1911 Encyclopædia Britannica/Number/Transfinite Numbers

20. Transfinite Numbers.—The theory of these numbers is quite recent, and mainly due to G. Cantor. The simplest of them, $\omega$, has been already defined (§ 4) as the order-type of the natural scale. Now there is no logical difficulty in constructing a scheme

$u_{1},u_{2},u_{3},\dots |v_{1}$,

indicating a well-ordered aggregate of type $\omega$ immediately followed by a distinct element $v_{1}$: for example, we may think of all positive odd integers arranged in ascending order of magnitude and then think of the even number $2$. A scheme of this kind is said to be of order-type $(\omega +1)$; and it will be convenient to speak of $(\omega +1)$ as the index of the scheme. Similarly we may form arrangements corresponding to the indices

$(\omega +2),(\omega +3)\dots (\omega +n)$

where $n$ is any positive integer. The scheme

$u_{1},u_{2},u_{3},\dots |v_{1},v_{2},v_{3}\dots$

is associated with $\omega +\omega =2\omega$;

$u_{11},u_{12},u_{13},\dots |u_{21},u_{22},u_{23},\dots |\dots |u_{n1},u_{n2},\dots |\dots$

with $\omega .\omega$ or $\omega ^{2}$; and so on. Thus we may construct arrangements of aggregates corresponding to any index of the form

$\phi (\omega )=a\omega ^{n}+b\omega ^{n-1}+\dots +k\omega +l$,

where $n,a,b,\dots l$ are all positive integers.

We are thus led to the construction of a scheme of symbols—

I.	$1,2,3,\dots n\dots$
II.	${\begin{cases}\omega ,\omega +1,\dots \omega +n,\dots \\2\omega ,2\omega +1,\dots 2\omega +n,\dots \\\vdots \\\omega ^{2},\omega ^{2}+1,\omega ^{2}+2\dots \omega ^{2}+n,\dots \\\vdots \\\phi (\omega ),\phi (\omega )+1,\dots \phi (\omega )+n,\dots \\\vdots \end{cases}}$
III.	${\begin{cases}\omega ^{\omega },\omega ^{\omega }+1,\dots \omega ^{\omega }+n,\dots \\\vdots \\\omega ^{\phi (\omega )},\omega ^{\phi (\omega )}+1,\dots \omega ^{\phi (\omega )}+n,\dots \\\vdots \end{cases}}$

The symbols $\phi (\omega )$ form a countable aggregate: so that we may, if we like (and in various ways), arrange the rows of block (II.) in a scheme of type $\omega$: we thus have each element $\alpha$ succeeded in its row by $(\alpha +1)$, and the row containing $\phi (\omega )$ succeeded by a definite next row. The same process may be applied to (III.), and we can form additional blocks (IV.), (V.), &c., with first elements $\omega _{4}=\omega ^{\omega ^{\omega }}$, $\omega _{5}=\omega ^{\omega _{4}}$, &c. All the symbols in which $\omega$ occurs are called transfinite ordinal numbers.

21. The index of a finite set is a definite integer however the set may be arranged; we may take this index as also denoting the power of the set, and call it the number of things in the set. But the index of an infinite ordinable set depends on the way in which its elements are arranged; for instance, ind. $(1,2,3,\dots )=\omega ,$ but ind. $(1,3,5,\dots |2,4,6,\dots )=2\omega$. Or, to take another example, the scheme—

$1,3,5,\dots (2n-1)\dots$

$2,6,10,\dots 2(2n-1)\dots$

$\;\vdots \quad \vdots \quad \vdots$

$2^{m},2^{m}.3,2^{m}.5,\dots 2^{m}(2n-1)\dots$

$\;\vdots$

Where each row is supposed to follow the one above it, gives a permutation of $(1,2,3,\dots )$, by which its index is changed from $\omega$ to $\omega ^{2}$. It has been proved that there is a permutation of the natural scale, of which the index is $\phi (\omega )$, any assigned clement of (II.); and that, if the index of any ordered aggregate is $\phi (\omega )$, the aggregate is countable. Thus the power of all aggregates which can be associated with indices of the class (II.) is the same as that of the natural scale; this power may be denoted by $a$. Since $a$ is associated with all aggregates of a particular power, independently of the arrangement of their elements, it is analogous to the integers, $1,2,3,$ &c., when used to denote powers of finite aggregates; for this reason it is called the least transfinite cardinal number.

22. There are aggregates which have a power greater than $a$: for instance, the arithmetical continuum of positive real numbers, the power of which is denoted by $c$. Another one is the aggregate of all those order-types which (like those in II. above) are the indices of aggregates of power $a$. The power of this aggregate is denoted by $\aleph _{1}$. According to Cantor's theory it is the transfinite cardinal number next superior to $a$, which for the sake of uniformity is also denoted by $\aleph _{0}$. It has been conjectured that $\aleph _{1}=c$, but this has neither been verified nor disproved. The discussion of the aleph-numbers is still in a controversial stage (November 1907) and the points in debate cannot be entered upon here.

23. Transfinite numbers, both ordinal and cardinal, may be combined by operations which are so far analogous to those of ordinary arithmetic that it is convenient to denote them by the same symbols. But the laws of operation are not entirely the same; for instance, $2\omega$ and $\omega 2$ have different meanings: the first has been explained, the second is the index of the scheme $(a_{1}b_{1}|a_{2}b_{2}|a_{3}b_{3}|\dots |a_{n}b_{n}|\dots )$ or any similar arrangement. Again if $n$ is any positive integer, $na=a^{n}=a$. It should also be observed that according to Cantor's principles of construction every ordinal number is succeeded by a definite next one; but that there are definite ordinal numbers (e.g. $\omega ,\omega ^{2}$) which have no ordinal immediately preceding them.