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An Address delivered at the Forum of the Society of Friends of Scripta Mathematica
by William Pepperell Montague, Professor of Philosophy, Barnard College, Columbia University, 1940
This paper has three parts.
Part I contains a description and proof of a curious equality relation pertaining to the simple harmonic series
1 + ½ + ⅓ + ¼ + + + ...
and the series derived from it by merely alternating the signs of the terms thus,
1 − ½ + ⅓ − ¼ + − + − ....
Part II contains a description (deduced from Part I) of a new type of series that I call the “Rainbow” which has a definite summation value but which lacks both a first term and a last term.
Part III contains the proof (deduced from Part II) of a new Cantorian (or “sub-Cantorian”) transfinite which is the logarithm of Aleph Null and the first of an endless series of logarithmic transfinites intervening between Aleph Null and the finite numbers.
With respect to the following pair of series, Series I,
1 + ½ + ⅓ + ¼ + ... + 1/n + 1/(n+1) + 1/(n+2) + ... + 1/2n
and Series II,
1 − ½ + ⅓ − ¼ + ... − 1/2n
we shall prove that the sum of the terms in the second half of Series I (namely, those from 1/(n+1) to 1/2n inclusive) is always equal to the sum of both halves of the terms in Series II (namely, those from 1 to −1/2n inclusive).
When we take 2n as equal in turn to 2, 4, and 6, we can see by inspection that the equality relation pertaining to the two series holds true, thus
When 2n = 2,
Series I is 1 + ½
Series II is 1 − ½ and
½ = 1 − ½
When 2n = 4,
Series I is 1 + ½ + ⅓ + ¼
Series II is 1 − ½ + ⅓ − ¼ and
⅓ + ¼ = 1 − ½ + ⅓ − 1/4
When 2n = 6,
Series I is 1 + ½ + ⅓ + ¼ + ⅕ + ⅙
Series II is 1 − ½ + ⅓ − ¼ + ⅕ − 1/6 and
¼ + ⅕ + ⅙ = 1 − ½ + ⅓ − ¼ + ⅕ − ⅙
Now that we have found that the equality relation pertaining to our two series does actually hold true when 2n, the number of terms in each series, is equal to either 2, 4, or 6, we shall show that when the equality relation holds for any given even number of terms, it will also hold for the next greater even number of terms. In other words, if it can ever be *assumed* to hold when the number of terms is 2n, it can be *proved* to hold when the number of terms is increased from 2n to 2n+2.
When 2n is the number of terms in each of the two series, they can be written thus
Series I
1 + ½ + ⅓ + ¼ + ... + 1/n + 1/(n+1) + 1/(n+2) + + + ... + 1/2n
Series II
1 − ½ + ⅓ − ¼ + ......... − 1/2n
And our assumption of an equality between the sum of the second half of the terms in Series I and the sum of all the terms in Series II can be written thus (1):
1/(n+1) + 1/(n+2) + ... + 1/2n = 1 − ½ + ⅓ − ¼ + ... − 1/2n
When the number of terms in each series is increased from 2n to 2n+2, they can be written thus
Series I
1 + ½ + ⅓ + ¼ + + + ... + 1/n + 1/(n+1) + 1/(n+2) + ... + 1/2n + 1/(2n+1) + 1/(2n+2)
Series II
1 − ½ + ⅓ − ¼ + .... + 1/(2n+1) − 1/(2n+2)
And the equality between the sum of the terms in the second half of Series I and the sum of all the terms in Series II can be written thus (2)
1/(n+2) + 1/(n+3) + + + .... + 1/2n + 1/(2n+1) + 1/(2n+2) = 1 − ½ + ⅓ − ¼ + ... − 1/2n + 1/(2n+1) − 1/(2n+2)
Comparing equation (2) with equation (1) we see in the first place that the left hand member of equation (2) has been derived from the left hand member of equation (1) by adding
1/(2n+1) + 1/(2n+2)
and subtracting 1/(n+1); and we see in the second place that the right hand member of equation (2) has been derived from the right hand member of equation (1) by adding 1/(2n+1) and subtracting 1/(2n+2). But the changes made in each member are equal because (3)
1/(2n+1) + 1/(2n+2) − 1/(n+1) = 1/(2n+1) − 1/(2n+2)
And inasmuch as when equals are added to equals the sums are equal, it follows that if equation (1) is true, equation (2) must also be true. In other words if the equality relation pertaining to Series I and Series II holds true when the number of terms in each series is any even number (2n) it must also hold true for each and all the succeeding even numbers of terms 2n+2, 2n+2+2, 2n+2+2+2, etc. And since we have already found by empirical inspection that the equality relation pertaining to the two series (when the number of their terms is the samee) holds true when that number (2n) is substituted in turn for 2, for 4, and for 6, it follows that the sum of the terms in the second half of Series I is always and for any even number whatever equal to the sum of all the terms in Series II.
