The Thomist 64 (2000): 101-25

CANTOR'S TRANSFINITE NUMBERS AND

TRADITIONAL OBJECTIONS TO ACTUAL INFINITY

Jean W. Rioux

Benedictine College

Atchison, Kansas

Georg Cantor was one of the most prominent mathemati-cians of the late nineteenth and early twentieth centuries. His development of a theory of transfinite numbers resur-rected philosophical questions about infinity and led to a division of mathematics into schools of thought such as formalism and intuitionism. Cantor's published attempts to justify his mathe-matical theories were directed not only toward the mathema-ticians of his day but also toward philosophers, both ancient and contemporary. His efforts on the latter front were rooted in his desire to deal with objections to the very idea of the actual infin-ite in quantity, and he attached great importance to those objec-tions that came from traditional philosophy. ⁽¹⁾ I intend to review the basic philosophical issues: Cantor's claims to a workable mathematics of real and actually infinite quantities, his response to Aristotelian objections to those claims, and my reflections on whether Cantor finally settled the matter, as he had hoped.

Cantor was well aware of the prevailing mathematical and philosophical climates of his day. ⁽²⁾ As he drew nearer to a completed theory of transfinite numbers, he became increasingly interested in justifying it, not only as a consistent and practical ⁽³⁾ exercise of mathematical thought, but also as one having a basis in the real world, ⁽⁴⁾ as somehow providing as real a view of the natural world as did the relatively unproblematic theory of finite integers. While other mathematicians seemed unconcerned with such metaphysical questions, Cantor devoted much of his time and effort to addressing them, especially later in his life. ⁽⁵⁾ Ironically, Cantor's fascination with these metaphysical aspects of the theory, so foreign to his contemporaries in mathematics, turned out to be somewhat prophetic, since the dubious ontological character of transfinite numbers was to figure prominently in later developments in mathematics. ⁽⁶⁾

I will begin by giving an overview of Cantor's transfinite number theory, focusing in particular upon his claim that transfinite numbers possess an objective, or real, infinity which is actual, not merely potential. ⁽⁷⁾

I. Summary of Cantor's Transfinite Number Theory

Cantor discovered that merely 'potential' infinity, the sort with which mathematicians were comfortable to that day, was not the only infinite that warranted their scrutiny. He saw the potential infinite, which he called the 'ideal' infinite, primarily in the instance of a variable that is allowed to increase or decrease without limits. Such a quantity, said Cantor, is at any point still finite. ⁽⁸⁾ He contrasted this sort of infinite to the actual infinite, which he first compared to the instance of an infinitely distant yet definite point. He then applied this first analogate of actual infinity to the transfinite numbers, initially the infinite ordinals. Unlike the variables mentioned above, and like the infinitely distant point, he claimed, these new numbers are fully determinate, yet infinite. Nor do they behave as finite series or potential infinities do: in fact, they appear to have very different relationships to each other, and are partially subject to different mathematical laws in these relationships. ⁽⁹⁾

Cantor's discovery amounted to this: though one might think that there could only be a single actual infinity (if any), namely, the 'many-without-limits' (since all infinites would seem to be equivalent in number), nevertheless it is possible to distinguish various infinities from one another. Crucial to his theory was that there are many arithmetically distinct infinities. Equally important for Cantor the mathematician was the actual construction-proof of these diverse infinites. ⁽¹⁰⁾

Cantor used three 'principles of generation' to create the transfinite numbers. ⁽¹¹⁾ The first principle is simply the adding of a unity to a given integer, by means of which any finite integer can be created. When he considered the unending series of finite integers (I), however, Cantor realized that the number of members in this class is infinite, and there is therefore no greatest number in the series. These observations led him to the 'second principle' of generation. As he says:

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 will be the expression for the idea that the entire assemblage (I) is given in its natural, orderly succession. (Just as v is an expression for the idea that a certain finite number of unities is united to form a whole.) It is accordingly permissible to think of the newly created number w as the limit to which numbers v approach, if by it nothing else be understood than that w is the first integer which succeeds all numbers v, that is, it is to be regarded as greater than every one of the numbers v. ⁽¹²⁾

