermore the
proposition "Every
set is
constructible"
(which
I abbreviate
by
"A")
can
be
proved
to be consistent with
the axioms of
T,
because A turns out to be
true for the model
consisting
of
the constructible sets. From A the
propo-
sitions 1-4 can be
deduced. In
particular, proposition
2 follows
from the
fact that all constructible
sets of
integers
are obtained
already
for orders
<
ci,
all constructible
sets of sets of
integers
for orders < o2
and so on.
VOL.
24,
1938 MATHEMATICS: BLUMENTHAL AND THURMAN
557
The
proposition
A added as a new axiom seems to
give
a natural com-
pletion
of
the
axioms of
set
theory,
in so far as it determines the
vague
notion of
an
arbitrary
infinite set in a definite
way.
In this connection it is
important
that the
consistency-proof
for A does not break down if
stronger
axioms of
infinity (e.g.,
the existence of inaccessible
numbers)
are
adjoined
to
T. Hence the
consistency
of A seems to be absolute in some
sense,
al-
though
it
is not
possible
in the
present
state of affairs to
give
a
precise
meaning
to this
phrase.
'
Cf. J. reine
angew. Math., 160, p.
227.
2
Cf. N.
Lusin, Lemons
sur les ensembles
analytiques,
Paris, 1930,
p.
270.
3
Cf. A.
Tarski,
Mh. Math.
Phys., 40, p.
97.
4 Cf. A.
Fraenkel,
Math.
Zeit., 22, p.
250.
5
This means that the model is constructed
by essentially
transfinite methods and
hence
gives only
a relative
proof
of
consistency, requiring
the
consistency
of T as a
hypothesis.
THE CHARA
CTERIZA TION OF
PSEUDO-S,
) SETS
BY
LEONARD M. BLUMENTHAL AND GEORGE R. THURMAN
DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF MISSOURI
Communicated November 8,
1938
I. If to each
pair
of elements
(points) p, q
of an abstract set there is
attached a
non-negative
real number
(distance) pq, independent
of the
order of the
elements,
while
pq
= 0 if and
only
if
p
=
q,
the
resulting space
is called semimetric.
A fundamental
problem
in the distance
geometry
of
the n-dimensional
spherical
surface
Sn,r
of radius
r(the
n-dimensional
surface of
a
sphere
of radius r in euclidean
space
of n
+
1
dimensions,
with
"shorter
arc"
distance)
consists in
characterizing
those semimetric
spaces
S
which
have the
following properties: (1)
S contains more than
n
+
3
points, (2)
if
p,
qeS,
then
pq
=
d =
Irr,
(3)
if
pi,
P2.,
, n
+
2
are elements
of
S,
then there exists a function
f
mapping
these n
+
2
points upon Sn,
r
with
preservation
of distances
(i.e., congruently), (4)
S cannot be
mapped
congruently upon
a subset of
S,,
,.
Reserving
the details of the
investiga-
tion for
publication
elsewhere,
we summarize in this note the
complete
solution of this
problem.
Semimetric
spaces
S with
properties (3), (4)
are
called
pseudo-S,,
sets.l
The
properties
of
the
Sn,
r,
by
virtue of which the characterization
theorems of
pseudo-Sn,
sets are
obtained,
are all
consequences
of the
following
metric ones:
(1)
the mutual distances of each set of n
+
2
points
of
Sn r satisfy
a relation of the form
I| s(pipj/r) I
=
O, (i, j
=
1,-
2,
. ..
,
n
+
2),
where
po(pq/r)
is a
real,
single-valued, monotonically
de-
VOL.
24,
1938 MATHEMATICS: BLUMENTHAL AND THURMAN
557
The
proposition
A added as a new axiom seems to
give
a natural com-
pletion
of
the
axioms of
set
theory,
in so far as it determines the
vague
notion of
an
arbitrary
infinite set in a definite
way.
In this connection it is
important
that the
consistency-proof
for A does not break down if
stronger
axioms of
infinity (e.g.,
the existence of inaccessible
numbers)
are
adjoined
to
T. Hence the
consistency
of A seems to be absolute in some
sense,
al-
though
it
is not
possible
in the
present
state of affairs to
give
a
precise
meaning
to this
phrase.
'
Cf. J. reine
angew. Math., 160, p.
227.
2
Cf. N.
Lusin, Lemons
sur les ensembles
analytiques,
Paris, 1930,
p.
270.
3
Cf. A.
Tarski,
Mh. Math.
Phys., 40, p.
97.
4 Cf. A.
Fraenkel,
Math.
Zeit., 22, p.
250.
5
This means that the model is constructed
by essentially
transfinite methods and
hence
gives only
a relative
proof
of
consistency, requiring
the
consistency
of T as a
hypothesis.
THE CHARA
CTERIZA TION OF
PSEUDO-S,
) SETS
BY
LEONARD M. BLUMENTHAL AND GEORGE R. THURMAN
DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF MISSOURI
Communicated November 8,
1938
I. If to each
pair
of elements
(points) p, q
of an abstract set there is
attached a
non-negative
real number
(distance) pq, independent
of the
order of the
elements,
while
pq
= 0 if and
only
if
p
=
q,
the
resulting space
is called semimetric.
A fundamental
problem
in the distance
geometry
of
the n-dimensional
spherical
surface
Sn,r
of radius
r(the
n-dimensional
surface of
a
sphere
of radius r in euclidean
space
of n
+
1
dimensions,
with
"shorter
arc"
distance)
consists in
characterizing
those semimetric
spaces
S
which
have the
following properties: (1)
S contains more than
n
+
3
points, (2)
if
p,
qeS,
then
pq
=
d =
Irr,
(3)
if
pi,
P2.,
, n
+
2
are elements
of
S,
then there exists a function
f
mapping
these n
+
2
points upon Sn,
r
with
preservation
of distances
(i.e., congruently), (4)
S cannot be
mapped
congruently upon
a subset of
S,,
,.
Reserving
the details of the
investiga-
tion for
publication
elsewhere,
we summarize in this note the
complete
solution of this
problem.
Semimetric
spaces
S with
properties (3), (4)
are
called
pseudo-S,,
sets.l
The
properties
of
the
Sn,
r,
by
virtue of which the characterization
theorems of
pseudo-Sn,
sets are
obtained,
are all
consequences
of the
following
metric ones:
(1)
the mutual distances of each set of n
+
2
points
of
Sn r satisfy
a relation of the form
I| s(pipj/r) I
=
O, (i, j
=
1,-
2,
. ..
,
n
+
2),
where
po(pq/r)
is a
real,
single-valued, monotonically
de-
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