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1. Cantor's
problem of the cardinal number of the continuum
Two systems, i.
e, two assemblages of ordinary real numbers or points, are said to be
(according to Cantor) equivalent or of equal cardinal number, if
they can be brought into a relation to one another such that to every
number of the one assemblage corresponds one and only one definite number
of the other. The investigations of Cantor on such assemblages of points
suggest a very plausible theorem, which nevertheless, in spite of the
most strenuous efforts, no one has succeeded in proving. This is the
theorem: Every system of
infinitely many real numbers, i. e., every assemblage of numbers (or
points), is either equivalent to the assemblage of natural integers, 1,
2, 3,... or to the assemblage of all real numbers and therefore to the
continuum, that is, to the points of a line; as regards equivalence
there are, therefore, only two assemblages of numbers, the countable
assemblage and the continuum. From this theorem
it would follow at once that the continuum has the next cardinal number
beyond that of the countable assemblage; the proof of this theorem would,
therefore, form a new bridge between the countable assemblage and the
continuum. Let me mention
another very remarkable statement of Cantor's which stands in the closest
connection with the theorem mentioned and which, perhaps, offers the key
to its proof. Any system of real numbers is said to be ordered, if for
every two numbers of the system it is determined which one is the earlier
and which the later, and if at the same time this determination is of
such a kind that, if a is before b and b is before c,
then a always comes before c. The natural arrangement of
numbers of a system is defined to be that in which the smaller precedes
the larger. But there are, as is easily seen infinitely many other ways
in which the numbers of a system may be arranged. If we think of a
definite arrangement of numbers and select from them a particular system
of these numbers, a so-called partial system or assemblage, this partial
system will also prove to be ordered. Now Cantor considers a particular
kind of ordered assemblage which he designates as a well ordered
assemblage and which is characterized in this way, that not only in the
assemblage itself but also in every partial assemblage there exists a
first number. The system of integers 1, 2, 3, ... in their natural order
is evidently a well ordered assemblage. On the other hand the system of
all real numbers, i. e., the continuum in its natural order, is
evidently not well ordered. For, if we think of the points of a segment
of a straight line, with its initial point excluded, as our partial
assemblage, it will have no first element. The question now
arises whether the totality of all numbers may not be arranged in another
manner so that every partial assemblage may have a first element, i.
e., whether the continuum cannot be considered as a well ordered
assemblage--a question which Cantor thinks must be answered in the
affirmative. It appears to me most desirable to obtain a direct proof of
this remarkable statement of Cantor's, perhaps by actually giving an
arrangement of numbers such that in every partial system a first number
can be pointed out.
2.
The compatibility of the arithmetical axioms
When we are engaged
in investigating the foundations of a science, we must set up a system of
axioms which contains an exact and complete description of the relations
subsisting between the elementary ideas of that science. The axioms so
set up are at the same time the definitions of those elementary ideas;
and no statement within the realm of the science whose foundation we are
testing is held to be correct unless it can be derived from those axioms
by means of a finite number of logical steps. Upon closer consideration
the question arises: Whether, in any way, certain statements of single
axioms depend upon one another, and whether the axioms may not therefore
contain certain parts in common, which must be isolated if one wishes to
arrive at a system of axioms that shall be altogether independent of one
another. But above all I
wish to designate the following as the most important among the numerous
questions which can be asked with regard to the axioms: To prove that
they are not contradictory, that is, that a definite number of logical
steps based upon them can never lead to contradictory results. In geometry, the
proof of the compatibility of the axioms can be effected by constructing
a suitable field of numbers, such that analogous relations between the
numbers of this field correspond to the geometrical axioms. Any
contradiction in the deductions from the geometrical axioms must
thereupon be recognizable in the arithmetic of this field of numbers. In
this way the desired proof for the compatibility of the geometrical
axioms is made to depend upon the theorem of the compatibility of the
arithmetical axioms. On the other hand a
direct method is needed for the proof of the compatibility of the
arithmetical axioms. The axioms of arithmetic are essentially nothing
else than the known rules of calculation, with the addition of the axiom
of continuity. I recently collected them4
and in so doing replaced the axiom of continuity by two simpler axioms,
namely, the well-known axiom of Archimedes, and a new axiom essentially
as follows: that numbers form a system of things which is capable of no
further extension, as long as all the other axioms hold (axiom of
completeness). I am convinced that it must be possible to find a direct
proof for the compatibility of the arithmetical axioms, by means of a
careful study and suitable modification of the known methods of reasoning
in the theory of irrational numbers. To show the
significance of the problem from another point of view, I add the
following observation: If contradictory attributes be assigned to a
concept, I say, that mathematically the concept does not exist. So,
for example, a real number whose square is -l does not exist
mathematically. But if it can be proved that the attributes assigned to
the concept can never lead to a contradiction by the application of a
finite number of logical processes, I say that the mathematical existence
of the concept (for example, of a number or a function which satisfies
certain conditions) is thereby proved. In the case before us, where we
are concerned with the axioms of real numbers in arithmetic, the proof of
the compatibility of the axioms is at the same time the proof of the
mathematical existence of the complete system of real numbers or of the
continuum. Indeed, when the proof for the compatibility of the axioms
shall be fully accomplished, the doubts which have been expressed
occasionally as to the existence of the complete system of real numbers
will become totally groundless. The totality of real numbers, i. e.,
the continuum according to the point of view just indicated, is not the
totality of all possible series in decimal fractions, or of all possible
laws according to which the elements of a fundamental sequence may
proceed. It is rather a system of things whose mutual relations are
governed by the axioms set up and for which all propositions, and only
those, are true which can be derived from the axioms by a finite number
of logical processes. In my opinion, the concept of the continuum is
strictly logically tenable in this sense only. It seems to me, indeed,
that this corresponds best also to what experience and intuition tell us.
