average range of vertical disturbance being about 00140 c.G.S., differences of this order may be obtained by supposing that the slabs of magnetic matter are 16 kiloms. (10 miles) thick, and that the upper surfaces are 4 kiloms. from the surface of the earth. We may, however, arrive at a similar result by assuming that iron exists in the crust of the earth chiefly in the form of magnetite and iron ores, and that the temperature at which these cease to be magnetic is 555° C. With a rise of temperature of 1° per 90 feet of depth this would give a depth of 15 kiloms, or 9 miles, as the thickness of the magnetic floor, a value which corresponds with that calculated from a knowledge of the range of vertical disturbance.

In concluding this description of the experiments, which were carried out in the Physical Laboratory of Birmingham University, I take the opportunity of expressing my thanks to Prof. Poynting for suggesting the investigation, providing the necessary apparatus and space, and for much encouragement and assistance during the progress of the experiments.

I am also indebted to all the members of the Geological Department of the same University for their cordial assistance with the geological side of the investigation.

Physical Laboratory,

Birmingham University,
May, 1903.

VII. On the Transfinite Cardinal Numbers of Well-ordered Aggregates. By PHILIP E. B. JOURDAIN, B.A., Trinity College, Cambridge*.

the memoir† in which the transfinite ordinal numbers

I first appeared in a form independent of the theory of

the derivatives of aggregates of points representing real numbers, the illustrious author, Georg Cantor, defined a series of "powers" which belong to various classes of the transfinite numbers, and which series possesses the remarkable property of being ordinally similar to the whole class of transfinite numbers ‡.

* Communicated by the Author.

Ueber unendliche, lineare Punktmannichfaltigkeiten, v.," Math. Ann. xxi. pp. 545-591 (1883) [dated December, 1882]; also as a separate pamphlet, Grundlagen einer allgemeinen Mannichfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen,' Leipzig, 1883. Pagen of the "Grundlagen" corresponds to page n +544 of the article mentioned first.

Grundlagen,' pp. 43-44.

It is essential for clearness that we should adopt the names and notations introduced by Cantor in 1895 and 1897 †. What were before called "powers" were then called (finite or transfinite) cardinal numbers, the transfinite numbers received the fuller designation of transfinite ordinal numbers, and finally, the above-mentioned series of cardinal numbers received the notation,

where the suffixes form the whole series of the finite and transfinite ordinal numbers.

It is now easy to state precisely the fundamental results of Cantor with respect to the series (1).

Ordinal numbers belong only to what Cantor has called "well-ordered" aggregates §, and every part of numbers of the whole aggregate of ordinal numbers up to any number, arranged in order of magnitude,

V, w, w +1, w +2, .. w + v, ... w. 2, w. 2+1,

...

The series of

itself forms a well-ordered aggregate. Alephs (1) is defined as the series of the cardinal numbers of well-ordered aggregates, and the Alephs are therefore the cardinal numbers of certain of the parts of (2) referred to. Cantor has, now, proved, in particular, that is the next greater cardinal number to No; and, in fact, in general ¶: If

is any cardinal number of the series (1), then y+1 is the next greater; and, inversely: if y is any number of (1), and has an immediate predecessor (that is, if y has an immediate predecessor in the series (2)), then &, is the next greater than this predecessor. If, on the other hand, &, has no immediate predecessor (that is, if y is a Limes-number of

"Beiträge zur Begrundung der transfiniten Mengenlehre," Math. Ann. xlvi. pp. 481-512 (1895), and xlix. pp. 207-246 (1897). See especially pp. 481-482, 492, 495, and 497 of the first article, and p. 216 of the second.

Cantor had, already in 1883, arrived at many of the formulations which were only published much later (cf. 'Zur Lehre vom Transfiniten,' Halle a. S., 1890, pp. 11, 12).

See 'Grundlagen,' pp. 35-39. Cf. Schönflies 'Die Entwickelung der Lehre von den Punktmannigfaltigkeiten,' Leipzig, 1900, pp. 44–50. § Grundlagen,' pp. 4-5; Math. Ann. xlix. pp. 207-208.

Cf. also Math. Ann. xlix. foot of p. 216, and §§ 2 and 3 below.

A detailed proof of this will be given in the continuation of this paper; cf. end of § 10.

Y

(2), like w, w.2, w.v, ww, L, ...), Ny is the next greater cardinal number to all the cardinal numbers Ng such that

B<7.

Further, is the smallest transfinite cardinal number † ; it is the cardinal number of any enumerable aggregate, an aggregate which can always be re-arranged (if a re-arrangement is necessary) in the form of a well-ordered aggregate of type w.

1.