If one conceives of the series of finite integers--not of any single integer (for each is finite) but of the set as a whole--the limiting number of this series must be greater than any one of them. But since any integer in such a set is finite, that which is greater than any such number must be infinite. And in this way is derived the first infinite ordinal number, w. It is both "the first integer which succeeds all numbers v" and "greater than every one of the numbers v." ⁽¹³⁾ The process of producing the number w is radically different from the process of 'adding a unity' first mentioned. Cantor defined the new method in this way:

If any definite succession of definite real integers is given for which no greatest exists, a new number is created on the basis of this second principle of generation, which is to be thought of as the limit of those numbers, that is, to be defined as the next greater number to all of them. ⁽¹⁴⁾

Given a definite and unending sequence of integers, then, the generation of a limiting integer is warranted, which integer must itself be not finite, but infinite (and therefore clearly not a member of that series).

At this point, however, Cantor had shown merely that the limiting number for the set of all finite integers is an infinite number he called w, something that could be easily confused with the potentially infinite, symbolized by . To show that w really differs from , Cantor had to prove that it is a definite number, standing in relation to other definite numbers of the same sort, and governed by certain arithmetical laws. ⁽¹⁵⁾ In short, Cantor had first to show that there are other numbers like w (that is, infinite numbers) that are clearly distinguishable from it.

Having produced the first infinite number w, Cantor then applied the first principle of generation, creating w + 1, w + 2, and so on: a series of infinite integers of which there is no greatest. Further, since the conditions of the second principle were met yet again, ⁽¹⁶⁾ a new number was created, 2w, at which time the first principle of generation was applied once more, yielding: 2w + 1, 2w + 2, and so on. ⁽¹⁷⁾ Cantor, then, had not only provided a means for generating infinite numbers, he had also created numbers that themselves fell out into distinct groups, or classes. Cantor called these the 'first' number class, the 'second' number class, and so on, the first number class being (I), the unending series of finite integers, the second being the unending series w, w + 1, w + 2, and so on, the third being 2w, 2w + 1, 2w + 2, and so on. He generated numbers within a number class by means of the first principle, and advanced to the next number class by applying the second principle of generation. The possibilities, Cantor admitted, are without limits. ⁽¹⁸⁾ And it appeared that not only were the new infinite numbers definite integers (and so essentially unlike ), but there were also infinitely many of them, all well arranged in their own proper classes.

Finally, Cantor had to demonstrate what he had insisted upon early on: that these various and distinct infinities are governed by mathematical laws that differ, at least in part, from the laws governing finite integers. This applies even to the fundamental and intuitively certain laws of association and commutation. So, as he noted, for finite numbers, the commutative law for addition (a + b = b + a) holds without exception, whereas it does not hold for infinite quantities. For instance, 1 + w does not equal w + 1: for 1 + w = w, while w + 1 is the second of the infinite ordinals, (as we have seen). The associative law for addition [a + (b + c) = (a + b) + c], however, holds for both finite and infinite numbers. Similarly, the commutative law for multiplication (ab = ba) holds without exception for finite numbers, whereas it does not hold for infinite numbers. The associative law of multiplication [a(bc) = (ab)c] holds for both types of number. ⁽¹⁹⁾

With these series of real integers, finite or not, Cantor associated a number of a different sort, and a theory of finite and infinite powers of the classes so created was developed. ⁽²⁰⁾ The notion of infinite powers, unlike the w numbers, corresponds to the size or quantity of such sets, independent of the ordering of the elements. Two sets are said to have the same power, so defined, if their elements can be placed in a one-to-one correspondence with each other, independently of their ordering.

