CANTOR'S PHILOSOPHICAL WRITING
§ 1 (p.
882) As for the mathematical infinite, to the extent that it has found a
justified application in science and contributed to is usefulness, it seems to me
that it has hitherto appeared principally in the role of a variable quantity ,
which either grows beyond all bounds or diminishes to any desired minuteness,
but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche].
… Infinity, in its first form
(the improper-infinite) presents itself as a variable finite [veranderliches
Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche])
it appears as a thoroughly determinate [bestimmtes] infinite.
[para.11]
For finite sets power coincides with the Anzahl of elements, because, as
everybody knows, such sets have the same Anzahl of elements in every ordering.
[para.12]
For infinite sets, on the other hand, until now nobody at all has talked of a
precisely defined Anzahl of their elements – even though a determinate
power, entirely independent of their ordering, can be ascribed to them.
§ 2 (Ewald
p. 885) [para.1] Another great gain ascribable to the new numbers
consists, for me, in a new concept which has not yet been mentioned –
namely the concept of the Anzahl of the elements of a well-ordered
infinite manifold. Because this concept is always expressed by a
completely determinate number of our widened number-domain (provided only that
the shortly-to-be-defined ordering of the elements of the set is determinate),
and because on the other hand the Anzahl concept obtains an immediate concrete
representation in our inner intuition [Anschauung], so, through this connection
between Anzahl and number [Zahl], the reality (which I emphasise) of the latter
is proved even in the cases where it is determinate-infinite.
[para.2] A
well-ordered set is a well-defined set in which the elements are bound
to one another by a determinate given succession such that (i) there is a first
element of the set; (ii) every single element (provided it is not the last in
the succession) is followed by another determinate element; and (iii) for any
desired finite or infinite set of elements there exists a determinate element
which is their immediate successor in the succession (unless there is
absolutely nothing in the succession following all of them). Two
"well-ordered" sets are now said to be of the same Anzahl
(with respect to their given successions) when a reciprocal one-to-one
correlation of them is possible such that, if E anf F are any two elements of
the one set, and E1 and F1 are the corresponding elements of the other, then
the position of E and F in the succession of the first set always agrees with
the position of E1 and F1 in the succession of the second set (i.e. when E
precedes F in the succession of the first set, then E1 also precedes F1 in the
succession of the second set). This correlation, if it is possible at
all, is, as one easily sees, always completely determinate; and since in the
widened number sequence there is always one and only one number alpha such that
the numbers preceding it (from 1 on) on the natural succession have the same
Anzahl, one must set the "Anzahl" of both these "well-ordered"
sets directly to alpha, if alpha is an infinitely large number, and to the
number alpha-1 which immediately precedes alpha, if alpha is a finite integer.
[para 3.] Der
wesentliche Unterschied zwischen den endlichen und unendlichen Mengen zeigt
sich nun darin, dass ein endliche Menge in jeder Sukzession, welche man
ihren Elementen geben kann, dieselbe Anzahl von Elementen darbietet;
dagegen werden einer aus unendlich vielen Elementen bestehenden Menge im
allgemeinen verscheidene Anzahlen zukommen, je nach der Sukzession,
welche man den Elementen gibt. Die Ma:chtigkeit einer Menge ist,
wie wir gesehen, ein von der Anordnung unabha:ngiges Attribut derselben; die Anzahl
der Menge weist sich aber als ein von einer gegebenen Sukzession der Elemente
im allgemeinen abha:ngiger Faktor aus, sobald man es mit unendlichen Mengen zu
tun hat.
[The essential
difference between finite and infinite sets now turns out to be that a finite
set presents the same Anzahl of elements for every succession which one can give
its elements; in contrast, a set consisting of infinitely many elements will in
general give rise to *different" Anzahlen, depending on the succession
that one gives the elements. The *power* of a set is, as we saw, a
property independent of the ordering; but the Anzahl of the set reveals itself
to be, in general (as soon as one has anything to do with infinite sets) a
property dependent on a given succession of elements. Nevertheless, there
is even for infinite sets a certain connection between the power of a set and
the Anzahl of its elements determined by a given succession.]
§ 4 (p. 887) The
question of the establishment of such interpolations [of finite numbers which
do not coincide with the rational and irrational numbers (which latter appear
as the limiting values of the sequence of rational numbers) but which might be
inserted into the supposed gaps amidst the real numbers], on which some authors
have expended much effort, can, in my opinion, only be clearly and distinctly
answered with the help of our new numbers …
(p. 889) I
believe, however, that I have proved above (and it will appear even more
clearly in what follows) that determinate countings can be carried out just as
well for infinite sets as for finite ones.
