Contributions to the Founding of the Theory of Transfinite Numbers/Article 1

by Georg Cantor

(First Article)

"Hypotheses non fingo."

"Neque enim leges intellectui aut rebus damus ad arbitrium nostrum, sed tanquam scribæ fideles ab ipsius naturæ voce latas et prolatas excipimus et describimus."

"Veniet tempus, quo ista quæ nunc latent, in lucem dies extrahat et longioris ævi diligentia."

§1
The Conception of Power or Cardinal Number

By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) $M$ of definite and separate objects $m$ of our intuition or our thought. These objects are called the "elements" of $M$.

In signs we express this thus:

We denote the uniting of many aggregates $M$, $N$, $P$, $...$, which have no common elements, into a single aggregate by

The elements of this aggregate are, therefore, the elements of $M$, of $N$, of $P$, $...$, taken together.

We will call by the name "part" or "partial aggregate" of an aggregate $M$ any other aggregate $M_{1}$ whose elements are also elements of $M$.

If $M_{2}$ is a part of $M_{1}$ and $M_{1}$ is a part of $M$, then $M_{2}$ is a part of $M$.

Every aggregate $M$ has a definite "power," which we will also call its "cardinal number."

We will call by the name "power" or "cardinal number" of $M$ the general concept which, by means of our active faculty of thought, arises from the aggregate $M$ when we make abstraction of the nature of its various elements $m$ and of the order in which they are given.

[482] We denote the result of this double act of abstraction, the cardinal number or power of $M$, by

Since every single element $m$, if we abstract from its nature, becomes a "unit," the cardinal number ${\overline {\overline {M}}}$ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate $M$.

We say that two aggregates $M$ and $N$ are "equivalent," in signs

if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. To every part $M_{1}$ of $M$ there corresponds, then, a definite equivalent part $N_{1}$ of $N$, and inversely.

If we have such a law of co-ordination of two equivalent aggregates, then, apart from the case when each of them consists only of one element, we can modify this law in many ways. We can, for instance, always take care that to a special element $m_{0}$ of $M$ a special element $n_{0}$ of $N$ corresponds. For if, according to the original law, the elements $m_{0}$and $n_{0}$ do not correspond to one another, but to the element $m_{0}$ of $M$ the element $n_{1}$ of $N$ corresponds, and to the element $n_{0}$ of $N$ the element $m_{1}$ of $M$ corresponds, we take the modified law according to which $m_{0}$ corresponds to $n_{0}$ and $m_{1}$ to $n_{1}$ and for the other elements the original law remains unaltered. By this means the end is attained.

Every aggregate is equivalent to itself:

If two aggregates are equivalent to a third, they are equivalent to one another; that is to say:

Of fundamental importance is the theorem that two aggregates M and N have the same cardinal number if, and only if, they are equivalent: thus,

and

Thus the equivalence of aggregates forms the necessary and sufficient condition for the equality of their cardinal numbers.

[483] In fact, according to the above definition of power, the cardinal number ${\overline {\overline {M}}}$ remains unaltered if in the place of each of one or many or even all elements $m$ of $M$ other things are substituted. If, now, $M\sim N$, there is a law of co-ordination by means of which $M$ and $N$ are uniquely and reciprocally referred to one another; and by it to the element $m$ of $M$ corresponds the element $n$ of $N$. Then we can imagine, in the place of every element $m$ of $M$, the corresponding element $n$ of $N$ substituted, and, in this way, $M$ transforms into $N$ without alteration of cardinal number. Consequently

${\overline {\overline {M}}}={\overline {\overline {N}}}$.

The converse of the theorem results from the remark that between the elements of $M$ and the different units of its cardinal number ${\overline {\overline {M}}}$ a reciprocally univocal (or bi-univocal) relation of correspondence subsists. For, as we saw, ${\overline {\overline {M}}}$ grows, so to speak, out of $M$ in such a way that from every element $m$ of $M$ a special unit of $M$ arises. Thus we can say that

(9)	$M\sim {\overline {\overline {M}}}$

In the same way $N\sim {\overline {\overline {N}}}$. If then ${\overline {\overline {M}}}={\overline {\overline {N}}}$, we have, by (6), $M\sim N$.

We will mention the following theorem, which results immediately from the conception of equivalence. If $M$, $N$, $P$, $...$ are aggregates which have no common elements, $M'$, $N'$, $P'$, $...$ are also aggregates with the same property, and if

$M\sim M'$, $N\sim N'$, $P\sim P'$, $...$,

then we always have

$(M,N,P,...)\sim (M',N',P',...)$.

§2
"Greater" and "Less" with Powers

If for two aggregates $M$ and $N$ with the cardinal numbers ${\mathfrak {a}}={\overline {\overline {M}}}$ and ${\mathfrak {b}}={\overline {\overline {N}}}$, both the conditions:

(a) There is no part of $M$ which is equivalent to $N$,
(b) There is a part $N_{1}$ of $N$, such that $N_{1}\sim M$,

are fulfilled, it is obvious that these conditions still hold if in them $M$ and $N$ are replaced by two equivalent aggregates $M'$ and $N'$. Thus they express a definite relation of the cardinal numbers ${\mathfrak {a}}$ and ${\mathfrak {b}}$ to one another.

[484] Further, the equivalence of $M$ and $N$, and thus the equality of ${\mathfrak {a}}$ and ${\mathfrak {b}}$, is excluded; for if we had $M\sim N$, we would have, because $N_{1}\sim M$, the equivalence $N_{1}\sim N$, and then, because $M\sim N$, there would exist a part $M_{1}$ of $M$ such that $M_{1}\sim M,$ and therefore we should have $M_{1}\sim N$; and this contradicts the condition (a).

Thirdly, the relation of ${\mathfrak {a}}$ and ${\mathfrak {b}}$ is such that it makes impossible the same relation of ${\mathfrak {b}}$ and ${\mathfrak {a}}$; for if in (a) and (b) the parts played by $M$ and $N$ are interchanged, two conditions arise which are contradictory to the former ones.

We express the relation of ${\mathfrak {a}}$ to ${\mathfrak {b}}$ characterized by (a) and (b) by saying: ${\mathfrak {a}}$ is "less" than ${\mathfrak {b}}$ or ${\mathfrak {b}}$ is "greater" than ${\mathfrak {a}}$; in signs

We can easily prove that,

Similarly, from the definition, it follows at once that, if $P_{1}$ is part of an aggregate $P$, from ${\mathfrak {a}}<{\overline {\overline {P_{1}}}}$ follows ${\mathfrak {a}}<{\overline {\overline {P}}}$ and from ${\overline {\overline {P}}}<{\mathfrak {b}}$ follows ${\overline {\overline {P_{1}}}}<{\mathfrak {b}}$.