Q.E.D.
The series which we have referred to as Series II, namely,
1 − ½ + ⅓ − ¼ + ⅕ − ⅙ ... − 1/2n
is convergent and as the number of its terms is increased the limiting value of the sum of the terms is the logarithm of 2 to the base e, the so-called “natural logarithm” of 2, symbolized by ln(2). Now the series which we have referred to as Series I, namely,
1 + ½ + ⅓ + ¼ + + + ... 1/2n
is divergent but inasmuch as the sum of the second half of its terms has just been proved to be equal to the sum of all the terms in Series I (this latter being convergent and when n is infinite equal to ln(2)), it follows that though Series I as a whole is divergent, yet the “second half” of it is convergent and (when n is infinite) is itself equal to the natural logarithm of 2. The limiting form of the “equality relation” pertaining to Series I and Series II can then be represented by the following equation (3′)
lim n→∞ [1/(n+1) + 1/(n+2) + + + ... + 1/2n] = lim n→∞ [1 − ½ + ⅓ − ¼ + ... − 1/2n] = ln(2)
This equation (3′) is simply the limiting form of the equation (3) which was proved true in part I. But the series
lim n→∞ [1/(n+1) + 1/(n+2) + + + ... + 1/2n]
is a most peculiar thing, for it is a series which has no proper first term and no proper last term yet which does nevertheless possess a definite finite value because it is equal to the limiting value of Series II which in its turn is equal to the natural logarithm of 2.
The expression 1/(n+1) though we have written it as a “first” term, is really no proper term at all; nor has it any proper or definite position in the whole array of terms in Series I. The same is true of the expression
lim n→∞ [1/2n]
(which we have written as the “last” term), and also, of course, of each and all of the infinity of terms that lies between the pseudo boundaries of the series. But what is most remarkable is that though the series
1/(n+1) + 1/(n+2) + ... + 1/2n
in its limiting form has boundaries that vanish into vagueness and indeterminacy, yet it is just in that limiting form and when the boundaries become indefinite that the sum of the terms becomes definite and equal to the natural logarithm of 2.
Now the horizon on the left in which a rainbow may be said to begin and the horizon on the right in which it may be said to end are no proper or definite places, yet the rainbow itself and as a whole possesses a perfectly definite length. Thought extending from “nowhere” to “nowhere”, its linear value has a definiteness that is untainted by the indefiniteness of its terminals.
Thus it is with the limiting form of my series
1/(n+1) + 1/(n+2) + + + ... 1/2n
and for this reason and in the hope of conveying by a metaphor some hint of its strange beauty, I have named it the Rainbow Series.
It is a property and from the modern standpoint it is a definite property of any aggregate called infinite that it possesses parts with which it can be put in one-to-one correspondence. This does not mean that an infinite can be put in one-to-one correspondence with every one of its parts, nor does it mean that infinite assemblages can always be put in one-to-one correspondence with each other. In fact those infinities which differ so greatly in magnitude that they cannot be put in one-to-one correspondence with one another, form the series of infinites discovered by Cantor and named by him the Transfinites.
The first or lowest of these transfinites is Aleph Null, exemplified by the number of all rational numbers or more simply by the number of all integers. The second transfinite called Gamma is exemplified by the aggregate of all the combinations of all the integers contained in Aleph or by the number of all real numbers both rational and irrational or by the number of points in a line or in any other continuum of higher order so long as the number of its dimensions remains finite. [sic] The transfinite Gamma can be attained by raising any finite number or aleph itself to the aleph power. And analogously a third transfinite can be attained by raising any finite number, or Aleph, or Gamma, to the Gamma power. And in general an endless series of successively higher transfinites can be generated by taking each in turn as an exponent, thus making it the logarithm of the next higher transfinite.