Beschaffenheit endlicher
Mengen, dass das Resultat der Za:hlung – die Anzahl - unabha:ngig ist
von der jeweiligung Abordnung; wa:hrend bei unendlichen Mengen, wie wir gesehen
haben, ein solche Unabha:ngigkeit im allgemeinen nicht zutrifft, sondern
die Anzahl einer unendlichen Menge eine durch das Gesetz der Za:hlung mitbestimmte
unendliche ganze Zahl ist; hierin liegt eben und hierin allein der in der Natur
slebst begru:ndete und daher niemals fortzuschaffende wesentliche Unterschied zwischen
dem endlichen und unendlichen; nimmermehr wird aber um dieses Unterschiedes
willen die Existenz des Unendlichen geleugnet, dagegen die des Endlichen
aufrecht erhalten werden ko:nnen; la:sst man das eine fallen, so muss man mit
dem andern auch aufra:umen; wo wu:rdern wir also auf diesem Wege hinkommen?
§ 5 What I
declare and believe to have demonstrated in this work as well as in earlier
papers is that following the finite there is a transfinite
(transfinitum)--which might also be called supra-finite (suprafinitum), that
is, there is an unlimited ascending ladder of modes, which in its nature is not
finite but infinite, but which can be determined as can the finite by
determinate, well-defined and distinguishable numbers. (Ewald p. 891)"
§ 7 When I
conceive the infinite … there follows for me a genuine pleasure … in seeing how
the concept of integer [der ganze Zahlbegriff], which in the finite has
only the background of number [Anzahl], as it were splits into *two*
concepts when we ascend to the infinite – one of power [Machtigkeit]
which is independent of the ordering which a set [Menge] is given and
one of number [Anzahl] which is necessarily bound to a lawlike ordering
of the set [Menge] by virtue of which it becomes well-ordered
(wohlgeordneten). And when I descend again from the infinite to the
finite I see clearly how the two concepts become one again and flow together to
from the concept of finite integer [endlichen ganzen Zahl].
§ 8 (p. 895)…
we may regard the integers as actual in so far as, on the basis of definitions,
they occupy an entirely determinate place in our understanding , [and] are well
distinguished from other parts of our thought … let us call this kind of
reality of our numbers their intrasubjective or immanent reality.
[which he contrasts with numbers of things in nature, their "transient
reality"].
(896)
Mathematics, in the development of its ideas, has only to take account
of the immanent reality of its concepts and has absolutely no
obligation to examine their transient reality.
… Mathematics is
in its development entirely free and is only bound in the self-evident respect
that its concepts must both be consistent with each other, and also stand in
exact relationships, ordered by definitions, to those concepts which have
previously been introduced and are already at hand and established. In
particular, in the introduction of new numbers, it is only obligated to give
definitions of them which will bestow such a determinacy and, in certain
circumstances, such a relationship to the other numbers that they can in any
given instance be precisely distinguished. As soon as a number satisfies
all these conditions, it can and must be regarded in mathematics as existent
and real.
"… the
essence of mathematics lies entirely in its freedom".
§ 9 [He argues
for the existence of limits]
§ 11 (Ewald p.
907)
I shall now show
[ie demonstrate - gezeigen] how one is led to the definitions of the new
numbers and how the natural cuts I call the number classes arise in the
absolutely infinite sequence of real integers.
The sequence (I)
of positive integers 1,2,3…,v,… has its ground of origin [Entstehungsgrund]
in the repeated positing [Setzung] and uniting [Vereinigung] of
underlying unities [Einheiten], which are regarded as alike; the
number v is the expression for a [definite] finite number [bestimmte
endliche Anzahl] of such positings following one another in a sequence; it
is also the expression for the unification [Vereinigung] of the posited
unities [gesetzten Einheiten] into a whole [zu einem Ganzen].
The formation of the finite real integers [i.e. natural numbers] thus rests
upon the principle of adding a unity to an already formed and existing number; I
call this principle (which, as we shall soon see, plays an essential role in
the generation of the higher integers) the first principle of generation.
The number [Anzahl] of the numbers v of class (I) formed in this
way is infinite and there is no greatest among them. However
contradictory it might be to speak of a greatest number of class (I), there is
nevertheless nothing offensive in thinking of a new number which we
shall call w, and which [my emphasis] will be the expression for the idea
[fact] that the entire assemblage [aggregate, Inbegriff] (I) is
given in its natural, orderly succession [natural succession according to a
law] (Just as v is an expression for the idea that a certain finite
number [Anzahl] of unities is united to form a whole [zu einem Ganzen
vereinigt wird].)
