§ 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].




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





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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