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Cantor's Absolute Infinite: Lead

Cantor's absolute infinite or the absolute is the infinite that transcends Georg Cantor's transfinite. Cantor introduced it in his 1883 article that also introduced the transfinite (infinite) ordinals. In this article, he pointed out that the absolute was discussed by earlier mathematicians and philosophers, who placed it after the finite. He added the transfinite to their work, which he placed between the finite and the absolute.

The absolute played an important role in Cantor's efforts to convince theologians that his transfinite numbers did not promote pantheism. Also, the absolute's property of being beyond human comprehension leads to reflection principles, which have been used to justify axioms extending Zermelo-Frankel set theory (ZFC).

In the 1890s, Cantor called a multiplicity (collection) "inconsistent" if the assumption that it is a set leads to a contradiction. For example, he proved that assuming the multiplicity of all ordinals is a set leads to a contradiction. Cantor identified these multiplicities as the absolutely infinite multiplicities and used the inconsistent multiplicity of all ordinals in his "proof" of the well-ordering theorem. Historians of mathematics disagree on when Cantor discovered that the multiplicity of all ordinals is not a set. Some historians argue that he discovered it while working on his 1883 article, while others give a later date.

Lack of knowledge about Cantor's absolute infinite led to the belief that Cantor developed a naive set theory that is susceptible to paradoxes, such as the Burali-Forti paradox and Russell's paradox. Historians of mathematics have studied the absolute infinite and have determined that it protects his theory from these paradoxes.

Finite, transfinite, and the absolute

The concept of the absolute predates Cantor's work. In 1883, Cantor introduced the absolute into his mathematical work in an article that is now called "Cantor's Grundlagen." This article is a mathematical and philosophical investigation of the infinite. In it, Cantor referenced philosophical works by Gottfried Wilhelm Leibniz, Baruch Spinoza, René Descartes, and John Locke. Then he stated:

However different the theories of these writers may be, in their judgment of the finite and infinite they essentially agree that finiteness is part of the concept of number and that the true infinite or absolute, which is in God, permits no determination whatsoever. As to the latter point I fully agree, and cannot do otherwise; …"

Cantor not only accepted the concept of the absolute but also adopted Leibniz's terminology: "The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts." By stating that the absolute is in God, Cantor agreed with Leibniz who had stated that the absolute is an attribute of God. Spinoza, Descartes, and Locke also had spoken of the infinite as characteristic of God. Cantor's statement that the absolute "permits no determination whatsoever" means that it is beyond human comprehension, so it cannot be understood using mathematics.

On the other hand, Cantor disagreed that finiteness was part of the number concept and that there was only the finite and the absolute:

I can find no justification for the assumption that besides the absolute (which is not attainable by any determination) and the finite there should be no modifications which, although not finite, nevertheless are determinable by numbers and are therefore what I call proper-infinite."

Cantor introduced transfinite (infinite) numbers using infinite well-ordered sets: each of these sets has a corresponding ordinal number. He used well-ordered sets to define addition and multiplication of ordinal numbers, explored the properties of these numbers, and discovered that some of their properties differ from those of finite numbers. For example, addition and multiplication are not commutative: n + ω = ω ≠ ω + n for 1 ≤ n < ω and nω = ω ≠ ωn for 2 ≤ n < ω. This resolved an objection of Aristotle to the infinite: "if the infinite existed then it would absorb the finite and destroy it." (NEED REF!) If ω is added to n, the result n + ω equals ω. However, if n is added to ω, the result ω + n is greater than ω. So the infinite absorbs the finite only if it is placed after the finite. Cantor said that this and similar facts were "entirely unrecognized by Aristotle".

Absolute infinite and theology

Cantor's belief in infinite sets that were accessible to mathematicians was criticized by some Christian theologians who thought that his theory supported pantheism, which was considered a heresy by the Catholic Church. Pantheism developed from the philosophy of Baruch Spinoza, who believed that there was one absolutely infinite, eternal substance that he called "God or Nature." This identification of God with nature contradicts the Christian belief in a God who transcends nature. It also implies that infinity is in nature. Since Cantor believed that transfinite infinities can be found in nature, some theologians argued that Cantor's theory supported pantheism.

In 1886, Cantor corresponded with Cardinal Johannes Franzelin, who had been the papal theologian to the First Vatican Council. Cardinal Franzelin was critical of Cantor's transfinite infinities and thought that they were connected to pantheism. In his correspondence, Cantor explained that although a transfinite infinity and the absolute infinity are both actual infinities, pantheism arises from believing that God's absolute infinity can be found in nature.