Symbolizing Aleph Null by T_0, Gamma by T_1, and the others correspondingly, we get the sequence
T_0, T_1, T_2, T_3 ...
in which each member will be the logarithm of its successor. And apart from any other exemplifications which may be found for it, any one of these higher transfinites, T_N, will exemplify the totality of combinations, 2^T(n−1), of the members of its predecessor T_(n−1). Aleph Null as the first member of this sequence, has no predecessor and hence, of course, can have no logarithm.
I wish now to prove that contrary to the accepted belief Aleph Null does have a logarithm and that this logarithm is not only a true transfinite predecessor of Aleph Null, but the first of an endless descending series of logarithmic transfinites which intervene between aleph and the domain of finite numbers.
Let us write the simple harmonic series which we have referred to above as Series I as a series of “segments”, the number of terms in the successive segments increasing with the successive powers of two; thus
1 + (½) + (⅓ + ¼) + (⅕ + ⅙ + ⅐ + ⅛) + ... + (1/(n+1) + 1/(n+2) + ... + 1/2n)
Now let us note that the number of terms in this Series I is the number of the totality of integers, inasmuch as the successive terms of the series have the successive integers as their denominators, their numerators being of course always unity.
Let us note, second, that the number of terms in our series is not only Aleph Null but when considered as arranged in m successive segments it is equal to
1 + 2^m − 1
For the number of terms in any succession of m segments beginning with the first segment containing only the single term (½) can be represented as
2^0 + 2^1 + 2^2 + 2^3 + ... + 2^(m−1) = 2^m−1
And when the number of terms in Series I is infinite and equal to the totality of integers or to Aleph Null, the number of segments which we denoted by “m” must also be infinite but an infinite logarithmically or transfinitely lower than the other [sic], and we can write
lim n→∞ [1 + 2^m − 1 = 2^m] = Aleph Null
And “m” becomes the logarithm of Aleph Null, whose “existence” we are seeking to prove.
Let us pause here for a moment to consider the meaning of the term “existence” in a mathematical context such as the one before us. Obviously the existence of the logarithm of Aleph is not to be proved merely by our ability to write the symbols “2^m = Aleph”. If m is to be proved to *exist* it must be shown to be something more than a “symbol ad hoc” having only a prima facie meaningfulness. The problem of establishing a mathematical existence is indeed significantly analogous to the problem of legitimizing a scientific hypothesis. Each must be demonstrated to be a vera causa, i.e., an entity capable of specific exemplification in a context other than the one for which it is invoked.
If at this point we are challenged as to our right to express the simple harmonic series when expanded to infinity as composed of m segments and hence as containing 2^m terms, we can reply that any number will differ from some integral power of 2 by a number less than itself. So even if Aleph were provisionally assumed not to be expressible as an integral power of 2 it would differ from some other transfinite that *was* expressible as an integral power of 2 by a number less than itself. And as Aleph plus or minus any number less than Aleph is nothing but Aleph, the supposed “other” tranfinite would be Aleph itself.
We may say then that Aleph is equal to 2^m where Aleph is the number of terms in the harmonic series. And we may say further that m, the logarithm of that number, can be exemplified in the number of segments of which the series can be regarded as compounded.
In a sense then the number of segments of our series does already provide the existentiality of the logarithm of Aleph for which we have been seeking.
If it should be said that these segments are artificial and abstract and hence do not furnish the degree of concretely specific exemplification demanded by “existence”, I might reply that they are no more abstract and artificial than the combinations of the elements of an aleph assemblage which in their totality undoubtedly exemplify the transfinite Gamma. Such a reply on my part would, however, savor too much of a tu quoque retort; and even at that it would leave untouched those who would refuse to admit that the existentiality of Gamma was established by using it merely as the number of Aleph’s combinations. Fortunately it is possible to go further in the direction of specific exemplification and to use the “segments” not as in themselves establishing the existentiality of the logarithm of Aleph but as a valid stepping-stone to that goal.
