Cantor’s
“Beiträge zur Begründung der transfiniten Mengenlehre”
(Contributions to The Founding of the Theory of Transfinite Numbers)
• English Translation •

This is a new (2024) English translation of Sections 1 to 6 of Cantor’s 1895 paper “Beiträge zur Begründung der transfiniten Mengenlehre”. In these sections Cantor summarizes the key foundations of his theory of transfinite cardinal numbers. There is an older translation (1915) of the complete papers by Philip Jourdain, viewable online at “Contributions to the Foundations of the Theory Of Transfinite Numbers”, but it is a scanned copy without cross-references, and the terminology is dated, and at one point Jourdain incorrectly translates an unproven proposition as a theorem. This translation uses current terminology wherever possible to make it easier to read for modern readers.

English translation by James R Meyer, copyright 2024 www.jamesrmeyer.com

Note: The footnotes are not Cantor’s, they are provided as additional comments to Cantor’s text.

INDEX

Contributions to The Founding of the Theory of Transfinite Numbers: First Article
by
Georg Cantor, 1895

§ 1
The Concept of Power or Cardinal Number

By a “set” we understand any collection into a whole M of definite and separate objects m of our intuition or our thought. These objects are called the “elements” of M. We express this symbolically by:

We denote the uniting of multiple sets M, N, P, … , which have no elements in common, into a single set by: (Footnote: Cantor’s original terminology was (M, N, P, …). Here the set union symbol is used with an underline to indicate that the sets are to have no common elements.)

The elements of this single set are, therefore, the elements of M, N, P, … all taken together.

A set M₁ is a “proper subset” of the set M if it is not the same set as M and if all of its elements are also elements of M. If M₂ is a proper subset of M₁ and M₁ is a proper subset of M, then M₂ is a proper subset of M.

Every set M has a definite “magnitude” which we will also call its “cardinal number.”

The term “magnitude” of M or “cardinal number” of M will be used to denote the general concept which, by means of our active faculty of thought, emerges from the set M by an abstraction from the nature of its various elements m and of the order in which they are given.

We denote the result of this double act of abstraction, the cardinal-number or magnitude of M, by: (Footnote: This should display with a double overline. If it does not show correctly, a different browser should be tried.)

Since every single element m, regardless of its nature, becomes a “unit,” the cardinal number M is a definite set composed entirely of units, and has an existence in our minds as a mental image or projection of the given set M.

To every proper subset M₁ of M there corresponds then a definite equivalent proper subset N₁ of N, and the inverse also applies.

If we have a rule for the one-to-one correspondence of two equivalent sets, then, apart from the case when each of them consists only of one element, we can modify this rule in various ways. We can, for example, always ensure that for some specific element m₀ of M, a specific element n₀ of N corresponds to it. For if by the original rule, the elements m₀ and n₀ do not correspond to one another, but the element n₁ of N corresponds to the element m₀ of M, and the element m₁ of M corresponds to the element n₀ of N, we create the modified rule, by which m₀ corresponds to n₀ and m₁ to n₁ and for the other elements the original rule remains unaltered. By this means the desired goal is attained.

Every set is equivalent to itself:

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

Of fundamental importance is the assertion that two sets M and N have the same cardinal number if, and only if, they are equivalent, i.e:

and

In this way, the equivalence of sets can give the necessary and sufficient conditions for the equality of their cardinal numbers.

In fact, according to the above definition of magnitude, the cardinal number 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 M ∼ N, then there exists a rule for the one-to-one correspondence whereby M and N are uniquely and reciprocally related to one another, where each element m of M corresponds an element n of N. We can then imagine, that in place of every element m of M, the corresponding element n of N is substituted, and in this way, M is transformed into N without any alteration of the cardinal number. Therefore:

M = N

The converse of the theorem results from the observation that there is a mutually unique relationship between the elements of M and the individual units of its cardinal number M. For, as we saw, M emerges from M in such a way that every element m of M becomes a specific unit of M. We can therefore say that:

Similarly N ∼ N, and if M ∼ N then it follows by (6) that M ∼ N.

We emphasize the following theorem, which follows immediately from the conception of equivalence.