Cardinal Franzelin accepted Cantor's explanation but found a serious flaw in another of Cantor's beliefs. Cantor agreed with Spinoza that the possibility of an idea, which arises from the idea being consistent and coherent, leads to its necessary creation in nature. Since Cantor believed the transfinite numbers were consistent and coherent, this implied that they were in nature. Cardinal Franzelin pointed out that this "necessity of creation would always be as it were a contradiction of the freedom of creation." Cantor replied that he was not speaking of "an objective, metaphysical necessity in the creative act … that would subjugate God's absolute freedom." Instead, he was speaking of a "subjective necessity for us (not issuing from God) of inferring, from God's absolute goodness and glory, the fact of an actually accomplished creation—not merely of a finite system, but of a transfinite system." Nevertheless, because of the cardinal's criticism, Cantor emphasized God's freedom of creation in later letters.

Cantor was proud that his theory gained Cardinal Franzeln's acceptance, and continued his correspondence with theologians. Cantor had deep religious beliefs and thought that his theory of the infinite would help Christian theology.

Reflection principles

Reflection principles can be derived from Cantor's view of the absolute: "The absolute can only be acknowledged but never known—and not even approximately known." This is a negative statement, but it can be used to obtain positive consequences. It has been pointed out that Cantor used the absolute in this way. For example, the finite ordinals (the natural numbers), which were previously thought to form an absolute infinity, is a "graspable idea" for Cantor. These ordinals form a multiplicity with the property of being closed under the successor operation. If this property defines the absolute, then the absolute is known. Since the absolute is unknowable, this property must also be true of some set of ordinal numbers, the smallest being the infinite set consisting of the finite ordinals. Cantor said that this infinity "seems to me to dwindle into nothingness by comparison [with the absolute]."

In general, a reflection principle states that any property true of the universe is true of some set since otherwise the absolute infinity of the universe is defined by this property. Hence, this property is "reflected" down to a set. Reflection principles can be generalized to hold for any proper class. In the example above, the proper class of all ordinals is closed under the successor operation, so reflection implies that there is a set of ordinals with this closure property. Reflection principles lead to large cardinal axioms. For example, the proper class of all cardinals has the property that it is closed under the operations of union and exponentiation. By reflection, there is a set of cardinals closed under these operations. The cardinality of this set is a strongly inaccessible cardinal.

Gödel stated that all the principles for generating axioms of set theory should come from the absolute being unknowable:

"All the principles for setting up the axioms of set theory should be reducible to a form of Ackermann's principle: The absolute is unknowable. The strength of this principle increases as we get stronger and stronger systems of set theory. The other principles are only heuristic principles. Hence, the central principle is the reflection principle …"

Reflection principles have been stated for nearly all of the large cardinal axioms. For example, there are reflection principles that imply the existence of n-huge and I3 rank-into-rank cardinals. These are among the cardinals having the greatest consistency strength and whose existence is believed to be consistent with ZFC. Conversely, the existence of these cardinals implies the consistency of their associated reflection principles.

In his statement, Gödel pointed out the connection between stronger reflection principles and stronger systems of set theory. The reflection principles for n-huge and I3 cardinals are an example of this because they are stated in a strong system of set theory that includes n-classes. A 2-class is a collection of classes; an (n +1)-class is a collection of n-classes.

From the absolute to inconsistent multiplicities

Cantor continued writing about the absolute after his 1883 Grundlagen. From 1885 to 1888, Cantor published philosophical articles on the transfinite and the absolute. In his 1887-1888 articles, Cantor emphasized that the transfinite, like the finite, is increasable—namely, every finite or transfinite ordinal α can be increased by adding n to obtain α + n. On the other hand, Cantor states that the absolute's "magnitude is capable of no increase or diminution, and is therefore to be looked upon quantitatively as an absolute maximum." By diminution, Cantor means subtracting a transfinite number of elements from the absolute. In his Grundlagen, Cantor states that "… every supra-finite [transfinite] number, however great … is followed by an aggregate of numbers … whose power [cardinality] is not in the slightest reduced compared to the entire absolutely infinite aggregate of numbers, …." This is equivalent to subtracting a transfinite number of elements from the front of the absolute sequence of all ordinal numbers. So removing a transfinite set of ordinals from this absolute sequence produces an absolute sequence of the same magnitude. Also, the magnitude of this absolute sequence cannot be increased since it contains all the ordinals.

A major change in Cantor's handling of the absolute occurred in the 1890's. This was motivated by his desire to prove the well-ordering theorem. In his 1883 Grundlagen, Cantor introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought." Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1897, he sent Hilbert his proof. This proof was lost, but it is believed to be similar to the proof of an equivalent theorem, which he sent Dedekind in 1899.