The concept of the continuum or even that of the system of all functions
exists, then, in exactly the same sense as the system of integral,
rational numbers, for example, or as Cantor's higher classes of numbers
and cardinal numbers. For I am convinced that the existence of the latter,
just as that of the continuum, can be proved in the sense I have
described; unlike the system of all cardinal numbers or of all
Cantor s alephs, for which, as may be shown, a system of axioms,
compatible in my sense, cannot be set up. Either of these systems is,
therefore, according to my terminology, mathematically non-existent. From the field of
the foundations of geometry I should like to mention the following
problem:
3.
The equality of two volumes of two tetrahedra of equal bases and equal
altitudes
In two letters to
Gerling, Gauss5
expresses his regret that certain theorems of solid geometry depend upon
the method of exhaustion, i. e., in modern phraseology, upon the
axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in
particular the theorem of Euclid, that triangular pyramids of equal
altitudes are to each other as their bases. Now the analogous problem in
the plane has been solved.6
Gerling also succeeded in proving the equality of volume of symmetrical
polyhedra by dividing them into congruent parts. Nevertheless, it seems
to me probable that a general proof of this kind for the theorem of
Euclid just mentioned is impossible, and it should be our task to give a
rigorous proof of its impossibility. This would be obtained, as soon as
we succeeded in specifying two tetrahedra of equal bases and equal
altitudes which can in no way be split up into congruent tetrahedra, and
which cannot be combined with congruent tetrahedra to form two polyhedra
which themselves could be split up into congruent tetrahedra.7
4.
Problem of the straight line as the shortest distance between two points
Another problem
relating to the foundations of geometry is this: If from among the axioms
necessary to establish ordinary euclidean geometry, we exclude the axiom
of parallels, or assume it as not satisfied, but retain all other axioms,
we obtain, as is well known, the geometry of Lobachevsky (hyperbolic
geometry). We may therefore say that this is a geometry standing next to
euclidean geometry. If we require further that that axiom be not
satisfied whereby, of three points of a straight line, one and only one
lies between the other two, we obtain Riemann's (elliptic) geometry, so
that this geometry appears to be the next after Lobachevsky's. If we wish
to carry out a similar investigation with respect to the axiom of
Archimedes, we must look upon this as not satisfied, and we arrive
thereby at the non-archimedean geometries which have been investigated by
Veronese and myself. The more general question now arises: Whether from
other suggestive standpoints geometries may not be devised which, with
equal right, stand next to euclidean geometry. Here I should like to
direct your attention to a theorem which has, indeed, been employed by
many authors as a definition of a straight line, viz., that the straight
line is the shortest distance between two points. The essential content
of this statement reduces to the theorem of Euclid that in a triangle the
sum of two sides is always greater than the third side--a theorem which,
as is easily seen, deals sole]y with elementary concepts, i. e.,
with such as are derived directly from the axioms, and is therefore more
accessible to logical investigation. Euclid proved this theorem, with the
help of the theorem of the exterior angle, on the basis of the congruence
theorems. Now it is readily shown that this theorem of Euclid cannot be
proved solely on the basis of those congruence theorems which relate to
the application of segments and angles, but that one of the theorems on
the congruence of triangles is necessary. We are asking, then, for a
geometry in which all the axioms of ordinary euclidean geometry hold, and
in particular all the congruence axioms except the one of the congruence
of triangles (or all except the theorem of the equality of the base
angles in the isosceles triangle), and in which, besides, the proposition
that in every triangle the sum of two sides is greater than the third is
assumed as a particular axiom. One finds that such
a geometry really exists and is no other than that which Minkowski
constructed in his book, Geometrie der Zahlen,8
and made the basis of his arithmetical investigations. Minkowski's is
therefore also a geometry standing next to the ordinary euclidean
geometry; it is essentially characterized by the following stipulations: In Minkowski's
geometry the axiom of parallels also holds. By studying the theorem of
the straight line as the shortest distance between two points, I arrived9
at a geometry in which the parallel axiom does not hold, while all other
axioms of Minkowski's geometry are satisfied. The theorem of the straight
line as the shortest distance between two points and the essentially
equivalent theorem of Euclid about the sides of a triangle, play an
important part not only in number theory but also in the theory of
surfaces and in the calculus of variations. For this reason, and because
I believe that the thorough investigation of the conditions for the
validity of this theorem will throw a new light upon the idea of distance,
as well as upon other elementary ideas, e. g., upon the idea of
the plane, and the possibility of its definition by means of the idea of
the straight line, the construction and systematic treatment of the
geometries here possible seem to me desirable. |
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