If M, N, P, … are sets which have no common elements, and the same applies to M′, N′, P′, … , and if

M ∼ M′,   N ∼ N′,   P ∼ P′,   …

then we always have:

M ∪ N ∪ P ∪ …   ∼   M′ ∪ N′ ∪ P′ ∪ …

§ 2:
“Greater” and “Less” for Magnitudes

If two sets M and N with the cardinal numbers 𝔞 = M and 𝔟 = N satisfy both of the conditions:

1. There is no proper subset of M which is equivalent to N, and
2. There is a proper subset N₁ of N, such that N₁ ∼ M,

then it is obvious that these conditions still hold if M and N are replaced by two equivalent sets M′ and N′.

Thus the sets give rise to a definite relationship of the cardinal numbers 𝔞 and 𝔟 to one another.

Further, the equivalence of M and N, that is, the equality of 𝔞 and 𝔟, is excluded; for if we had M ∼ N, we would have N₁ ∼ N since N₁ ∼ M, and then, since M ∼ N, there would exist a proper subset M₁ of M such that M₁ ∼ M and also M₁ ∼ N, and this contradicts the condition (1).

And thirdly, the relationship of 𝔞 to 𝔟 is such that it renders impossible the same relationship of 𝔟 to 𝔞, since if in (1) and (2) the roles of M and N are interchanged, two conditions then arise which are contradictory to the former ones.

We express the relation of 𝔞 to 𝔟, as characterized in (1) and (2) by:

𝔞 is “less” than 𝔟, or 𝔟 is “greater” than 𝔞.

Symbolically we have:

We can easily prove that:

Similarly, it follows immediately from the definition that, if P₁ is a proper subset of a set P, then 𝔞 < P₁ implies that 𝔞 < P, and P < 𝔟 implies that P₁ < 𝔟.

We have seen that, of the three relations

𝔞 = 𝔟,   𝔞 < 𝔟,   𝔟 < 𝔞

each one excludes the two others. On the other hand, the theorem that, with any two cardinal numbers 𝔞 and 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 an overview of the ascending sequence of the transfinite cardinal numbers and an insight into their connection, will the truth of the following theorems be revealed:

Theorem A: If 𝔞 and 𝔟 are any two cardinal numbers, then either 𝔞 = 𝔟 or 𝔞 < 𝔟 or 𝔟 < 𝔞.

From this theorem the following theorems can be derived, although they will not be used at this stage:

Theorem B: If two sets M and N are such that M is equivalent to a proper subset N₁ of N and N is equivalent to a proper subset M₁ of M, then M and N are equivalent.

Theorem C: If M₁ is a proper subset of a set M and M₂ is a proper subset of the set M₁, and if the sets M and M₂ are equivalent, then M₁ is equivalent to both M and M₂.

Theorem D: For two sets M and N, if N is not equivalent to either M nor to a proper subset of M, then there is a proper subset N₁ of N that is equivalent to M.

Theorem E: If two sets M and N are not equivalent, and there is a proper subset N₁ of N that is equivalent to M, then there is no proper subset of M that is equivalent to N.

§ 3
The Addition and Multiplication of Magnitudes

The union of two sets M and N was given in Section 1 (2) by M ∪ N. We call it the “union” of M and N. If M and N have no common elements, and M′ and N′ are two other sets without common elements, and if M ∼ M′ and N ∼ N′, we saw that we have:

M ∪ N ∼ M′ ∪ N′.

Hence the cardinal number of M ∪ N depends only upon the cardinal numbers M = 𝔞 and N = 𝔟.