In his letter to Dedekind, Cantor stated:

For a multiplicity can be such that the assumption that all of its elements "are together" leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as "one finished thing". Such multiplicities I call absolutely infinite or inconsistent multiplicities.

Multiplicities that can consistently be conceived as a unity, he called consistent multiplicities or "sets." The idea that not every multiplicity can be conceived of as a unity dates back to Cantor's 1883 definition of set: "In general, by a 'manifold' or 'set' I understand every multiplicity which can be thought of as one, i.e. every aggregate of determinate elements which can be united into a whole by some law." Cantor's letter goes further than his 1883 article: it provides a method for recognizing that certain multiplicities are not sets—namely, the assumption that they are sets leads to a contradiction.

In his letter, Cantor proved the aleph theorem: The cardinality of every infinite set is an aleph—that is, an $\aleph _{\alpha }$. Since every $\aleph _{\alpha }$ is the cardinality of a well-ordered set, this theorem is equivalent to the well-ordering theorem. After defining consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities), Cantor considered the sequence Ω = 0, 1, 2, 3, …, ω, ω + 1, …, α, …. This sequence contains all the ordinals and every ordinal has the order type of the sequence of ordinals preceding it. If Ω is a set, then it is a well-ordered set, so there is ordinal δ corresponding to this set that is greater than all the ordinals in Ω. Since Ω contains all the ordinals, δ is in Ω. Therefore, δ > δ, which contradicts Ω being a well-ordered set. Cantor concluded that the sequence of all ordinals is an inconsistent, absolutely infinite multiplicity.

Cantor then used the inconsistent multiplicity Ω to prove the aleph theorem. He stated that "one easily sees" that if M is an infinite multiplicity that cannot be put into one-to-one correspondence with an $\aleph _{\alpha },$ then there is a one-to-one function mapping the inconsistent multiplicity Ω into M. This implies that M contains a submultiplicity that is inconsistent, so M is an inconsistent multiplicity. Therefore, if M cannot be put into one-to-one correspondence with an $\aleph _{\alpha },$ then it is an inconsistent multiplicity. This implies the aleph theorem.

Zermelo analyzed Cantor's proof and stated that Cantor probably constructed the one-to-one function using successive choices as he did in his proof that every infinite set has a countable subset. Assume that a multiplicity M cannot be put into one-to-one correspondence with an $\aleph _{\alpha }.$ Construct a function from the multiplicity of all ordinals to M by successively choosing different elements of M for each ordinal. If this construction runs out of elements of M, then M is well-ordered by this function and thus can be placed into one-to-one correspondence with an $\aleph _{\alpha }$, contradicting the assumption. Otherwise, a one-to-one function mapping the inconsistent multiplicity of all ordinals into M has been constructed. Zermelo criticized this construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make successive arbitrary choices."

Cantor never published his proof. In 1903, Philip Jourdain developed a similar proof and sent it to Cantor. In his reply, Cantor encouraged Jourdain to publish this proof. Cantor's letter also contained his proof. When Jourdain asked for permission to publish the letter, Cantor refused. Cantor probably did not want his proof published because he was not comfortable with it.

Although Cantor used an absolutely infinite multiplicity in a proof, it appears that he still believed in the theological aspects of the absolute. In fact, one development that came from Cantor's theology is still with us today—namely, reflection principles, which come from the absolute being unknowable to humans.

Cantor's discovery that some multiplicities cannot be sets

There is disagreement on when Cantor discovered that a contradiction occurs when certain multiplicities are assumed to be sets. The historian Alejandro Garciadiego wrote:

Historians have suggested that Cantor discovered the paradoxes in one of the following years: 1883, 1890, 1892, 1895, 1896 and 1899. There is evidence both for and against all of these dates. The clues contained in Cantor's public works and private correspondence are brief and scarce.

Garciadiego gives evidence only for the 1899 date in his article. The evidence for this date is in Cantor's 1899 letter to Dedekind, which is discussed above. This letter shows that by 1899 Cantor had discovered that assuming the multiplicity of all ordinals is a set leads to a contradiction. It also shows that Cantor had found a way to prevent this contradiction from implying his set theory is inconsistent–namely, by classifying this multiplicity as "an inconsistent multiplicity" and not allowing an inconsistent multiplicity to be an element of a multiplicity.

As for the earlier dates, only evidence for the 1883 date will be considered here–this is the earliest date in Garciadiego's list and the date that Cantor's Grundlagen appeared. The evidence for this date comes from this article, from later letters in which Cantor mentions the article, from simple mathematical arguments that Cantor supposedly would not miss, and from one of the works cited by Cantor in his article.