We have seen that, of the three relations

${\mathfrak {a}}={\mathfrak {b}}$, ${\mathfrak {a}}<{\mathfrak {b}}$, ${\mathfrak {b}}<{\mathfrak {a}}$,

each one excludes the two others. On the other hand, the theorem that, with any two cardinal numbers ${\mathfrak {a}}$ and ${\mathfrak {b}}$, one of those three relations must necessarily be realized, is by no means self-evident and can hardly be proved at this stage.

Not until later, when we shall have gained a survey over the ascending sequence of the transfinite cardinal numbers and an insight into their connexion, will result the truth of the theorem:

A. If ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are any two cardinal numbers, then either ${\mathfrak {a}}={\mathfrak {b}}$ or ${\mathfrak {a}}<{\mathfrak {b}}$ or ${\mathfrak {a}}>{\mathfrak {b}}$.

From this theorem the following theorems, of which, however, we will here make no use, can be very simply derived:

B. If two aggregates $M$ and $N$ are such that $M$ is equivalent to a part $N_{1}$ of $N$ and $N$ to a part $M_{1}$ of $M$, then $M$ and $N$ are equivalent;

C. If $M_{1}$ is a part of an aggregate $M$, $M_{2}$ is a part of the aggregate $M_{1}$, and if the aggregates $M$ and $M_{2}$ are equivalent, then $M_{1}$ is equivalent to both $M$ and $M_{2}$;

D. If, with two aggregates $M$ and $N$, $N$ is equivalent neither to $M$ nor to a part of $M$, there is a part $N_{1}$ of $N$ that is equivalent to $M$;

E. If two aggregates $M$ and $N$ are not equivalent, and there is a part $N_{1}$ of $N$ that is equivalent to $M$, then no part of $M$ is equivalent to $N$.

The Addition and Multiplication of Powers

The union of two aggregates $M$ and $N$ which have no common elements was denoted in § 1, (2), by $(M,N)$. We call it the "union-aggregate (Vereinigungsmenge) of $M$ and $N$."

If $M'$ and $N'$ are two other aggregates without common elements, and if $M\sim M'$ and $N\sim N'$, we saw that we have

$(M,N)\sim (M',N')$.

Hence the cardinal number of $(M,N)$ only depends upon the cardinal numbers ${\overline {\overline {M}}}={\mathfrak {a}}$ and ${\overline {\overline {N}}}={\mathfrak {b}}$.

This leads to the definition of the sum of a and b. We put

Since in the conception of power, we abstract from the order of the elements, we conclude at once that

and, for any three cardinal numbers ${\mathfrak {a}}$, ${\mathfrak {b}}$, ${\mathfrak {c}}$, we have

We now come to multiplication. Any element $m$ of an aggregate $M$ can be thought to be bound up with any element $n$ of another aggregate $N$ so as to form a new element $(m,n)$; we denote by $(M.N)$ the aggregate of all these bindings $(m,n)$, and call it the "aggregate of bindings (Verbindungsmenge) of $M$ and $N$." Thus

We see that the power of $(M.N)$ only depends on the powers ${\overline {\overline {M}}}={\mathfrak {a}}$ and ${\overline {\overline {N}}}={\mathfrak {b}}$; for, if we replace the aggregates $M$ and $N$ by the aggregates

$M'={m'}$ and $N'={n'}$

respectively equivalent to them, and consider $m$, $m'$ and $n$, $n'$ as corresponding elements, then the aggregate

$(M'.N')={(m',n')}$

is brought into a reciprocal and univocal correspondence with $(M.N)$ by regarding $(m,n)$ and $(m',n')$ as corresponding elements. Thus

We now define the product ${\mathfrak {a}}.{\mathfrak {b}}$ by the equation

[486] An aggregate with the cardinal number ${\mathfrak {a}}.{\mathfrak {b}}$ may also be made up out of two aggregates $M$ and $N$ with the cardinal numbers ${\mathfrak {a}}$ and ${\mathfrak {b}}$ according to the following rule: We start from the aggregate $N$ and replace in it every element $n$ by an aggregate $M_{n}\sim M$; if, then, we collect the elements of all these aggregates $M$ to a whole $S$, we see that

and consequently

${\overline {\overline {S}}}={\mathfrak {a}}.{\mathfrak {b}}$.

For, if, with any given law of correspondence of the two equivalent aggregates $M$ and $M_{n}$, we denote by $m$ the element of $M$ which corresponds to the element $m_{n}$ of $M_{n}$, we have

and thus the aggregates $S$ and $(M.N)$ can be referred reciprocally and univocally to one another by regarding $m_{n}$ and $(m,n)$ as corresponding elements.

From our definitions result readily the theorems:

because:

$(M.N)\sim (N.M)$,

$(M.(N.P))\sim ((M.N).P)$,

$(M.(N,P))\sim ((M.N),(M.P))$.

Addition and multiplication of powers are subject, therefore, to the commutative, associative, and distributive laws.

§4
The Exponentiation of Powers

By a "covering of the aggregate $N$ with elements of the aggregate $M$," or, more simply, by a "covering of $N$ with $M$," we understand a law by which with every element $n$ of $N$ a definite element of $M$ is bound up, where one and the same element of $M$ can come repeatedly into application. The element of $M$ bound up with $n$ is, in a way, a one-valued function of $n$, and may be denoted by $f(n)$; it is called a "covering function of $n$." The corresponding covering of $N$ will be called $f(N)$.

[487] Two coverings $f_{1}(N)$ and $f_{2}(N)$ are said to be equal if, and only if, for all elements $n$ of $N$ the equation

is fulfilled, so that if this equation does not subsist for even a single element $n=n_{0}$,$f_{1}(N)$ and $f_{2}(N)$ are characterized as different coverings of $N$. For example, if $m_{0}$ is a particular element of $M$, we may fix that, for all $n$'s

this law constitutes a particular covering of $N$ with $M$. Another kind of covering results if $m_{0}$ and $m_{1}$ are two different particular elements of $M$ and $n_{0}$ a particular element of $N$, from fixing that

$f(n_{0})=m_{0}$

$f(n)=m_{1}$

for all $n$'s which are different from $n_{0}$.