This leads to the definition of the sum of 𝔞 and 𝔟 by:

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

and, for any three cardinal numbers 𝔞, 𝔟, 𝔠 we have

We now come to multiplication. Any element m of a set M can be considered as being connected to any element n of another set N so as to form a new element (m, n). We denote the set of all possible such pairings (m, n) by (M • N), and we call it the “set of all pairings of M and N”. So: (Footnote: The intention here is that this represents the concept: (M • N) = {(m₁, n₁), (m₁, n₂), … , (m₂, n₁), (m₂, n₂), … , (m₃, n₁), (m₃, n₂), … })

We see that the magnitude of (M • N) depends only on the magnitudes M = 𝔞 and N = 𝔟 since if we replace the sets M and N by the equivalent sets:

M′= {m′} and N′={n′}

and if we consider each (m, n) and (m′, n′) as corresponding elements, then the set:

(M′ • N′) = {(m′, n′)}

is thereby brought into a definite one-to-one correspondence to (M • N) by such consideration of each (m, n) and (m′, n′) as corresponding elements. Therefore:

We can now define the product 𝔞 • 𝔟 by the equation:

A set with the cardinal number 𝔞 • 𝔟 may also be given from two sets M and N with the cardinal numbers 𝔞 and 𝔟 according to the following rule: Start with the set N and replace every element n in it by a set M_n ∼ M. If the elements of each of these sets M_n are now combined to make a new set S, we can easily see that: (Footnote: There is an implied intention here that there are no elements common to any two M_n.)

and hence

S ∼ 𝔞 • 𝔟

If, for any one-to-one correspondence of the two equivalent sets 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 sets S and (M • N) can be set in a one-to-one correspondence to one another by considering m_n and (m, n) as corresponding elements. From our definitions follow these results:

since:

(M • N) ∼ (N • M)

(M • (N • P)) ∼ ((M • N) • P)

(M • (N ∪ P)) ∼ ((M • N) ∪ (M • P))

Hence the addition and multiplication of magnitudes are subject to the commutative, associative, and distributive laws

§ 4
The Exponentiation of Magnitudes

By a “covering of the set N with elements of the set M” or, more simply by a “covering of N with M”, we mean a definition by which a specific element of M is bound to each element n of N , and where that same element of M can be bound repeatedly to each element of N. The element of M bound to n is in a certain sense a one-valued function of n and may be denoted by f (n), and it is called a “covering function of n”. The corresponding covering of N is denoted by f (N). (Footnote: For example, if N = {x, y, z} and M = {0, 1} the coverings of N may be given as: f₁(N) = 000, f₂(N) = 001, f₃(N) = 010, f₄(N) = 100, etc, so that the covering of N with M, i.e, the covering of {x, y, z} with {0, 1} is {000, 001, 010, 100, 011, 101, 110, 111}.)

Two coverings f₁(N) and f₂(N) are said to be equal if, and only if, for all elements n of N the equation

is satisfied, so that if for even one single element n = n₀ this equation does not hold, f₁(N) and f₂(N) are characterized as different coverings of N.

For example, if m₀ is a particular element of M, we may define that, for all n’s that:

f (n) = m₀

so that this definition constitutes a specific covering of N with M.

Another kind of covering results if m₀ and m₁ are two different specific elements of M and n₀ a specific element of N, by defining that:

f (n₀) = m₀
f (n) = m₁

for all n’s which are different to n₀.

The totality of different coverings of N with M forms a definite set with the elements f (n). We call it the “covering-set of N with M” and denote it by (N | M). So we have:

If M ∼ M′ and N ∼ N we easily find that:

Therefore the cardinal number of (N | M) depends only on the cardinal numbers M = 𝔞 and N = 𝔟, and we can use it for the definition of 𝔞^𝔟:

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

from which, if we put P = 𝔠, then by (4) and by consideration of the previous Section 3, we have the following theorems for any three cardinal numbers, 𝔞, 𝔟, and 𝔠:

The following example shows how comprehensive and far-reaching these simple formulas are when extended to the magnitudes:

§ 5
The Finite Cardinal Numbers

First, we will show how the principles presented, and on which the theory of actually infinite or transfinite cardinal numbers is to be built, also provide the most natural, shortest and most rigorous foundation for the theory of finite numbers.