To show how this advance can be made we turn from a consideration of the segments merely in themselves to a consideration of the actual sum of the terms of the harmonic series when viewed in the light of those segments. And writing the series again
1st segment, 2nd segment, 3rd segment, ... mth segment
lim n→∞ [1 + (½) + (⅓ + ¼) + (⅕ + ⅙ + ⅐ + ⅛) + ... (1/(n+1) + 1/(n+2) + + + ... /2n)] [sic]
we observe that the sums of the terms in each one of the segments is a finite quantity that increases from ½ which is the minimum sum in the first segment to
(1/(n+1) + 1/(n+2) + + + ... − 1/2n) [sic]
which is the maximum sum of the terms in the mth segment. But when n is infinite this maximum sum which is also the limiting sum becomes the sum of the terms in the Rainbow Series which as we proved above in Part I and Part II is equal to the natural logarithm of 2. If the sums of the terms in the successive segments were all equal we could get the sum of the terms in the whole series by multiplying the sum in each segment by m, which is the number of segments, and adding one to this product. As it is we can only say that the sum of the terms in the whole series is a quantity greater than
m*(½) + 1
and less than
m*(ln2) + 1
When n, the number of terms in the series, is infinite and equal to Aleph Null, m, the number of segments of the series is also infinite, but it is the infinite of next lower potency [sic] and is therefore, at least in a formal sense, the logarithm of Aleph. And this “formal” logarithm of Aleph multiplied by ½ or by ln2 or by any intermediate finite quantity is not changed in its “potency” or even in its “order” and thus remains simply the logarithm of Aleph. But this logarithm of Aleph, when taken as lying between
m*(½) + 1
and
m*(ln2) + 1
truly and concretely expresses the sum of the simple harmonic series,
lim n→∞ [1 + ½ + ⅓ + ¼ + ... + 1/2n]
and thereby loses its merely formal character and acquires that “specific exemplification” or “existentiality” which we have sought to establish.
It remains now for us to prove that the existence of the logarithm of Aleph entails the existence of a further transfinite of next lower potency which would be the logarithm of the logarithm of Aleph; and that this latter entails a third logarithmic transfinite which in its turn entails by the same process a fourth, and so on without end.
To achieve this demonstration we shall replace Aleph by its own logarithm, m, as the infinite number of terms in an infinite section of the harmonic series. We shall now let m = 2^r, with the same justification that we let Aleph = 2^m, and divide this new and still infinite section of the series into segments, the number of terms in each of these successive segments to be equal, as before, to the successive powers of 2. Since the number of terms in the new series is to be m = 2^r, the number of its segments will be r, which is the logarithm of m. Taking this new logarithm in the same formal sense that we took m itself we multiply it as we multiplied m by the natural logarithm of 2 (or by any other finite quantity between ln2 and ½) and thus get in a new form the sum of the harmonic series. But the number of terms of the series as we are now taking it is no longer Aleph, but rather m or the logarithm of Aleph. And consequently the sum of the series is no longer equal to m or to m*ln2, but rather to r = log m = log log Aleph. This is still infinite but it is an infinite of next lower potency to m, even as m itself was the infinite of next lower potency to Aleph. Thus by using r first in a quasi-formal sense as the number of segments in the harmonic series of m = log Aleph terms, we have been able to go further and give it specific exemplification or existentiality by the same procedure that we had already used with m.
This procedure is obviously perfectly uniform and general and could be repeated with r, thus entailing as a lower transfinite predecessor the logarithm of r: which could in turn be shown to entail a further still lower predecessor and so on without end.
Substituting the symbols T_(−1) for m, the logarithm of Aleph, and T_(−2) for r, the logarithm of m, we can write the series of new or logarithmic transfinites below Aleph (whose existence we claim to have established) in conjunction with the series of old transfinites above Aleph (whose existence was already established) in the following form:
... T_(−3), T_(−2), T_(−1), T_0, T_1, T_2, T_3, ...