Cantor then went on to note that the power of a finite set is the same as its ordinal number, despite the ordering of its ele-ments. However (and in this Cantor identified what he believed to be the essential difference between finite and infinite sets), among infinite sets, the ordinal number of the set changes depen-ding upon the ordering, even though its power remains the same. ⁽²¹⁾ For example, suppose the set of finite real integers (I). ⁽²²⁾ The ordinal number associated with this set will vary depending upon the principle of order used. To the set of finite real integers: 1, 2, 3, . . . v, . . . corresponds w, the first infinite ordinal. How-ever, to the set of finite real integers: 1, 3, 4, . . . v + 1, . . . 2 corresponds w + 1, the second infinite ordinal. Similarly, to the well-ordered set of finite real integers: 1, 3, 5, . . . v, . . . 2, 4 corresponds w + 2, and so on. Depending upon the order of its elements, the same infinite series can be measured by several different infinite ordinal numbers. Yet any ordering of the elements of this series will result in the same power. An infinite power, in Cantor's sense, corresponds to our own notions of an infinite cardinal number, of which ₀, the power of the set of all finite integers, is demonstrably the least. In addition to the infinite ordinal numbers, then, are the infinite cardinals.

Cantor next pointed out that a set having the power of one class (say, the class of finite integers) would have ordinal numbers belonging to the next higher class (in this case, w, w + 1, w + 2, and so on). Not only are there many distinct infinite numbers, and distinct classes of infinite numbers, as well as arithmetical laws that partially differ from those of finite numbers, there are also different types of infinite numbers altogether (ordinals and cardinals), themselves having definite relationships to one another. ⁽²³⁾ Cantor had provided the mathematical community with more infinity than they could possibly have expected. ⁽²⁴⁾

Mathematicians had varied reactions to Cantor's claims, from Bertrand Russell and David Hilbert, who defended them, to Ernst Kronecker ⁽²⁵⁾ and Henri Poincaré, who strongly opposed them. ⁽²⁶⁾ As we noted above, Cantor was very well aware of the objections his theory of transfinite numbers would inevitably raise among mathematicians and philosophers alike. Despite the sympathetic hearing of mathematicians of note, Cantor seemed to think that the philosophical objections simply had to be addressed. Though he maintained that this justification of transfinite numbers is not a specifically mathematical obligation, he felt compelled to provide it all the same. ⁽²⁷⁾ The philosophical questions, Cantor saw, were of the greatest importance even for mathematics. In truth, as the implications of Cantor's work became clearer, Gottlob Frege noted that the issue would result in a damaging conflict within mathematics itself:

Here is the reef on which it [mathematics] will founder. For the infinite will eventually refuse to be excluded from arithmetic, and yet it is irreconcilable with that [finitist] epistemological direction. Here, it seems, is the battlefield where a great decision will be made. ⁽²⁸⁾

As the objections began to surface, other mathematicians joined the fray. Frege's analogy of a battle was becoming all too true. Most famous, perhaps, is David Hilbert's war-cry: "No one shall drive us out of the paradise which Cantor has created for us." ⁽²⁹⁾ It is to his credit that Cantor anticipated the revolutionary character of his discoveries. It remains to be seen whether he would successfully address the very root of many of the objections to his theory, Aristotle himself.

II. Cantor and Aristotle

In section four of the Grundlagen Cantor deals with certain difficulties associated with his theory of transfinite numbers. One is whether there is such a thing as the infinitely small; another is whether there really are numbers apart from the finite integers. It is the latter question with which I am directly concerned, since here are found the objections made by traditional philosophy to transfinite numbers: for traditional philosophy objects to such things on the grounds that they are not numbers, that 'number' properly so called describes the finite integers and nothing more. Cantor, well aware of the origins of his opponents' views, places his consideration of Aristotle here. ⁽³⁰⁾

Cantor first notes that some of those who deny transfinite numbers would admit the existence of the rationals, since they come directly from the integers and are expressed in terms of the integers. ⁽³¹⁾ He goes on to say, though, that traditional mathematicians are somewhat squeamish when it comes to the irrational numbers:

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. ⁽³²⁾

Though he does see some benefits to mathematics in such a view, Cantor finally discounts it as erroneous and overly restrictive. ⁽³³⁾ Not all numbers, then, can be reduced to the finite integers.