If we subsume a single object e₀ under the concept of a set E₀ = {e₀ }, the cardinal number corresponds to what we call “one” and we designate it by 1, we have:

If we now unite E₀ with another object e₁ and call the union-set E₁ so that:

The cardinal number of E₁ is called “two” and is denoted by 2:

By addition of new elements we obtain the series of sets:

E₂ = E₁ ∪ e₁, E₃ = E₂ ∪ e₃, …

which give us successively, in an limitless sequence, the rest of what we call the finite cardinal numbers, denoted by 3, 4, 5, … . We justify the use of these numbers here as suffixes by the fact that a number is only used here as a suffix when it has already been defined as a cardinal number. If v - 1 is understood to be the number immediately preceding v in the above series, we have:

in other words, every cardinal number except 1, is the sum of the immediately preceding cardinal number and 1. The following three theorems now come to the fore:

Theorem A: The terms of the limitless series of finite cardinal numbers

1, 2, 3, … , v, …

are all different to one another (i.e: the condition of equivalence established in Section 1 is not satisfied between the corresponding sets).

Theorem B: Every one of these numbers v is greater than the preceding numbers and less than the following numbers (Section 2).

Theorem C: There are no cardinal numbers which in magnitude are between two consecutive numbers v and v + 1 (Section 2).

We base the proofs of these theorems on the following two theorems D and E, which will be proven below:

Theorem D: If M is a set such that none of its proper subsets has the same magnitude as M, then the set M ∪ {e}, which arises from M by the addition of a single new element, has that same property that none of its proper subsets has the same magnitude as it.

Theorem E: If N is a set with the finite cardinal number v, and N₁ is any proper subset of N, the cardinal number of N₁ is equal to one of the preceding numbers 1, 2, 3, … , v - 1.

Proof of Theorem D:

Suppose that the set M ∪ {e} is equivalent to one of its proper subsets which we will call N. Then two cases are possible, both of which lead to a contradiction:

1. The set N contains e as element ; let N = M₁ ∪ {e}; then M₁ is a proper subset of M because N is a proper subset of M ∪ {e}. As we saw in Section 1, the definition of a one-to-one correspondence between the two equivalent sets M ∪ {e} and M₁ ∪ {e} can be modified so that the element e of the one corresponds to the same element e of the other. Hence, M and M₁ are in a one-to-one correspondence to one another. But this contradicts the requirement that M is not equivalent to its proper subset M₁.
2. The proper subset N of M ∪ {e} does not contain e as element, so that N is either M or a proper subset of M. In the definition of the one-to-one correspondence between M ∪ {e} and N, which is the basis of our supposition, then some element f of N must correspond to the element e of M ∪ {e}. Let N = M₁ ∪ { f }. Then the set M will also be in a one-to-one correspondence with M₁. But M₁ is a proper subset of N and hence it is also a proper subset of M. So in this case M would also be equivalent to one of its proper subsets, which contradicts the original supposition.

Proof of Theorem E:

We assume the theorem to be true up to a certain v and then prove it for the number v₁ + 1 which immediately follows, in the following manner:

We start with the set E_v = {e₀ } ∪ {e₁} ∪ … ∪ {e_v} as having the cardinal number v₁ + 1. If the theorem is true for this set, its truth for any other set with the same cardinal number v₁ + 1 follows at once by Section 1. Let E′ be any proper subset of E_v and we consider the following cases:

1. E′ does not contain e_v as element, then E′ is either E_v - 1 or a proper subset of E_v - 1 and so has as cardinal number either v or one of the numbers 1, 2, 3, … , v₁ - 1, because we assumed the theorem to be true for the set E_v - 1 with the cardinal number v.
2. E′ consists of the single element e_v then E′ = 1.
3. E′ consists of e_v and a set E′′ so that E′ = E′′ ∪ {e_v}. So E′′ is a proper subset of E_v - 1 and therefore, by the supposition, its cardinal number is one of the numbers 1, 2, 3, … , v₁ - 1. But now E′ = E′′ + 1, and thus the cardinal number of E′ is one of the numbers 2, 3, … , v.

Proof of Theorem A (The terms of the limitless series of finite cardinal numbers 1, 2, 3, … , v, … are all different to one another):

Every one of the sets which we have denoted by E_v has the property of not being equivalent to any of its proper subsets. If we suppose that this applies up to a certain v, it follows from theorem D that it also applies for the immediately following number v₁ + 1. For v = 1, we see at once that the set E₁ = {e₀ } ∪ {e₁} is not equivalent to any of its proper subsets, which are {e₀ } and {e₁}. Consider now any two numbers μ and v of the series 1, 2, 3, … . If μ is earlier in the series and v is later, then E_μ - 1 is a proper subset of E_v - 1. Therefore E_μ - 1 and E_v - 1 are not equivalent, and accordingly their cardinal numbers μ = E_μ - 1 and E_v - 1 are not equal.