The totality of different coverings of N with M forms a definite aggregate with the elements $f(N)$; we call it the "covering-aggregate (Belegungsmenge) of $N$ with $M$" and denote it by $(N|M)$. Thus:

If $M\sim M'$ and $N\sim N'$, we easily find that

Thus the cardinal number of $(N|M)$ depends only on the cardinal numbers ${\overline {\overline {M}}}={\mathfrak {a}}$ and ${\overline {\overline {N}}}={\mathfrak {b}}$; it serves us for the definition of ${\mathfrak {a}}^{\mathfrak {b}}$ :

For any three aggregates, $M,N,P$, we easily prove the theorems:

from which, if we put ${\overline {\overline {P}}}={\mathfrak {c}}$, we have, by (4) and by paying attention to § 3, the theorems for any three cardinal numbers, ${\mathfrak {a}}$, ${\mathfrak {b}}$, and ${\mathfrak {c}}$:

[488] We see how pregnant and far-reaching these simple formulæ extended to powers are by the following example. If we denote the power of the linear continuum $X$ (that is, the totality $X$ of real numbers $x$ such that $x\geq$ and $\leq 1$) by ${\mathfrak {o}}$, we easily see that it may be represented by, amongst others, the formula:

where § 6 gives the meaning of $\aleph _{0}$. In fact, by (4), $2^{\aleph _{0}}$ is the power of all representations

(where $f(\nu )=0$ or $1$)

of the numbers $x$ in the binary system. If we pay attention to the fact that every number $x$ is only represented once, with the exception of the numbers $x={\frac {2\nu +1}{2^{\mu }}}<1$, which are represented twice over, we have, if we denote the "enumerable" totality of the latter by ${s_{\nu }}$,

$2^{\aleph _{0}}=({\overline {\overline {\{s_{\nu }\},X}}})$.

If we take away from $X$ any "enumerable" aggregate ${t_{\nu }}$ and denote the remainder by $X_{1}$, we have:

$X=(\{t_{\nu }\},X_{1})=(\{t_{2\nu -1}\},\{t_{2\nu }\},X_{1})$,

$(\{s_{\nu }\},X)=(\{s_{\nu }\},\{t_{\nu }\},X_{1})$,

$\{t_{2\nu -1}\}\sim \{s_{\nu }\},\quad \{t_{2\nu }\}\sim \{t_{\nu }\},\quad X_{1}\sim X_{1}$;

so

$X\sim (\{s_{\nu }\},X)$,

and thus (§1)

$2^{\aleph _{0}}={\overline {\overline {X}}}={\mathfrak {o}}$.

From (11) follows by squaring (by § 6, (6))

${\mathfrak {o}}\cdot {\mathfrak {o}}=2^{\aleph _{0}}\cdot 2^{\aleph _{0}}=2^{\aleph _{0}+\aleph _{0}}={\mathfrak {o}}$,

and hence, by continued multiplication by ${\mathfrak {o}}$,

where $\nu$ is any finite cardinal number.

If we raise both sides of (11) to the power^[1] $\aleph _{0}$ we get

${\mathfrak {o}}^{\aleph _{0}}=(2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}\cdot \aleph _{0}}$.

But since, by § 6, (8), $\aleph _{0}\cdot \aleph _{0}=\aleph _{0}$, we have

The formulae (13) and (14) mean that both the $\nu$-dimensional and the $\aleph _{0}$-dimensional continuum have the power of the one-dimensional continuum. Thus the whole contents of my paper in Crelle's Journal, vol. lxxxiv, 1878,^[2] are derived purely algebraically with these few strokes of the pen from the fundamental formulæ of the calculation with cardinal numbers.

The Finite Cardinal Numbers

We will next show how the principles which we have laid down, and on which later on the theory of the actually infinite or transfinite cardinal numbers will be built, afford also the most natural, shortest, and most rigorous foundation for the theory of finite numbers.

To a single thing $e_{0}$, if we subsume it under the concept of an aggregate $E_{0}=(e_{o})$, corresponds as cardinal number what we call "one" and denote by 1; we have

Let us now unite with $E_{0}$ another thing $e_{1}$ and call the union-aggregate $e_{1}$, so that

The cardinal number of $E_{1}$ is called "two" and is denoted by 2:

By addition of new elements we get the series of aggregates

$E_{2}=(E_{1},e_{2}),\quad E_{3}=(E_{2},e_{3}),...$,

which give us successively, in unlimited sequence, the other so-called "finite cardinal numbers" denoted by $3$, $4$, $5$, ... The use which we here make of these numbers as suffixes is justified by the fact that a number is only used as a suffix when it has been defined as a cardinal number. We have, if by $\nu -1$ is understood the number immediately preceding $\nu$ in the above series,

From the definition of a sum in § 3 follows:

that is to say, every cardinal number, except 1, is the sum of the immediately preceding one and 1.

Now, the following three theorems come into the foreground: A. The terms of the unlimited series of finite cardinal numbers

$1,2,3,...,\nu ,...$

are all different from one another (that is to say, the condition of equivalence established in § 1 is not fulfilled for the corresponding aggregates).

[490] B. Every one of these numbers $\nu$ is greater than the preceding ones and less than the following ones (§ 2).

C. There are no cardinal numbers which, in magnitude, lie between two consecutive numbers $\nu$ and $\nu +1$ (§ 2).

We make the proofs of these theorems rest on the two following ones, D and E. We shall, then, in the next place, give the latter theorems rigid proofs.

D. If $M$ is an aggregate such that it is of equal power with none of its parts, then the aggregate $(M,e)$ which arises from $M$ by the addition of a single new element $e$, has the same property of being of equal power with none of its parts.

E. If $N$ is an aggregate with the finite cardinal number $\nu$, and $N_{1}$ is any part of $N$, the cardinal number of $N_{1}$ is equal to one of the preceding numbers $1,2,3,...,\nu -1$.

Proof of D. Suppose that the aggregate $(M,e)$ is equivalent to one of its parts which we will call $N$. Then two cases, both of which lead to a contradiction, are to be distinguished:

(a) The aggregate $N$ contains $e$ as element; let $N=(M_{1},e)$; then $M_{1}$ is a part of $M$ because $N$ is a part of $(M,e)$. As we saw in § 1, the law of correspondence of the two equivalent aggregates $(M,e)$ and $(M_{1},e)$ can be so modified that the element $e$ of the one corresponds to the same element $e$ of the other; by that, then, $M$ and $M_{1}$ are referred reciprocally and univocally to one another. But this contradicts the supposition that $M$ is not equivalent to its part $M_{1}$.