Having raised the issue in this way, Cantor next takes up Aristotle himself. He cites book 11 of the Metaphysics as his reference for Aristotle's arguments against the infinite. ⁽³⁴⁾ He specifically addresses two of Aristotle's arguments in the Grund-lagen, and later deals with what he sees as the source of Aris-totle's error. ⁽³⁵⁾ The general problem, he claims, is that Aristotle begs the question--that he assumes that all numbers must be countable by means of finite numbers, and thereby proves that infinite numbers are not numbers. As Cantor says:

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. ⁽³⁶⁾

Apparently, Cantor is referring to Aristotle's argument that an actually infinite number is impossible, since every number, or whatever has a number, can be numbered. ⁽³⁷⁾ Of course, the actually infinite cannot be numbered in this way: that is, one cannot enumerate the members of an infinite number one at a time. ⁽³⁸⁾ This, says Cantor, is at the heart of finitists' arguments against the infinite: for they expect the infinite to have the same properties as finite numbers do. ⁽³⁹⁾ To their credit, neither Aristotle nor Thomas Aquinas takes the argument in question as having much weight. ⁽⁴⁰⁾ In his comments on identical passages from the Physics, Aquinas says the argument is merely probable, since it does not have a necessary conclusion. Moreover, he anticipates what someone defending infinite numbers might say in response:

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. ⁽⁴¹⁾

Cantor insists upon keeping certain elements of the traditional notion of number when speaking of his transfinite numbers, yet he clearly wishes to dissociate from them the fundamental notion of being 'able to be gone through', or of being enumerated in the traditional sense. Apparently Cantor sees Aristotle as allowing, among the principles of mathematics, Cantor's first principle of generation alone. ⁽⁴²⁾ It would be no surprise, then, if one who adopts such a position would see infinity in quantity as essentially indeterminate and ultimately infinite in a merely potential way, as something similar to Cantor's 'ideal' infinity, mentioned above. If the transfinite numbers are in fact real, Aristotle's argument would certainly be subject to the charge of question-begging. For one might ask how he knows that there are no infinite numbers, to which his response, as Cantor reports it, would simply be "because all numbers are finite," which, of course, is the point in question.

Still, even as Cantor presents it there is something more to Aristotle's argument, namely, the claim that numbers are numerable. It is not so much Aristotle's claim that all numbers are finite that Cantor attacks here, but the notion that all enumerations are finite ones, or are made in terms of finite numbers, or, most precisely, that one can make determinate (or measure) the quantity of some number using finite numbers alone. The charge of petitio principii stands only on the supposition that one could show that not all numbers are finite ones. Cantor is well aware of this. Immediately after presenting what he believed to be the basic flaw in the Aristotelian arguments, Cantor says:

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. ⁽⁴³⁾

Cantor's point is that, despite Aristotle, infinite sets can be measured, and their elements can be enumerated. They are not measured, however, as finite sets are, that is, by means of the unit and in terms of some finite number of such units; rather, they are measured by means of a limiting number which succeeds all in the set, and which is greater than any finite quantity of units. ⁽⁴⁴⁾ Such a number is generated (or rather, discovered) by means of a principle altogether different from that which generates enumerations among finite numbers. This infinite enumeration depends upon the principle whereby the members of the set being numbered are ordered; enumerations of finite sets, though they also require an ordering, are the same despite the particular ordering chosen. ⁽⁴⁵⁾ Members of the higher number classes differ from themselves in determinate ways acceptable to mathe-maticians (for example, w + 1 is one greater than w), and laws of arithmetic can be formulated to deal with these numbers. Thus, Cantor would say, not all numbers are measured by counting up units, but only finite ones. Infinite numbers are measured by correlating a series of arithmetically different infinite integers with certain orderings of their members, such that a different ordering yields a different enumeration. These enumerations, being determinate, can therefore be a legitimate subject of our understanding:

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. ⁽⁴⁶⁾

Cantor next turns to the second Aristotelian argument, which he summarizes in this way: "the finite would be dissolved and destroyed by the infinite if it (the infinite) existed, since the finite number is allegedly destroyed by the infinite." ⁽⁴⁷⁾ As presented, the argument has some cogency to it: how could one increase infinity? How could one go beyond that which is beyond all? How could one infinite be greater than another? For, apparently, to be greater implies that the other is less and so has a definite limit, or term: and if something is less than infinite, then how could it be said to be infinite at all? The logical consequence, of which Cantor, once again, is well aware, is that if there were such a thing as the infinite, there could be only one. Nor would such an infinite be at all determinate or, especially, numerable, since to number it (or to measure it) would be to establish some point at which it ended, and beyond which it did not go.

Cantor's response to this argument is straightforward, and comes down to asking whether an infinite number, though it measure an unlimited number of members in a series, ⁽⁴⁸⁾ need itself be unlimited in Aristotle's sense of the word. ⁽⁴⁹⁾ As we saw earlier, though any number within the series of finite integers is a limit, in some sense, with respect to those that precede it, w is not such a limit: it exceeds all numbers within the series, but is not itself a member of that series. So a limit need not be finite. Nor is it the case that an infinite limit be itself without limits, for though it is the limit for integers in the first number class (and so is without a finite limit), w is itself surpassed by countless integers in its own number class, whose limit is the first number within the third number class (2w), and so on. There are many numbers, then, that are 'beyond' w, though no finite number can surpass it. It is a limit (and so a definite quantity), and is itself both limited and unlimited, but in different respects. ⁽⁵⁰⁾

To return to the argument proper, then, while one could not add a finite quantity to Aristotle's actually infinite number (that is, to a quantity that surpasses all quantities and is strictly unlimited,) and in this sense the finite would be destroyed by the infinite, still, Cantor argues,

to an infinite number (if it is thought of as determinate and complete) a finite number can indeed be adjoined and united without effecting the dissolution of the latter (the finite number)--the infinite number is itself modified by such an adjunction of a finite number. ⁽⁵¹⁾

What sense does Cantor make of adding a finite number to an infinite number? Recall his qualification of the commutative law for addition when it came to infinite numbers. According to Cantor, 1 + w = w, while w + 1 > w; in fact, w + 1 is the second of the infinite ordinal numbers. Returning to the example of the series of finite integers mentioned above, let us suppose the well-ordered series of finite integers: 1, 2, 3, . . . v, . . . which has, as its limiting number, the first infinite ordinal w. Let us now suppose that we displace one member of the series, obtaining: 2, 3, 4, . . . v + 1, . . . 1. Since we are dealing with a series in which there is no greatest, and since the numbers in the series are reciprocally well-ordered (up to the last element of the second series, that is, to 1 in the first corresponds 2 in the second, to 2 in the first corresponds 3 in the second, to v in the first corresponds, and in the same position, v + 1 in the second, and so on), there are as many members up to 1 in the second series as there are in the first series, taking their ordering as the operative principle. Therefore the ordinal number of the second series is one greater than that of the first, or w + 1. ⁽⁵²⁾ Nevertheless, it is crucial that the finite number be added to the infinite number, and not conversely. For, in comparing the well-ordered series of finite integers: 1, 2, 3, . . . v, . . . to the series 2, 1, 3, . . . v, . . . one can see that there is, in fact, a one-to-one correspondence throughout. In other words, v + w = w (where v is any finite integer). ⁽⁵³⁾

In this way, Cantor meets Aristotle's objection:

If w is the first number of the second number class, then 1 + w = w, but w + 1 = (w + 1), where (w + 1) is a number entirely distinct from w. Everything depends, as is here clearly seen, upon the position of the finite relative to the infinite; in the first case, the finite is absorbed into the infinite and vanishes, but if it modestly takes its place after the infinite it remains intact and unites with the infinite to form a new (since modified) infinite. ⁽⁵⁴⁾