Proof of Theorem B (Every one of these numbers v is greater than the preceding numbers and less than the following numbers):

If μ is earlier in the of the series 1, 2, 3, … , and v is later, then μ < v, since both of the two sets M = E_μ - 1 and N = E_v - 1 satisfy the two conditions for M < N as in Section 2. Condition (1) is satisfied because, by theorem E a proper subset of M = E_μ - 1 can only have one of the cardinal numbers 1, 2, 3, … , μ − 1 and therefore, by theorem A it cannot be equivalent to the set N = E_v - 1. Condition (2) is satisfied because M itself is a proper subset of N.

Proof of Theorem C (There are no cardinal numbers which in magnitude are between two consecutive numbers v and v + 1):

Let 𝔞 be a cardinal number which is less than v + 1. By condition (2) of Section 2, there is a proper subset of E_v with the cardinal number 𝔞. By theorem E, a proper subset of E_v can only have one of the cardinal numbers in 1, 2, 3, … , v. Hence 𝔞 is equal to one of the cardinal numbers 1, 2, 3, … , v. By theorem B, none of these is greater than v. Consequently there is no cardinal number 𝔞 which is less than v + 1 and greater than v.

The following theorem is important for what follows:

Theorem F: If K is any set of different finite cardinal numbers, there is one x₁ amongst them which is smaller than all the rest, and which therefore the smallest of them all.

Proof:

The set K either contains the number 1, in which case x₁ is the smallest cardinal number in the set and x₁ = 1. Otherwise, let J be the set of all those cardinal numbers of our series 1, 2, 3, … which are all smaller than any occurring in K. So, if a number v belongs to J, all numbers less than v also belong to J. But J must have one element v₁ such that v₁ + 1 and all greater numbers do not belong to J, because otherwise J would contain all finite numbers, but the numbers belonging to K do not occur in J. Hence J is the subset {1, 2, 3, … , v₁} and the number v₁ + 1 = x₁ is necessarily an element of K and smaller than the other numbers in K.

From theorem F we conclude:

Theorem G: Every set K = {x} of different finite cardinal numbers can be brought into the form of a series:

K = x₁, x₂, x₃, …

such that

x₁ < x₂ < x₃ …

§ 6
The Smallest Transfinite Cardinal Number Aleph-Zero

Sets with finite cardinal numbers are called “finite sets” and all others we will call “transfinite sets”, and their cardinal numbers “transfinite cardinal numbers”. The first example of a transfinite set is given by the totality of finite cardinal numbers v and we call its cardinal number “Aleph-zero”, denoting it by:

The fact that ℵ₀ is a transfinite number, i.e: not equal to any finite number μ, follows from the simple fact that if a new element e₀ is added to the set {v} the resultant union set {v₁} ∪ e₀ is equivalent to the original set {v}. This is because we can think of a reciprocal one-to-one correspondence between them where the element e₀ of the first set corresponds the element 1 of the second set , and where the element v of the first set corresponds to the element v + 1 of the other set. We therefore have, according to Section 3:

But we showed in Section 5 that μ + 1 is always different to μ and therefore ℵ₀ is not equal to any finite number μ, and the number ℵ₀ is greater than any finite number μ :

By Section 3, this follows from the fact that:

• μ = {1, 2, 3, … , μ}, and
• no proper subset of the set {1, 2, 3, … , μ} is equivalent to the set {v}, and
• {1, 2, 3, … , μ} is itself a proper subset of {v}.

On the other hand ℵ₀ is the smallest transfinite cardinal number. If 𝔞 is any transfinite cardinal number different to ℵ₀ then:

This result is proved by the following theorems A and B:

Theorem A: Every transfinite set T has proper subsets with the cardinal number ℵ₀.