(b) The part $N$ of $(M,e)$ does not contain $e$ as element, so that $N$ is either $M$ or a part of $M$. In the law of correspondence between $(M,e)$ and $N$, which lies at the basis of our supposition, to the element $e$ of the former let the element $f$ of the latter correspond. Let $N=(M_{1},f)$; then the aggregate $M$ is put in a reciprocally univocal relation with $M_{1}$. But $M_{1}$ is a part of $N$ and hence of $M$. So here too $M$ would be equivalent to one of its parts, and this is contrary to the supposition.

Proof of E.—We will suppose the correctness of the theorem up to a certain $\nu$ and then conclude its validity for the number $\nu +1$ which immediately follows, in the following manner:—We start from the aggregate $E_{\nu }=(e_{0},e_{1},...,e_{\nu })$ as an aggregate with the cardinal number $\nu +1$. If the theorem is true for this aggregate, its truth for any other aggregate with the same cardinal number $\nu +1$ follows at once by § 1. Let $E'$ be any part of $E_{\nu }$; we distinguish the following cases:

(a) $E'$ does not contain $e_{\nu }$ as element, then $E$ is either $E_{\nu -1}$ [491] or a part of $E_{\nu -1}$, and so has as cardinal number either $\nu$ or one of the numbers $1,2,3,...,\nu -1$, because we supposed our theorem true for the aggregate $E_{\nu -1}$, with the cardinal number $\nu$,

(b) $E'$ consists of the single element $e_{\nu }$, then ${\overline {\overline {E'}}}=1$.

(c) $E'$ consists of $e_{\nu }$ and an aggregate $E''$, so that $E'=(E'',e_{\nu })$. $E''$ is a part of $E_{\nu -1}$ and has therefore by supposition as cardinal number one of the numbers $1,2,3,...,\nu -1$. But now ${\overline {\overline {E'}}}={\overline {\overline {E''}}}+1$, and thus the cardinal number of $E'$ is one of the numbers $2,3,...,\nu$.

Proof of A.—Every one of the aggregates which we have denoted by $E_{\nu }$ has the property of not being equivalent to any of its parts. For if we suppose that this is so as far as a certain $\nu$, it follows from the theorem D that it is so for the immediately following number $\nu +1$. For $\nu =1$, we recognize at once that the aggregate $E_{1}=(e_{0},e_{1})$ is not equivalent to any of its parts, which are here $(e_{0})$ and $(e_{1})$. Consider, now, any two numbers $\mu$ and $\nu$ of the series $1,2,3,...$; then, if $\mu$ is the earlier and $\nu$ the later, $E_{\mu -1}$ is a part of $E_{\nu -1}$. Thus $E_{\mu -1}$ and $E_{\nu -1}$ are not equivalent, and accordingly their cardinal numbers $\mu ={\overline {\overline {E}}}_{\mu -1}$ and $\nu ={\overline {\overline {E}}}_{\nu -1}$ are not equal.

Proof of B.—If of the two finite cardinal numbers $\mu$ and $\nu$ the first is the earlier and the second the later, then $\mu <\nu$. For consider the two aggregates $M=E_{\mu -1}$ and $N=E_{\nu -1}$; for them each of the two conditions in § 2 for ${\overline {\overline {M}}}<{\overline {\overline {N}}}$ is fulfilled. The condition (a) is fulfilled because, by theorem E, a part of $M=E_{\mu -1}$ can only have one of the cardinal numbers $1,2,3,...,\mu -1$, and therefore, by theorem A, cannot be equivalent to the aggregate $N=E_{\mu -1}$. The condition (b) is fulfilled because $M$ itself is a part of $N$.

Proof of C.— Let ${\mathfrak {a}}$ be a cardinal number which is less than $\nu +1$. Because of the condition (b) of §2, there is a part of $E_{\nu }$ with the cardinal number ${\mathfrak {a}}$. By theorem E, a part of $E_{\nu }$ can only have one of the cardinal numbers $1,2,3,...,\nu$. Thus ${\mathfrak {a}}$ is equal to one of the cardinal numbers $1,2,3,...,\nu$. By theorem B, none of these is greater than $\nu$. Consequently there is no cardinal number ${\mathfrak {a}}$ which is less than $\nu +1$ and greater than $\nu$.

Of importance for what follows is the following theorem:

F. If $K$ is any aggregate of different finite cardinal numbers, there is one, $\kappa _{1}$, amongst them which is smaller than the rest, and therefore the smallest of all.

[492] Proof—The aggregate $K$ either contains the number $1$, in which case it is the least, $\kappa _{1}=1$, or it does not. In the latter case, let $J$ be the aggregate of all those cardinal numbers of our series, $1,2,3,...$, which are smaller than^those occurring in $K$. If a number $\nu$ belongs to $J$, all numbers less than $\nu$ belong to $J$. But $J$ must have one element $\nu _{1}$ such that $\nu _{1}+1$, and consequently all greater numbers, do not belong to $J$, because otherwise $J$ would contain all finite numbers, whereas the numbers belonging to $K$ are not contained in $J$. Thus $J$ is the segment (Abschnitt) ($1,2,3,...,\nu _{1}$). The number $\nu _{1}+1=\kappa _{1}$ is necessarily an element of $K$ and smaller than the rest.

From F we conclude:

G. Every aggregate of different finite cardinal numbers can be brought into the form of a series

$K=(\kappa _{1},\kappa _{2},\kappa _{3},...)$

such that

$\kappa _{1}<\kappa _{2}<\kappa _{3},...$

§6
The Smallest Transfinite Cardinal Number Aleph-Zero

Aggregates with finite cardinal numbers are called "finite aggregates," all others we will call "transfinite aggregates" and their cardinal numbers "transfinite cardinal numbers."

The first example of a transfinite aggregate is given by the totality of finite cardinal numbers $\nu$; we call its cardinal number (§1) "Aleph-zero" and denote it by $\aleph _{0}$ thus we define

That $\aleph _{0}$ is a transfinite number, that is to say, is not equal to any finite number $\mu$, follows from the simple fact that, if to the aggregate $\{\nu \}$ is added a new element $e_{0}$, the union-aggregate $(\{\nu \},e_{0})$ is equivalent to the original aggregate $\{\nu \}$. For we can think of this reciprocally univocal correspondence between them: to the element $e_{0}$ of the first corresponds the element $1$ of the second, and to the element $\nu$ of the first corresponds the element $\nu +1$ of the other. By §3 we thus have

But we showed in §5 that $\mu +1$ is always different from $\mu$ and therefore $\aleph _{0}$ is not equal to any finite number $\mu$.