Finally, Cantor raises a point that gets at the very heart of Aristotle's difficulties with infinite numbers in general, which difficulty Cantor called the prw'ton yeu'do", the initial falsehood, upon which all finitistic reasoning is based:

All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on general grounds as well. It is their prw'ton yeu'do" 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. ⁽⁵⁵⁾

Transfinite numbers are not extensions of the finite integers in the sense that an infinity has been added to a given finite integer to produce them. A transfinite number is a different kind of number. (This point is at the heart of Cantor's charge of question-begging, above.) One cannot expect the properties of one species within a genus to be applicable to another within that same genus. Aristotle's basic error was in taking one species of number, namely the finite integers, to be the genus itself, number. One might just as mistakenly take the species of rectilinear figures to be the genus itself, thereby excluding the whole class of curved figures from consideration. But what holds for one among the species within a genus does not necessarily hold for the others:

It is here [by finitists] tacitly assumed that properties which for numbers as we have previously understood them are disjunct, are equally so for the new numbers, and one accordingly concluded the impossibility of infinite numbers. Who fails to see this fallacy at a glance? Isn't every generalization or extension of concepts associated with the abandonment of certain special properties, even unthinkable without it? ⁽⁵⁶⁾

Cantor likens his introduction of the theory of transfinite numbers to previous extensions of the concept of 'number', such as the rationals, the irrationals, and the complex numbers. ⁽⁵⁷⁾ Such extensions, he notes, are regarded as mathematically legitimate:

It (mathematics) is obligated when new numbers are introduced to give definitions of them by which such a determinacy and, under certain conditions such a relationship to older numbers is granted them, that they can in any case be definitely distinguished from each other. As soon as a number satisfies all these conditions [consistency, and standing in determinate, orderly relationships to other numbers] it must be regarded as mathematically existent and real. It is in this that I see the reason given in paragraph 4 why the rational, irrational and the complex numbers are to be considered as much existent as the finite positive integers. ⁽⁵⁸⁾

Cantor's strategy is to defend his transfinite numbers as legitimate extensions of the concept of real integer by establishing their consistency and definite relationships to the finite integers. To that extent, they would then be regarded as just as mathematically legitimate as previous extensions of the same sort and included within a distinct species of number.

The charge of taking the species to be the genus is a serious one, since the mistake would have occurred in the very principles of mathematics, affecting the remainder of that study. Yet how might this have occurred? Bertrand Russell offers a distinct account of mathematical reasoning in general. He recognizes, of course, the basic distinction between arriving at the general prin-ciples of a science and making deductions from such principles. Thus one might arrive at Euclid's axioms and postulates by generalizing from practices in land-surveying, and then turn around and deduce other propositions from the principles so discovered. With respect to the question of where mathematical reasoning begins, Russell gives a surprising answer: "The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle." ⁽⁵⁹⁾ One would not naturally begin mathematical reasoning with the axioms and then make deductions from them; rather, one might begin with some intermediate proposition, deduce something from it, or, conversely, ask in what principle that proposition itself is grounded.

To apply this to what Cantor calls the prw'ton yeu'do", one might imagine Aristotle abstracting the notion of finite integer (which would be somewhere in the middle, logically, between the genus, number, and the species of finite integer) and seeing it as the starting point of mathematical reasoning, much as one might mistakenly take a theorem to be an axiom. If finite number is number, then infinite numbers cannot be numbers at all. If Russell is correct, however, to regard what one first arrives at in mathematical reasoning as logically first also is to neglect another form of mathematical thought altogether. The process of extending the concept of 'number' (which originally was taken to mean 'finite number') so as to include other species (such as the rationals and the mixed numbers) can be seen as an attempt, through history, to engage in Russell's second type of reasoning. As he says: "We shall find that by analysing our ordinary mathematical notions we acquire fresh insight, new powers, and the means of reaching whole new mathematical subjects by adopting fresh lines of advance after our backward journey." ⁽⁶⁰⁾