Here, T_0 symbolizing Aleph Null, stands in the middle with T_1 symbolizing Gamma, and T_2, T_3, etc., symbolizing the higher successors of Gamma. The T’s with the succession of negative subscripts stretch persistently and hopelessly down toward the domain of the finite numbers as their unattainable limit. An analogous “hopeless persistence” is shown by the series of proper fracftions stretching down from unity to their unattainable limit of zero. And as the proper fractions in getting ever closer to zero get ever closer to one another so too do the logarithmic transfinites get ever closer to one another in approaching as their “zero” the Nirvana of finitude. In the light of the infinite the finite is as nothing. Yet that “nothing” is the actuality to which even the infinite aspires.
In conclusion let us remember that the new transfinites, pygmy members of a giant race, and so at once both huge and squat, are true Children of the Rainbow. For they owe their existence (in a technical sense) to the Rainbow Series which thus proves itself to be as good and as useful as it was valid and beautiful. Indeed it was only by exhibiting the harmonic series as itself composed of an infinity of series, each of the rainbow type, that our “logarithms of infinity” could pass from the shadow status of mere formal symbols to the solid status of specific and therefore existential exemplification.
1. A gentleman who heard this paper when read wrote me a letter in which he claimed that another and in some ways simpler proof of the “Equality Relation” between the two series set forth in the first section of this paper had been published about ten years ago. Unfortunately, the letter was lost, and a careful search for the particle referred to has been unsuccessful.
2. Note by Professor W. M. Whyburn, Head of Department of Mathematics, University of California: Euler’s constant is defined as
lim n→∞ [H_n − log(n)] = c = .577,215,664,901...
where
H_n = 1 + ½ + ... + 1/n
Now
lim n→∞ [H_2n − log 2n] = c
∴
lim n→∞ [H_2n − log n] = c + log(2)
The Rainbow Series
1/(n+1) + ..... 1/2n = H_2n − H_n = [H_2n − log(n)] − [H_n − log(n)]
Hence
lim[H_2n − H_n] = lim [H_2n − log(n)] − lim [H_n − log(n)] = c + log(2) − c = log(2)
This may be generalized in the following way.
Let
H_jn − H_kn = 1/jn + 1/(jn−1) + ... 1/kn (j < k)
We shall have then
lim H_kn − log(kn) = c
or
lim n→∞ [H_jn − log(j) − log(n)] = c lim n→∞ [H_kn − log(k) − log(n)] = c
Hence
lim n→∞ [H_jn − log(n)] = c + log(j) lim n→∞ [H_kn − log(n)] = c + log(k)
∴
lim n→∞ [H_kn − H_jn] = lim n→∞ [H_kn − log(n)] − lim [H_jn − log(n)] = (c + log(k)) − (c + log(j)) = log(k) − log(j) = log(k/j)
3. The transfinites are defined or conceived by Cantor, their discoverer, in two ways: (1) positively as a series of infinites each of which is the logarithm of the successor thus, 2^T_n = T_n + 1; (2) negatively as a series of infinites separated by abysses which no relation of one-to-one correspondence can bridge.
I have taken the transfinites in the first, or positive, sense, as I have a right to do; but as Professor Whyburn and others have already observed, in the course of my proof by means of the Rainbow Series, that a logarithm of Aleph Null existed and that there was consequently no first or lowest transfinite there emerged as an incident of the argument a kind of one-to-one correspondence between the successive transfinites. This would appear to indicate, if my demonstration is valid, that between the negative and the positive definitions of the transfinites there is a contradiction. Not knowing what else to do I shall deposit this baby on the doorstep of the mathematicians and proceed to run away as fast as my legs will carry me.
4. This paper was included by the author in a volume entitled ‘The Ways of Thungs, An Experimental Introduction to Philosophy’ (Prentice-Hall) as an illustration of the application of certain logical methods.
1. The transfinite denoted in this paper as “Gamma” is a conflation of Aleph One, the next higher transfinite after Aleph Null, and Beth One, the cardinality of the reals. These two might be the same, or they might not; I’m not here to prove the continuum hypothesis. In Prof. Montague’s defense, I have no idea how these terms were used or defined in 1940.
2. Unfortunately, it’s pretty clear that the number of segments of Series I denoted by m can, in fact, be put in one-to-one correspondence with the integers; and thus m is just equal to Aleph Null. The dog wasn’t *that* shaggy.