The number $\aleph _{0}$ is greater than any finite number $\mu$:

[493] This follows, if we pay attention to §3, from the three facts that $\mu ={\overline {\overline {(1,2,3,...,\mu )}}}$, that no part of the aggregate $(1,2,3,...,\mu )$ is equivalent to the aggregate $\{\nu \}$, and that $(1,2,3,...,\mu )$ is itself a part of $\{\nu \}$.

On the other hand, $\aleph _{0}$ is the least transfinite cardinal number. If ${\mathfrak {a}}$ is any transfinite cardinal number different from $\aleph _{0}$, then

This rests on the following theorems:

A. Every transfinite aggregate $T$ has parts with the cardinal number $\aleph _{0}$.

Proof.—If, by any rule, we have taken away a finite number of elements $t_{1},t_{2},t_{3},...,t_{\nu -1}$, there always remains the possibility of taking away a further element $t_{\nu }$. The aggregate $\{t_{\nu }\}$, where $\nu$ denotes any finite cardinal number, is a part of $T$ with the cardinal number $\aleph _{0}$, because $\{t_{\nu }\}\sim \{\nu \}$ (§1).

B. If $S$ is a transfinite aggregate with the cardinal number $\aleph _{0}$, and $S_{1}$ is any transfinite part of $S$, then ${\overline {\overline {S_{1}}}}=\aleph _{0}$.

Proof.—We have supposed that $S\sim \{\nu \}$. Choose a definite law of correspondence between these two aggregates, and, with this law, denote by $s_{\nu }$ that element of $S$ which corresponds to the element $\nu$ of $\{\nu \}$, so that

$S=\{s_{\nu }\}$

The part $S_{1}$ of $S$ consists of certain elements $s_{\kappa }$ of $S$, and the totality of numbers $\kappa$ forms a transfinite part $K$ of the aggregate $\{\nu \}$. By theorem G of §5 the aggregate $K$ can be brought into the form of a series

$K=\{\kappa _{\nu }\}$

where

$\kappa _{\nu }<\kappa _{\nu +1}$;

consequently we have

$S_{1}=\{s_{\kappa _{\nu }}\}$.

Hence follows that $S_{1}\sim S$, and therefore ${\overline {\overline {S_{1}}}}=\aleph _{0}$.

From A and B the formula (4) results, if we have regard to §2.

From (2) we conclude, by adding $1$ to both sides,

$\aleph _{0}+2=\aleph _{0}+1=\aleph _{0}$,

and, by repeating this

We have also

[494] For, by (1) of §3, $\aleph _{0}+\aleph _{0}$ is the cardinal number ${\overline {\overline {(\{a_{\nu }\},\{b_{\nu }\})}}}$ because

${\overline {\overline {\{a_{\nu }\}}}}={\overline {\overline {\{b_{\nu }\}}}}=\aleph _{0}$.

Now, obviously

$\{\nu \}=(\{2\nu -1\},\{2\nu \})$,

$(\{2\nu -1\},\{2\nu \})\sim (\{a_{\nu }\},\{b_{\nu }\})$,

and therefore

${\overline {\overline {(\{a_{\nu }\},\{b_{\nu }\})}}}={\overline {\overline {\{\nu \}}}}=\aleph _{0}$.

The equation (6) can also be written

$\aleph _{0}\cdot 2=\aleph _{0}$;

and, by adding $\aleph _{0}$ repeatedly to both sides, we find that

We also have

Proof.—By (6) of §3, $\aleph _{0}\cdot \aleph _{0}$ is the cardinal number of the aggregate of bindings

$\{(\mu ,\nu )\}$,

where $\mu$ and $\nu$ are any finite cardinal numbers which are independent of one another. If also $\lambda$ represents any finite cardinal number, so that $\{\lambda \}$, $\{\mu \}$, and $\{\nu \}$ are only different notations for the same aggregate of all finite numbers, we have to show that

$\{(\mu ,\nu )\}\sim \{\lambda \}$.

Let us denote $\mu +\nu$ by $\rho$; then $\rho$ takes all the numerical values $2,3,4,...$, and there are in all $\rho -1$ elements $(\mu ,\nu )$ for which $\mu +\nu =\rho$, namely:

$(1,\rho -1),(2,\rho -2),...,(\rho -1,1)$.

In this sequence imagine first the element $(1,1)$, for which $\rho =2$, put, then the two elements for which $\rho =3$, then the three elements for which $\rho =4$, and so on. Thus we get all the elements $(\mu ,\nu )$ in a simple series:

$(1,1);(1,2),(2,1);(1,3),(2,2),(3,1);(1,4),(2,3),...$,

and here, as we easily see, the element $(\mu ,\nu )$ comes at the $\lambda$th place, where

The variable $\lambda$ takes every numerical value $1,2,3,...$, once. Consequently, by means of (9), a reciprocally univocal relation subsists between the aggregates $\{\nu \}$ and $\{(\mu ,\nu )\}$.

[495] If both sides of the equation (8) are multiplied by $\aleph _{0}$, we get $\aleph _{0}^{3}=\aleph _{0}^{2}=\aleph _{0}$, and, by repeated multiplications by $\aleph _{0}$, we get the equation, valid for every finite cardinal number $\nu$:

The theorems E and A of §5 lead to this theorem on finite aggregates:

C. Every finite aggregate $E$ is such that it is equivalent to none of its parts.

This theorem stands sharply opposed to the following one for transfinite aggregates:

D. Every transfinite aggregate $T$ is such that it has parts $T_{1}$ which are equivalent to it.

Proof.—By theorem A of this paragraph there is a part $S=\{t_{\nu }\}$ of $T$ with the cardinal number $\aleph _{0}$. Let $T=(S,U)$, so that $U$ is composed of those elements of $T$ which are different from the elements $t_{\nu }$. Let us put $S_{1}=\{t_{\nu +1}\}$, $T_{1}=(S-1,U)$; then $T_{1}$ is a part of $T$, and, in fact, that part which arises out of $T$ if we leave out the single element $t_{1}$. Since $S\sim S_{1}$, by theorem B of this paragraph, and $U\sim U$, we have, by §1, $T\sim T_{1}$.

In these theorems C and D the essential difference between finite and transfinite aggregates, to which I referred in the year 1877, in volume lxxxiv [1878] of Crelle's Journal, p. 242, appears in the clearest way.