The logical
function which gave us the two numbers w and 2w is obviously
different from the first principle of generation. I call it the second
principle of generation of whole real numbers [integers], which I define
more exactly as follows: if there is some determinate succession of defined
whole real numbers, among which there exists no greatest, on the basis of this
second principle of generation a new number is obtained which is regarded as
the limit of those numbers, i.e. is defined as the next greater number than all
of them.
The
Correspondence
Cantor to
Hilbert (2 October 1897). [I said]
"The totality of all alephs cannot be conceived as a determinate,
well-defined, and also a finished set". This is the punctum
saliens, and I venture to say that this completely certain theorem,
provable rigorously from the definition of the totality of all alephs, is
the most important and noblest theorem of set theory. One must only
understand the expresion "finished" correctly. I say of a set
that it can be thought of as finished (and call such a set, if it contains
infinitely many elements, "transfinite" or "suprafinite")
if it is possible without contradiction (as can be done with finite sets) to
think of all its elements as existing together, and to to think of the
set itself as a compounded thing for itself; or (in other words) if it
is possible to imagine the set as actually existing with the totality of
its elements.
So the "transfinite"
coincides with what has since antiquity been called the "actual
infinite", and is to be considered as an ajwrismenon (something
determinate).
So too I have translated the word
"set" [Menge] (when it is finite or transfinite) into
French with "ensemble" and into Italian with "insieme".
And so too, in the first article of the work, "Contributions to the
founding of transfinite set theory" I define a "set" (meaning
thereby only the finite or transfinite) at the very beginning as an
"assembling together" [Zusammenfassung]. But an "assembling
together" is only possible if an existing together [Zusammenseins]
is possible.
In contrast, infinite sets such
that the totality of their elements cannot be thought of as
"existing together" or as a "thing for itself" or an
ajwrismenon, and that therefore also in this totality are absolutely not
an object of further mathematical contemplation, I call "absolutely
infinite sets", and to them belongs the "set of all alephs".
Cantor to
Dedekind (3 August 1899). As I wrote
to you last week [in a letter dated 28 July], it is important to me to know
your opinion on certain fundamental points in set theory; please forgive me for
the trouble this causes you.
[The next is
quoted by Zermelo as if from the 28 July letter].
If we start from
the notion of a definite multiplicity [bestimmte Vielheit] (a system [System],
a totality [Inbegriff]) of things, it is necessary, as I discovered, to
distinguish two kinds of multiplicities (by this I always mean definite
multiplicities [bestimmte Vielheiten]).
For a
multiplicity can be such that the assumption that all of its elements
"are together" [die Annahme eines "Zusammenseins" aller
ihre Elemente] leads to a contradiction, so that it is impossible to
conceive of the multiplicity [Vielheit] as a unity [Einheit], as
"one finished thing" [ein fertiges Ding]. Such
multiplicities I all absolutely infinite [absolut undendliche] or
inconsistent multiplicities [inkonsistente Veilheiten].
As we can
readily see, the "totality of everything thinkable" [Inbegriff
alles Denbaren], for example, is such a multiplicity; later still other
examples will turn up.
If on the other
hand the totality [Gesamtheit] of the elements of a multiplicity can be
thought of without contradiction as "being together" [zusammenseiend],
so that they can be gathered together into "one thing" [ihr
Zusammengefasstwerden su einem Ding moglich ist], I call it a consistent
multiplicity [konsistente Veilheit] or a "set" [Menge].
[snip]
A multiplicity [Vielheit]
is called well-ordered [wohlgeordnet] if it fulfills the condition that
every sub-multiplicity [Teilvielheit] has a first element; such a
multiplicity [Vielheit] I call for short a sequence
[Folge].
[snip]
Now I envisage
the system of all numbers [System aller Zahlen] and denote it W.
The system W in its natural-ordering-according-to-magnitude [naturlichen
Grossenordnung] is a sequence" [Folge]. Now let us adjoin 0 as
an additional element to this sequence, and certainly if we set this 0 in the
first position then W' is still a sequence ... of which one can readily
convince oneself that every number occurring in it is the [ordinal number] of
the sequence of all its preceding elements.
Now W' (and
therefore also W) cannot be a consistent multiplicity [konsistente Vielheit].
For if W' were consistent, then as a well-ordered set [wohlgeordneten Menge],
a number [Zahl] d would belong to it which would be greater than all
numbers [Zahlen] of the system [System]
W; the number d, however, also
belongs to the system [System] W, because it comprises all numbers. Thus
d would be greater than d, which is a contradiction.