The error in finitist arguments against infinite quantities can therefore be seen as a mistaken insistence upon one's first (or 'ordinary') mathematical concepts as being logically first as well. To define number as a "multitude measured by the unit," as Aristotle does, is to associate properties belonging to a single species of number with the genus, or to take finite numbers as being the genus of number itself. The fundamental error in Aristotle's arguments against the actual infinite is a logical one. ⁽⁶¹⁾

III. Reflections and Observations

It first appears that Cantor and Aristotle are speaking of very different things: the actual infinite Cantor affirms is not the actual infinite Aristotle denies. For one thing, Aristotle regards the actual infinite as being unbounded in any respect: it is entirely without limits. ⁽⁶²⁾ This is why arguments such as that given against a finite number of elements in an infinitely large universe have the cogency they do. Cantor's transfinite numbers, on the other hand, are not entirely unlimited: for since infinite numbers are generated in classes, such that one advances from the second number class ⁽⁶³⁾ to the third by applying the second principle of generation, numbers within such classes are arranged as a series. So, the numbers of the first number class are: w, w + 1, w + 2, and so on. Clearly, any of these numbers has a limit: for w + 1 > w, and so w + 1 is a sort of limiting quantity for w. Further, the first number of the third number class, 2w, is set down as the limiting number for all numbers in the second number class. Therefore, there is always a number greater than any transfinite number, or no transfinite number has an unlimited quantity. And this seems to accord with what Cantor himself says:

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. ⁽⁶⁴⁾

Aristotle's arguments against an actually infinite number were not directed against transfinite numbers as Cantor describes them. Only a number that exceeded all limits would 'absorb' finite numbers added to it; transfinite numbers do not, precisely because they are limited in that respect.

Yet one might take Aristotle's arguments as also applying to any but finite numbers. For Aristotle saw number as a type of quantity (more precisely, as a type of discrete quantity,) and quantity is predicated in answer to the question how much or how many. For Aristotle, one answers such a question in the case of discrete quantities by 'counting', or enumeration. To determine attributes of things in other ways is to ask (and answer) other sorts of question about them, such as where, when, of what sort, and so on. ⁽⁶⁵⁾ Although, as Bertrand Russell points out, definitions of things by extension (that is, through enumeration) differ from those by intension (through specifying some proper character-istic), in the case of infinite numbers enumeration is not possible, and we are left with the possibility of intensional definitions alone. ⁽⁶⁶⁾ The difficulty is that intensional definitions seem to belong more properly to questions and manners of answering questions that differ from the how much or the how many. In truth, it is clear that an intensional definition may be given independently of any considerations concerning the how much or the how many of some thing. The first infinite ordinal number, w, may say something about the unending series of finite integers, but it is not clearly an answer to the question of how many such integers there are. ⁽⁶⁷⁾ It may be something more akin to quality or relation. If defining by extension, that is, counting, is how one answers the question how many, then properly speaking only finite numbers could be found in the category of quantity. To use the word 'number' to describe something in a category other than quantity would be to equivocate. If transfinite numbers are numbers, that is, one among the species within the genus number, as Cantor claimed, the word 'number' would be used of them and of finite integers without such equivocation.