After we have introduced the least transfinite cardinal number $\aleph _{0}$ and derived its properties that lie the most readily to hand, the question arises as to the higher cardinal numbers and how they proceed from $\aleph _{0}$. We shall show that the trans- finite cardinal numbers can be arranged according to their magnitude, and, in this order, form, like the finite numbers, a "well-ordered aggregate" in an extended sense of the words. Out of $\aleph _{0}$ proceeds, by a definite law, the next greater cardinal number $\aleph _{1}$, out of this by the same law the next greater $\aleph _{2}$ and so on. But even the unlimited sequence of cardinal numbers

$\aleph _{0},\aleph _{1},\aleph _{2},...,\aleph _{\nu },...$

does not exhaust the conception of transfinite cardinal number. We will prove the existence of a cardinal number which we denote by $\aleph _{\omega }$ and which shows itself to be the next greater to all the numbers $\aleph _{\nu }$; out of it proceeds in the same way as $\aleph _{1}$ out of $\aleph$ a next greater $\aleph _{\omega +1}$, and so on, without end.

[496] To every transfinite cardinal number ${\mathfrak {a}}$ there is a next greater proceeding out of it according to a unitary law, and also to every unlimitedly ascending well-ordered aggregate of transfinite cardinal numbers, $\{{\mathfrak {a}}\}$, there is a next greater proceeding out of that aggregate in a unitary way.

For the rigorous foundation of this matter, discovered in 1882 and exposed in the pamphlet Grundlagen einer allgemeinen Mannichfaltigkeitslehre (Leipzig, 1883) and in volume xxi of the Mathematische Annalen we make use of the so-called "ordinal types" whose theory we have to set forth in the following paragraphs.

§7
The Ordinal Types of Simply Ordered Aggregates

We call an aggregate $M$ "simply ordered" if a definite "order of precedence" (Rangordnung) rules over its elements $m$, so that, of every two elements $m_{1}$ and $m_{2}$, one takes the "lower" and the other the "higher" rank, and so that, if of three elements $m_{1}$, $m_{2}$, and $m_{3}$, $m_{1}$, say, is of lower rank than $m_{2}$, and $m_{2}$ is of lower rank than $m_{3}$, then $m_{1}$ is of lower rank than $m_{3}$.

The relation of two elements $m_{1}$ and $m_{2}$, in which $m_{1}$ has the lower rank in the given order of precedence and $m_{2}$ the higher, is expressed by the formulæ:

Thus, for example, every aggregate $P$ of points defined on a straight line is a simply ordered aggregate if, of every two points $p_{1}$ and $p_{2}$ belonging to it, that one whose co-ordinate (an origin and a positive direction having been fixed upon) is the lesser is given the lower rank.

It is evident that one and the same aggregate can be "simply ordered" according to the most different laws. Thus, for example, with the aggregate $R$ of all positive rational numbers $p/q$ (where $p$ and $q$ are relatively prime integers) which are greater than $0$ and less than $1$, there is, firstly, their "natural" order according to magnitude; then they can be arranged (and in this order we will denote the aggregate by $R_{0}$) so that, of two numbers $p_{1}/q_{1}$ and $p_{2}/q_{2}$ which the sums $p_{1}+q_{1}$ and $p_{2}+q_{2}$ have different values, that number for which the corresponding sum is less takes the lower rank, and, if $p_{1}+q_{1}=p_{2}+q_{2}$ then the smaller of the two rational numbers is the lower. [497] order of precedence, our aggregate, since to one and the same value of $p+q$ only a finite number of rational numbers $p/q$ belongs, evidently has the form

$R_{0}=(r_{1},r_{2},...,r_{\nu },...)=({\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},{\tfrac {2}{3}},{\tfrac {1}{5}},{\tfrac {1}{6}},{\tfrac {2}{5}},{\tfrac {3}{4}},...)$,

where

$r_{\nu }<r_{\nu +1}$.

Always, then, when we speak of a "simply ordered" aggregate $M$, we imagine laid down a definite order or precedence of its elements, in the sense explained above.

There are doubly, triply, $\nu$-ply and ${\mathfrak {a}}$-ply ordered aggregates, but for the present we will not consider them. So in what follows we will use the shorter expression "ordered aggregate" when we mean "simply ordered aggregate."

Every ordered aggregate $M$ has a definite "ordinal type," or more shortly a definite "type," which we will denote by

By this we understand the general concept which

results from $M$ if we only abstract from the nature of the elements $m$, and retain the order of precedence among them. Thus the ordinal type ${\overline {M}}$ is itself an ordered aggregate whose elements are units which have the same order of precedence amongst one another as the corresponding elements of $M$, from which they are derived by abstraction.

We call two ordered aggregates $M$ and $N$ "similar" (ähnlich) if they can be put into a biunivocal correspondence with one another in such a manner that, if $m_{1}$ and $m_{2}$ are any two elements of $M$ and $n_{1}$ and $n_{2}$ the corresponding elements of $N$, then the relation of rank of $m_{1}$ to $m_{2}$ in $M$ is the same as that of $n_{1}$ to $n_{2}$ in $N$. Such a correspondence of similar aggregates we call an "imaging" (Abbildung) of these aggregates on one another. In such an imaging, to every part—which obviously also appears as an ordered aggregate—$M_{1}$ of $M$ corresponds a similar part $N_{1}$ of $N$.

We express the similarity of two ordered aggregates $M$ and $N$ by the formula:

Every ordered aggregate is similar to itself.

If two ordered aggregates are similar to a third, they are similar to one another.

[498] A simple consideration shows that two ordered aggregates have the same ordinal type if, and only if, they are similar, so that, of the two formulæ

one is always a consequence of the other.

If, with an ordinal type ${\overline {M}}$ we also abstract from the order of precedence of the elements, we get (§1) the cardinal number ${\overline {\overline {M}}}$ of the ordered aggregate $M$, which is, at the same time, the cardinal number of the ordinal type ${\overline {M}}$. From ${\overline {M}}={\overline {N}}$ always follows ${\overline {\overline {M}}}={\overline {\overline {N}}}$, that is to say, ordered aggregates of equal types always have the same power or cardinal number; from the similarity of ordered aggregates follows their equivalence. On the other hand, two aggregates may be equivalent without being similar.

We will use the small letters of the Greek alphabet to denote ordinal types. If a is an ordinal type, we understand by

its corresponding cardinal number.