Thus the system
W of all ordinal numbers is an inconsistent, absolutely infinite multiplicity.
[Das System W aller Zahlen ist eine inkonsistente, eine absolut unendliche
Vielheit].
Dedekind
to Cantor: (Letter 29 August 1899) A
visit from you will always be welcome to me and my sister, but I am not at all
ripe for a discussion of your communication: it would for the time being be
quite fruitless! You will certainly sympathise with me if I frankly
confess that, although I have read through your letter of 3 August many
times [my emphasis], I am utterly unclear [my emphasis] about your
distinction of totalities [Inbegriffe] into consistent and inconsistent;
I do not know what you mean by the "co-existence of all elements of a
multiplicity", and what you mean by its opposite [my
emphasis]. I do not doubt that with a more thorough study of your letter
a light will go on for me; for I have great trust in your deep and perceptive
research. But until now … I have not had the time or the necessary mental
energy to immerse myself in these things.
Cantor,
"On an Elementary Question", from Ewald.
"..
[it is asked] how I know that the well-ordered multiplicities or sequences to
which I ascribe the cardinal numbers À0, À1, ... Àw,
...ÀÀ, ... are really “sets” in the sense of the word I have
explained, i.e. “consistent multiplicities”. Is it not thinkable that these
multiplicities are already “inconsistent”, and that the contradiction arising
from the assumption of a “being together of all their elements” has simply not
yet been made noticeable? My answer to this is that the same question can just
as well be raised about finite multiplicities, and that a careful consideration
will lead one to the conclusion that even for finite multiplicities no “proof”
of their consistency is to be had. In other words, the fact of the
“consistency” of finite multiplicities is a simple, unprovable truth; it
is (in the old sense of these words) “the axiom of finite arithmetic”.
And in just the same way, the “consistency” of those multiplicities to which I
attribute the alephs as cardinal numbers is “the axiom of extended transfinite
arithmetic. [Cantor, Letter to Dedekind, 28th August, 1899; GA 447-8]
In the article
entitled: "On a property of the set of all real algebraic numbers", (Journ
Math. 77, p.258, 1874 – Ewald p.839 [this was the epochal paper]), a proof
is given, probably for the first time, of the theorem that there are infinite
manifolds (Mannigfalltigkeiten) which cannot be correlated in a
reciprocal one-to-one way with the totality (Gesamtheit) of all (aller)
finite integers 1, 2, 3,…, v, …; or, as I am accustomed to saying, which
do not have the power (Machtigkeit) of the number-sequence 1, 2, 3,…, v,
… That is, from the propositions proved in §2 it follows immediately
that, for example, the totality (Gesamtheit) of all real numbers of an
arbitrary interval (a…b) cannot be presented in the sequential form w1,
w1, …, wv, … But it is
possible to give a much simpler proof of that theorem which does not depend on
considering the irrational numbers.
Cantor: On
Infinite, Linear Point-Manifolds ("Uber unendliche, lineare
Punktmannigfaltigkeiten," Mathematische Annalen 21: 545-86,
published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre,
Leipzig 1883).
Cantor.
" that from the outset they expect or even impose all the properties of
finite numbers upon the numbers in question, while on the other hand the
infinite numbers, if they are to be considered in any form at all, must (in
their contrast to the finite numbers) constitute an entirely new kind of
number, whose nature is entirely dependent upon the nature of things and is an
object of research, but not of our arbitrariness or prejudices. "
Letter to Gustac Enestrom, quoted in Dauben Georg Cantor p. 125
The actual
material of analysis is composed, in this opinion, exclusively of finite, real
integers and all truths in arithmetic and analysis already discovered or still
to be discovered must be looked upon as relationships of the finite integers to
each other; the infinitesimal analysis and with it the theory of functions are
considered to be legitimate only in so far as their theorems are demonstrable
through laws holding for the finite integers. [Punktmannigfaltigkeiten §
4 p. 103]
If one considers
the arguments which Aristotle presented against the real existence of the
infinite (vid. his Metaphysics, Book XI, Chap. 10), it will be found that they
refer back to an assumption, which involves a petitio principii, the
assumption, namely, that there are only finite numbers, from which he concluded
that to him only enumerations of finite sets were recognizable.