Apart from whether Aristotle and Cantor are speaking of the same thing, however, is another issue: whether Cantor has in fact established that there are such things as transfinite numbers. Note that, for Cantor, existence is of two sorts. ⁽⁶⁸⁾ The sort with which mathematicians per se are concerned he calls 'intra-subjective' or 'immanent' existence. For mathematicians are concerned not with 'transsubjective' reality, what is actually found "in corporeal and intellectual nature," ⁽⁶⁹⁾ but with consis-tency and determinate relations among mathematical concepts in the mind: "Mathematics, in the construction of its ideas, has only and solely to take account of the immanent reality of its concepts and has no obligation whatever to make tests for their transient reality." ⁽⁷⁰⁾ It is on account of this distinction that mathematics deserves the name 'free mathematics'. ⁽⁷¹⁾ Nevertheless, Cantor claims throughout the Grundlagen that his transfinite numbers are real in the second sense also: that the concepts in the mind are tokens of separate natural or intellectual realities. In truth, Aristotle's arguments against an actual infinite would pose no threat to Cantor's theory unless he were making such a claim also: for the arguments are clearly directed not against the logical consistency of such concepts but against their actual existence in the world.

Cantor does hold that the transfinite numbers, in virtue of being well-defined concepts, ⁽⁷²⁾ have existence in the mind, but he is also convinced they have existence outside the mind:

Reality can be ascribed to numbers in so far as they must be taken as an expression or image of the events and relationships of that outer world which is exterior to the intellect, as, for instance, the various number-classes (I) (II) (III) etc. are representative of powers which are actually found in corporeal and intellectual nature. ⁽⁷³⁾

And, in the same place:

In lieu of the thoroughly realistic but at the same time none the less idealistic basis of my considerations, there is no doubt in my mind that these two spheres of reality [intrasubjective and transsubjective] are always found together, in the sense that a concept said to exist in the first way always also possesses in certain and even in an infinity of ways a transient reality.

His reason for this claim, which he immediately provides, is rooted in the inseparable unity of all things. "The connection of both realities has its peculiar foundation in the unity of the All, to which we ourselves belong." He does not claim that the extramental existence of numbers is a thing easy to grasp; ⁽⁷⁴⁾ nevertheless, he is confident that they are there.

As regards establishing the legitimacy of transfinite numbers as concepts, Cantor made some clear advances. He argued that the transfinite numbers are clearly distinct from the finite numbers and from the potentially infinite. He also argued that they have a definite character, and stand in definite relationships both to each other and to other numbers, including finite numbers. He even outlined a rudimentary arithmetic which applies to transfinite numbers alone. Nevertheless, at the conceptual level, transfinite numbers have not been entirely free of difficulty. ⁽⁷⁵⁾ But even if one were to grant the free use of transfinites at the conceptual level, does it follow that what one conceives of in this way thereby exists?

Among the arguments given in favor of the infinite, Aristotle notes one for which he claims a special status:

Most important of all [among the reasons for a belief in the infinite] is one which raises a difficulty for everyone: for it seems that number and mathematical magnitudes and what is outside the heavens are infinite because they do not cease in our thought. ⁽⁷⁶⁾

He addresses this argument at the end of his account of the infinite, noting that thinking and what thinking is about may not correspond:

To trust to thinking is absurd, for the excess or the deficiency is not in the thing but in the thought. For one of us might think that someone is bigger than he is, increasing him ad infinitum: but it is not because something thinks this that he is bigger than we are, rather, it is because he is [bigger], and the thought is accidental. ⁽⁷⁷⁾

For Aristotle, number is real insofar as there is a multiplicity of things that are numbered. This is why Aristotle calls the infinity of number a 'potential' one, since it is consequent upon the division of continuous quantity, and such a division results in numerically distinct units. By one act of division I produce two things, by two acts three, and so on. Since the continuous is divisible ad infinitum, but never all at once, the number that is consequent upon such a kind of division is also infinite in the same way. ⁽⁷⁸⁾ It is also clear, then, that, for Aristotle, there could not be such an infinite among numbers unless there were an infinite magnitude as well. ⁽⁷⁹⁾ The same is also clear when one considers Aristotle's notion of the objects of mathematics. For he says in book 2 of the Physics ⁽⁸⁰⁾ that the mathematician considers physical things not insofar as they are physical (that is the work of the physicist) but insofar as they are mathematical. To study infinite numbers would be to study infinite substances having such a number, not as physical substances, but precisely insofar as they are so many. ⁽⁸¹⁾