The ordinal types of finite ordered aggregates offer no special interest. For we easily convince ourselves that, for one and the same finite cardinal number $\nu$, all simply ordered aggregates are similar to one another, and thus have one and the same type. Thus the finite simple ordinal types are subject to the same laws as the finite cardinal numbers, and it is allowable to use the same signs $1,2,3,...,\nu ,...$ for them, although they are conceptually different from the cardinal numbers. The case is quite different with the transfinite ordinal types; for to one and the same cardinal number belong innumerably many different types of simply ordered aggregates, which, in their totality, constitute a particular "class of types" (Typenclasse), Every one of these classes of types is, therefore, determined by the transfinite cardinal number a which is common to all the types belonging to the class. Thus we call it for short the class of types $[{\mathfrak {a}}]$. That class which naturally presents itself first to us, and whose complete investigation must, accordingly, be the next special aim of the theory of transfinite aggregates, is the class of types $[\aleph _{0}]$ which embraces all the types with the least transfinite cardinal number $\aleph _{0}$. From the cardinal number which determines the class of types $[{\mathfrak {a}}]$ we have to distinguish that cardinal number ${\mathfrak {a}}'$ which for its part [499] is determined by the class of types $[{\mathfrak {a}}]$. The latter is the cardinal number which (§1) the class $[{\mathfrak {a}}]$ has, in so far as it represents a well-defined aggregate whose elements are all the types a with the cardinal number ${\mathfrak {a}}$. We will see that ${\mathfrak {a}}'$ is different from ${\mathfrak {a}}$, and indeed always greater than ${\mathfrak {a}}$.

If in an ordered aggregate M all the relations of precedence of its elements are inverted, so that "lower" becomes "higher" and "higher" becomes "lower" everywhere, we again get an ordered aggregate, which we will denote by

and call the "inverse " of $M$. We denote the ordinal type of $^{*}\!M$, if $\alpha ={\overline {M}}$, by

It may happen that $^{*}\!\alpha =\alpha$, as, for example, in the

case of finite types or in that of the type of the aggregate of all rational numbers which are greater than $0$ and less than $1$ in their natural order of precedence. This type we will investigate under the notation $\eta$.

We remark further that two similarly ordered aggregates can be imaged on one another either in. one manner or in many manners; in the first case the type in question is similar to itself in only one way, in the second case in many ways. Not only all finite types, but the types of transfinite "well-ordered aggregates," which will occupy us later and which we call transfinite "ordinal numbers," are such that they allow only a single imaging on themselves. On the other hand, the type $\eta$ is similar to itself in an infinity of ways.

We will make this difference clear by two simple examples. By $\omega$ we understand the type of a well-ordered aggregate

$(e_{1},e_{2},...,e_{\nu },...)$,

in which

$e_{\nu }<e_{\nu +1}$,

and where $\nu$ represents all finite cardinal numbers in turn. Another well-ordered aggregate

$(f_{1},f_{2},...,f_{\nu },...)$,

with the condition

$f_{\nu }<f_{\nu +1}$,

of the same type $\omega$ can obviously only be imaged on the former in such a way that $e_{\nu }$ and $f_{\nu }$ are corresponding elements. For $e_{1}$ the lowest element in rank of the first, must, in the process of imaging, be correlated to the lowest element $f_{1}$ of the second, the next after $e_{1}$ in rank $(e_{2})$ to $f_{2}$, the next after $f_{1}$, and so on. [500] Every other bi-univocal correspondence of the two equivalent aggregates $\{e_{\nu }\}$ and $\{f_{\nu }\}$ is not an "imaging" in the sense which we have fixed above for the theory of types.

On the other hand, let us take an ordered aggregate of the form

$\{e_{\nu }\}$,

where $\nu$ represents all positive and negative finite integers, including $0$, and where likewise

$e_{\nu }<e_{\nu +1}$.

This aggregate has no lowest and no highest element in rank. Its type is, by the definition of a sum given in §8,

$^{*}\!\omega +\omega$.

It is similar to itself in an infinity of ways. For let us consider an aggregate of the same type

$\{f_{\nu '}\}$,

where

$f_{\nu '}<f_{\nu '+1}$.

Then the two ordered aggregates can be so imaged on one another that, if we understand by $\nu _{0}'$ a definite one of the numbers $\nu '$, to the element $e_{\nu '}$ of the first the element $f_{\nu _{0}'+\nu '}$ of the second corresponds. Since $\nu _{0}'$ is arbitrary, we have here an infinity of imagings.

The concept of "ordinal type" developed here, when it is transferred in like manner to "multiply ordered aggregates," embraces, in conjunction with the concept of "cardinal number" or "power" introduced in §1, everything capable of being numbered (Anzahlmässige) that is thinkable, and in this sense cannot be further generalized. It contains nothing arbitrary, but is the natural extension of the concept of number. It deserves to be especially emphasized that the criterion of equality (4) follows with absolute necessity from the concept of ordinal type and consequently permits of no alteration. The chief cause of the grave errors in G. Veronese's Grundzüge der Geometrie (German by A. Schepp, Leipzig, 1894) is the non-recognition of this point.

On page 30 the "number (Anzahl oder Zahl) of an ordered group" is defined in exactly the same way as what we have called the "ordinal type of a simply ordered aggregate" (Zur Lehre vom Transfiniten, Halle, 1890, pp. 68-75; reprinted from the Zeitschr. für Philos. und philos. Kritik for 1887). [501] But Veronese thinks that he must make an addition to the criterion of equality. He says on page 31: "Numbers whose units correspond to one another uniquely and in the same order and of which the one is neither a part of the other nor equal to a part of the other are equal."^[3] This definition of equality contains a circle and thus is meaningless. For what is the meaning of "not equal to a part of the other" in this addition? To answer this question, we must first know when two numbers are equal or unequal. Thus, apart from the arbitrariness of his definition of equality, it presupposes a definition of equality, and this again presupposes a definition of equality, in which we must know again what equal and unequal are, and so on ad infinitum. After Veronese has, so to speak, given up of his own free will the indispensable foundation for the comparison of numbers, we ought not to be surprised at the lawlessness with which, later on, he operates with his pseudo-transfinite numbers, and ascribes properties to them which they cannot possess simply because they themselves, in the form imagined by him, have no existence except on paper. Thus, too, the striking similarity of his "numbers" to the very absurd "infinite numbers" in Fontenelle's Géométrie de l'Infini (Paris, 1727) becomes comprehensible. Recently, W. Killing has given welcome expression to his doubts concerning the foundation of Veronese's book in the Index lectionum of the Münster Academy for 1895-1896.^[4]

§8
Addition and Multiplication of Ordinal Types

The union-aggregate $(M,N)$ of two aggregates $M$ and $N$ can, if $M$ and $N$ are ordered, be conceived as an ordered aggregate in which the relations of precedence of the elements of $M$ among themselves as well as the relations of precedence of the elements of $N$ among themselves remain the same as in $M$ or $N$ respectively, and all elements of $M$ have a lower rank than all the elements of $N$. If $M'$ and $N'$ are two other ordered aggregates, $M\simeq M'$ and $N\simeq N'$, [502] then $(M,N)\simeq (M',N')$; so the ordinal type of $(M,N)$ depends only on the ordinal types $M=\alpha$ and $N=\beta$. Thus, we define:

In the sum $\alpha +\beta$ we call $\alpha$ the "augend" and $\beta$ the "addend."