Proof:

If we have a definition which removes a finite number of elements t₁, t₂, … , t_v - 1 there always remains the possibility of taking away a further element t_v. The set {t_v } where v denotes any finite cardinal number is a proper subset of T with the cardinal number ℵ₀  since from Section 1 we have that {t_v } ∼ {t_v }.

Theorem B: If S is a transfinite set with the cardinal number ℵ₀ and S₁ is any transfinite proper subset of S, then S = ℵ₀.

Proof:

From the theorems A and B the above equation (4) follows, if we take account of Section 2.

From equation (2) above we have, by adding 1 to both sides:

ℵ₀+ 2 = ℵ₀ + 1 = ℵ₀

and by repeating this we have:

We have also

since, by equation Section 3 (1), ℵ₀ + ℵ₀ is the cardinal number {a_v} ∪ {b_v} because

{a_v} ∪ {b_v} = ℵ₀

Now, obviously, we have:

{v} = {2v − 1} ∪ {2v}

{2v − 1} ∪ {2v} ∼ {a_v} ∪ {b_v}

and therefore:

{a_v} ∪ {b_v} = {v} = ℵ₀

The equation (6) can also be written

ℵ₀ • 2 = ℵ₀

and, by adding ℵ₀ repeatedly to both sides, we get:

We also have that:

Proof:

Theorem A and Theorem E of Section 5 imply the following theorem for finite sets:

Theorem C: Every finite set E is such that it is equivalent to none of its proper subsets.

That theorem stands sharply opposed to the following one for transfinite sets:

Theorem D: Every transfinite set T is such that it has proper subsets T₁ which are equivalent to it.

Proof:

By theorem A of this Section there is a proper subset S ={t_v} of T with the cardinal number ℵ₀ . Let T = S ∪ U so that U is composed of those elements of T that are different to the elements t_v . Let S₁ = {t_v + 1} and T₁ = S₁ ∪ U. Then T₁ is a proper subset of T, and is in fact the proper subset that arises out of T if we leave out the single element t₁ . Since S ∼ S₁, then by theorem B of this Section and by the fact that U ∼ U, we have by Section 1 that T ∼ T₁.

In these theorems C and D the essential difference between finite and transfinite sets, to which I referred in Volume 84 (1878) of Crelle’s journal, p. 242, (Footnote: “Ein Beitrag zur Mannigfaltigkeitslehre”, 1878, online English translation at A Contribution to the Theory of Sets.) appears in the clearest manner.

After we have introduced the smallest transfinite cardinal number ℵ₀ and derived its properties that lie the most readily to hand, the question arises as to how the higher cardinal numbers originate from ℵ₀. It will be shown that the transfinite cardinal numbers can be ordered according to their magnitude, and that in this ordering they form a “well-ordered set”, like the finite numbers, but in an extended sense. The next largest cardinal number ℵ₁ arises from ℵ₀ by a definite rule, and the next largest cardinal number ℵ₂ arises from ℵ₁ by the same rule.

But even the limitless sequence of cardinal numbers

ℵ₀, ℵ₁, ℵ₂, … ℵ_v

does not exhaust the concept of a transfinite cardinal number. We will demonstrate the existence of a cardinal number which we will denote by ℵ_ω and which is the next largest after all ℵ_v . From it comes the next largest ℵ_ω + 1, in the same way as ℵ₁ comes out of ℵ₀ , and so on endlessly.

For every transfinite cardinal number 𝔞 there is a next largest number arising out of it according to a uniform rule. But it is also the case that for every limitlessly ascending well-ordered set of transfinite cardinal numbers {𝔞} there is a next largest number that arises uniformly out of that set.

For the rigorous foundation of this matter, discovered in 1882 and set out in the paper Grundlagen einer allgemeinen Mannigfaltigkeits­lehre, in Volume 21 of the Mathematische Annalen, Leipzig, 1883, (Footnote: Pages 545-591 of Volume 21 of Mathematische Annalen, online English Translation at Foundations of a general theory of sets.) we make use of the so-called “ordinal types”, the theory of which we will first need to explain in the following paragraphs.

Translator’s note: This is the end of Section 6, and this is as far through Cantor’s paper that this translation covers

Translator’s notes