We have thus an illimitable series of ascending cardinal numbers. But there are cardinal numbers, such as the cardinal number of the real-number-continuum (0 . . . . 1), which are not defined as cardinal numbers of well-ordered aggregates, and of which we cannot therefore immediately say that they occur in the series (1). However, Cantor showed that

though the latter equality has never yet been proved. But since is the next cardinal number to N, it is possible to take elements from the number-continuum corresponding to all the numbers of Cantor's first and second classes of ordinal numbers. For if this process were to stop we should have

which is contradicted by (3). Using, then, the theorem of Schröder and Bernstein, we can state that

1

If, now [>No1, we can conclude similarly that

+ Math. Ann. xlvi. p. 492. (Cf. Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre," Journ. für Math. lxxxiv. 1878, p. 242.)

66

Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen," Journ. für Math. lxxvii. 1874, pp. 258-263.

§ Cf. "Ein Beitrag zur Mannigfaltigkeitslehre," Journ. für Math. lxxxiv. 1878, p. 257; Grundlagen,' p. 7.

I See below, §§ 7 and 8.

¶ The fact that 2, follows, perhaps even more simply, from the consideration that 2 is the cardinal number of the transfinite classes which can be formed out of N. members, while, is the cardinal number of some of these classes.

0

and so on for all the numbers of (1). From this reasoning, which first appeared in a published form in a paper in which Hardy constructed an aggregate of points of cardinal number & in the continuum, it follows that every cardinal number is either contained in the series of Alephs (1), or is greater than any Aleph.

If, now, Cantor's view that every cardinal number is an Aleph-which he expressed in the equivalent form that every well-defined aggregate can be put, by re-arrangement if necessary, in the form of a well-ordered aggregate-is to be substantiated, we must prove that the supposition that a cardinal number is greater than any Aleph leads to a contradiction.

2.

If a cardinal number were greater than any Aleph, it would be equal to or greater than the cardinal number of the series (1) of all Alephs. For to this series can be correlated the series (2) of all ordinal numbers, and every Aleph is the cardinal number of some segment of the series (2); the cardinal number in question must, then, be at least equal to the cardinal number of the whole series (2), and, consequently, to that of the whole series (1).

If, now, the series (2) and consequently (1), be wellordered, the ordinal type of (2) (or (1)) is an ordinal number, B, and the cardinal number of (1) (or (2)) is an Aleph, s. Further, this ordinal number 8 must be the greatest ordinal number, and, consequently, & must be the greatest Aleph. But there can be neither a greatest ordinal nor a greatest. Aleph; for, given B, the type of the aggregate (1...) is the ordinal number + 1, and

This contradiction was first published by Burali-Forti ¶,

"A Theorem concerning the Infinite Cardinal Numbers," Quart. Journ. of Math. 1903, pp. 87-94.

‡ ‘Grundlagen,' p. 6.

This word is to imply that the two aggregates are similarly ordered in the sense of Math. Ann. xlvi. p. 497.

I use this word to translate Cantor's 'Abschnitt' of Math. Ann. xlix. p. 210.

Una questione sui numeri transfiniti," Rend. del circolo mat. di Palermo, xi. (1897).

who concluded from it that one must deny both Cantor's † fundamental theorem in the theory of ordinal numbers that: if a1 and a, are any two ordinal numbers, then either

and the corresponding theorem for Alephs. This conclusion is, in fact, necessary if one admits Burali-Forti's premisses; and, since Cantor's demonstration of the above theorem is beyond all possible objection, Russell avoided the contradiction by denying the premiss that the series of all ordinal numbers, arranged in order of magnitude, is well-ordered. Then the ordinal type (B) of (1) or (2) ceases to be an ordinal number, and we can no longer assert, in general, that B+1>8.

Further, the cardinal number ceases to be necessarily an Aleph, and we cannot therefore assert, as we could before, that one of the Alephs surpasses it.

But it appears possible to prove that this series of all ordinal numbers is well-ordered; to this proof the following section is devoted.

3.

If we adopt, as definition of a well-ordered aggregate (M) the property which Cantor § has proved to be the characteristic of well-ordered aggregates among all simply-ordered aggregates; namely, that both it and everyone of its partial aggregates should have a first element; it becomes evident that no part of M can be of ordinal type

*w.

But in no publication known to me does it appear to have been remarked explicitly that this gives a sufficient, as well as a necessary, condition that the simply-ordered aggregate M should be well-ordered. However, if M contains no part of ordinal type

*w,

it is well-ordered; for if not, at least one of its parts would have no first element, and in this part a part of type

*W

can always be found. Thus, in order that a simply-ordered

Math. Ann. xlix. P. 216.

The Principles of Mathematics,' Cambridge, 1903, p. 323. Cf.

Cantor, Math. Ann. xlix. foot of p. 216.

§ Math. Ann. xlix. pp. 208-209. This property has been taken as the definition of a well-ordered aggregate by Schönflies (op. cit. p. 36) and Russell (op. cit. p. 319, last note).

Phil. Mag. S. 6. Vol. 7. No. 37. Jan. 1904.

F