For any three types we easily prove the associative law:

On the other hand, the commutative law is not valid, in general, for the addition of types. We see this by the following simple example.

If $\omega$ is the type, already mentioned in §7, of the well-ordered aggregate

$E=(e_{1},e_{2},...,e_{\nu },...),\quad e_{\nu +1}$,

then $1+\omega$ is not equal to $\omega +1$. For, if $f$ is a new

element, we have by (1):

$1+\omega =({\overline {f,E}})$,
$\omega +1=({\overline {E,f}})$.

But the aggregate

$(f,E)=(f,e_{1},e_{2},...,e_{\nu },...)$

is similar to the aggregate E, and consequently

$1+\omega =\omega$.

On the contrary, the aggregates $E$ and $(E,f)$ are not similar, because the first has no term which is highest in rank, but the second has the highest term $f$. Thus $\omega +1$ is different from $\omega =1+\omega$.

Out of two ordered aggregates $M$ and $N$ with the types $\alpha$ and $\beta$ we can set up an ordered aggregate $S$ by substituting for every element $n$ of $N$ an ordered aggregate $M_{n}$ which has the same type $\alpha$ as $M$, so that

and, for the order of precedence in

we make the two rules:

(1) Every two elements of $S$ which belong to one and the same aggregate $M_{n}$ are to retain in $S$ the same order of precedence as in $M_{n}$;

(2) Every two elements of $S$ which belong to two different aggregates $M_{n_{1}}$ and $M_{n_{2}}$ have the same relation of precedence as $n_{1}$ and $n_{2}$ have in $N$.

The ordinal type of $S$ depends, as we easily see, only on the types $\alpha$ and $\beta$; we define

[503] In this product $\alpha$ is called the "multiplicand" and $\beta$ the "multiplier."

In any definite imaging of $M$ on $M_{n}$ let $m_{n}$ be the element of $M_{n}$ that corresponds to the element $m$ of $M$; we can then also write

Consider a third ordered aggregate $P=\{p\}$ with

$\alpha \,.\beta ={\overline {\{m_{n}\}}},\quad \beta \,.\gamma ={\overline {\{n_{p}\}}},\quad (\alpha \,.\beta )\,.\gamma ={\overline {\{(m_{n})_{p}\}}},\quad \alpha \,.(\beta \,.\gamma )={\overline {\{m_{(n_{p})}\}}}$.

But the two ordered aggregates $\{(m_{n})_{p}\}$ and $\{m_{(n_{p})}\}$ are similar, and are imaged on one another if we regard the elements $(m_{n})_{p}$ and $m_{(n_{p})}$ as corresponding.

Consequently, for three types $\alpha$, $\beta$, and $\gamma$ the associative law

subsists. From (1) and (5) follows easily the distributive law

but only in this form, where the factor with two terms is the multiplier.

On the contrary, in the multiplication of types as in their addition, the commutative law is not generally valid. For example, $2\,.\omega$ and $\omega \,.2$ are different types; for, by (5),

$2\,.\omega ={\overline {(e_{1},f_{1};e_{2},f_{2};...;e_{\nu },f_{\nu };...)}}=\omega$;

while

$\omega \,.2={\overline {(e_{1},e_{2},...,e_{\nu },...;f_{1},f_{2},...,f_{\nu },...)}}$

is obviously different from $\omega$.

If we compare the definitions of the elementary operations for cardinal numbers, given in §3, with those established here for ordinal types, we easily see that the cardinal number of the sum of two types is equal to the sum of the cardinal numbers of the single types, and that the cardinal number of the product of two types is equal to the product of the cardinal numbers of the single types. Every equation between ordinal types which proceeds from the two elementary operations remains correct, therefore, if we replace in it all the types by their cardinal numbers.

The Ordinal Type $\eta$ of the Aggregate $R$ of all Rational Numbers which are Greater than $0$ and Smaller than $1$, in their Natural Order of Precedence

By $R$ we understand, as in §7, the system of all rational numbers $p/q$ ($p$ and $q$ being relatively prime) which $>0$ and $<1$, in their natural order of precedence, where the magnitude of a number determines its rank. We denote the ordinal type of $R$ by $\eta$:

But we have put the same aggregate in another order of precedence in which we call it $R_{0}$. This order is determined, in the first place, by the magnitude of $p+q$, and in the second place—for rational numbers for which $p+q$ has the same value—by the magnitude of $p/q$ itself. The aggregate $R_{0}$ is a well-ordered aggregate of type $\omega$:

Both $R$ and $R_{0}$ have the same cardinal number since they only differ in the order of precedence of their elements, and, since we obviously have ${\overline {\overline {R_{0}}}}=\aleph _{0}$, we also have

Thus the type $\eta$ belongs to the class of types $[\aleph _{0}]$.

Secondly, we remark that in $R$ there is neither an element which is lowest in rank nor one which is highest in rank. Thirdly, $R$ has the property that between every two of its elements others lie. This property we express by the words: $R$ is "everywhere dense" (überalldicht).

We will now show that these three properties characterize the type $\eta$ of $R$, so that we have the following theorem: Page:Cantortransfinite.djvu/143 Page:Cantortransfinite.djvu/144 Page:Cantortransfinite.djvu/145 Page:Cantortransfinite.djvu/146 Page:Cantortransfinite.djvu/147 Page:Cantortransfinite.djvu/148 Page:Cantortransfinite.djvu/149 Page:Cantortransfinite.djvu/150 Page:Cantortransfinite.djvu/151 Page:Cantortransfinite.djvu/152 Page:Cantortransfinite.djvu/153 Page:Cantortransfinite.djvu/154 Page:Cantortransfinite.djvu/155

1. [In English there is an ambiguity.]
2. [See Section V of the Introduction.]
3. In the original Italian edition (p. 27) this passage runs : "Numeri le unità dei quali si corrispondono univocamente e nel medesimo ordine, e di cui l' uno non è parte o uguale ad una parte dell' altro, sono uguali."
4. [Veronese replied to this in Math. Ann. vol. xlvii, 1897, pp. 423-432. Cf. Killing, ibid., vol. xlviii, 1897, pp. 425-432.]