[Punktmannigfaltigkeiten § 4 p. 104-5]
These arguments
are probable, and proceed from things which are commonly said. For they do not
conclude of necessity: for . . . someone who said that some multitude is
infinite would not say that it is a number or that it has a number. For
'number' adds to 'multitude' the notion of a measure: for number is a multitude
measured by the unit, as is said in the tenth book of the Metaphysics. And
because of this number is said to be a species of discrete quantity, but not
multitude, which pertains to the transcendentals. [Aquinas III, Physics §
8 ]
I believe that I
have proven above, and it will appear even more clearly in what follows in this
paper, that determinate enumerations of infinite sets can be made just as well
as for finite ones, assuming that a definite law is given the sets by means of
which they become well-ordered. [Punktmannigfaltigkeiten § 4 p. 105]
The assumption
that besides the Absolute (which is not obtainable by any determination) and
the finite there are no modifications which, although not finite, nevertheless
are determinable by numbers and are therefore what I call the actual infinite -
this assumption I find to be thoroughly untenable as it stands. [Punktmannigfaltigkeiten
§ 5 p. 107]
What I declare
and believe to have demonstrated in this work as well as in earlier papers is
that following the finite there is a transfinite (transfinitum)--which might
also be called supra-finite (suprafinitum), that is, there is an unlimited
ascending ladder of modes, which in its nature is not finite but infinite, but
which can be determined as can the finite by determinate, well-defined and
distinguishable numbers. [Punktmannigfaltigkeiten § 5 p. 107]
The old and oft-repeated
proposition “Totum est majus sua parte” may be applied without proof
only in the case of entities that are based upon whole and part; then
and only then is it an undeniable consequence of the concepts “totum”
and “pars”. Unfortunately, however, this “axiom” is used innumerably often
without any basis and in neglect of the necessary distinction between “reality”
and “quantity”, on the one hand, and “number” and “set”, on the other,
precisely in the sense in which it is generally false. [An]
example may help to explain. Let M be the totality (n) of all finite numbers n,
and M¢ the totality (2n) of all even numbers 2n. Here it is undeniably correct
that M is richer in its entity, than M¢; M contains not only the even numbers,
of which M¢ consists, but also the odd numbers M¢¢ . On the other hand it
is just as unconditionally correct that the same cardinal number belongs to
both the sets M and M¢. Both of these are certain, and neither stands in the
way of the other if one heeds the distinction between reality and number.
("Uber
unendliche, lineare Punktmannigfaltigkeiten", 2, Mathematische Annalen 20,
1882, pp 113-121. Quoted in Tait "Cantor's Grundlagen and the
paradoxes of Set Theory")
The potential infinite
means nothing other than an undetermined, variable quantity, always
remaining finite, which has to assume values that either become smaller than
any finite limit no matter how small, or greater than any finite limit no
matter how great. [Cantor, “Mitteilungen” (1887-8), GA 409.]
As an example of
the latter, where one has an undetermined, variable finite quantity increasing
beyond all limits, we can think of the so-called time counted from a
determinate beginning moment, whereas an example of a variable finite quantity
which decreases beneath every finite limit of smallness would be, for example,
the correct presentation of [your] so-named differential. [Cantor, ibid.,
GA 401]
There is no
doubt that we cannot do without variable quantities in the sense of the
potential infinite. But from this very fact the necessity of the actual
infinite can be demonstrated. [“Über die verschiedenen Ansichten in Bezug
auf die actualunendlichen Zahlen” (Bihand Till Koniglen Svenska Vetenskaps
Akademiens Handigar 11 (19), 1-10 (1886), p. 9; cf. Hallett, p. 25.
… in order for
there to be a variable quantity in some mathematical study, the domain of its
variability must strictly speaking be known beforehand through a definition.
However, this domain cannot itself be something variable, since otherwise each
fixed support for the study would collapse. Thus this domain is a definite,
actually infinite set of values. Hence each potential infinite, if it is
rigorously applicable mathematically, presupposes an actual infinite.
[ibidem]
Definitions
of set.
Unter einer
"Menge" verstehen wir jede Zusammenfassung M von bestimmten
wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens … zu
einem Ganzen. [By a "set" we understand any gathering-together
M of determined well-distinguished objects m of our intuition or of our
thought, into a whole] (Cantor, Beitrage 1895b GG p. 112)
… jedes Viele,
welches sich als Eines denken laesst, d.h., jeden Inbegriff bestimmter
Elemente, welcher durch ein Gesetz zu einem Ganzen verbunden werden kann.
[ a many which can be thought of as one, i.e., a totality of definite elements
that can be combined into whole by a law] (Cantor 1932 p. 204, quoted Boolos
13.)
Source:
Ewald, W., From Kant to Hilbert, Oxford 1996.
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LOGIC MUSEUM Copyright © E.D.Buckner
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