In his article, Cantor states that the transfinite follows the finite and that it "like the finite can be determined by well-defined and distinguishable numbers. Following the transfinite is "the absolute (which is not attainable by any determination)." This implies that the absolute cannot be determined by numbers like the finite and transfinite. In his endnotes, Cantor defines "set": "In general, by a 'manifold' or 'set' [*** LOOKUP! Does it have a comma after 'set'?] I understand every multiplicity which can be thought of as one, i.e. every aggregate of determinate elements which can be united into a whole by some law." Concerning the transfinite numbers and the absolute, Cantor states of the transfinite that "we shall never reach a boundary that cannot be crossed;" and states of the absolute: "that we shall also never achieve even an approximate conception of the absolute." He also states "The absolutely infinite sequence of [ordinal] numbers thus seems to me to be an appropriate symbol of the absolute."

So in 1883, Cantor made a clear distinction between the transfinite and the absolute. He also gave two examples of absolutely infinite multiplicities: the absolutely infinite sequence of all ordinals and the absolutely infinite sequence of all number classes, which are sets of ordinals, each set having the next greater cardinality than the cardinalities of the previous number classes. Cantor used number classes in his theory of cardinal numbers (their cardinalities are the aleph numbers). To prove that the number classes form an absolutely infinite sequence, Cantor states, but does not prove, that there is a number class associated with each ordinal. Hence, there is a one-to-one correspondence between the two multiplicities.

The historian Joseph Dauben states "… it is possible that Cantor may have been aware of the paradoxes of set theory … as early as the 1880s …" Dauben points out that in his Grundlagen, Cantor refers to "collections that are too large to be a well-defined, completed, unified entity." He also says that Cantor obscurely referred to these absolute sets in theological terms. Dauben then asks: "Was this a hint that he already understood that the collection of all transfinite ordinal numbers was inconsistent, and therefore not to be regarded as a set?" Dauben also mentions that later Cantor said that when writing the Grundlagen, he did know of the contradiction that occurs when the multiplicity of all ordinals is assumed to be a set.

Dauben is referring to letters that Cantor later wrote. In an 1897 letter to David Hilbert, Cantor wrote: "Totalities that cannot be regarded as sets …, I have already many years ago called absolute infinite totalities …, which I sharply distinguish from transfinite sets." In an 1899 letter, Cantor told Hilbert that absolute infinite totalities are fundamental for his theory: "You can find this, my fundament, in the anno 1883 published 'Grundlagen' …, actually in the notes where it appears as already completely clear, but also intentionally hidden." In a 1907 letter to Grace Chisholm Young, Cantor wrote that W [the multiplicity of all ordinals] "is not a 'set' in my sense of the word, but an 'inconsistent multiplicity.' When I wrote the 'Grundlagen' I already saw this clearly, as is evident from the remarks (1) and (2) in its conclusion, where I referred to W as the 'absolutely infinite number sequence.' In (1) I said explicitly that I designate as 'sets' only those multiplicities that can be conceived as unities, i.e., as objects." (Moore & Garciadiego 1981, p. 342.)}} NEED EVIDENCE THAT CANTOR KNEW IT WOULD GENERATE A CONTRADICTION!!

The mathematician and historian Walter Purkert believes that Cantor intentionally hid his discovery of contradictions because he was concerned that the majority of mathematicians would not accept his philosophical justifications for the absolute infinite. Without acceptable justification for this concept, the contradiction generated by the set of all transfinite numbers would be a "serious blow against his theory, which was ignored and even under attack as it was." Purkert's argument is consistent with Cantor's cautious handling of previous articles. For example, in his 1874 article, he gave a constructive proof of the existence of transcendental numbers rather than the non-constructive proof found in his earlier letters. Akihiro Kanamori states that "This presentation is suggestive of Cantor’s natural caution in overstepping mathematical sense at the time."

Purkert also believes that Cantor would have seen the following mathematical argument: If the collection of all ordinals form a set, it would be well-ordered set, which implies it has an ordinal number Ω. However, Ω would be the greatest ordinal, which contradicts the fact that each ordinal α has a successor α + 1. Purkert comments: "It is almost unthinkable that Cantor, who reflected so deeply on these matters, would have taken no notice of such a simple consequence."

The philosopher William W. Tait agrees with Purkert and states that "it would be difficult to believe that Cantor was not aware that it would be contradictory to assume that Ω [the class of all finite and transfinite numbers] is a set." However, Tait criticizes Purkert's suggested proof because it uses well-orderings and is not based on Cantor's definition of the transfinite numbers. Tait also gives his version of an argument that Cantor may have used.

The historian Alejandro Garciadiego disagrees with Purkert and Tait. He states that "It is possible that Cantor had intuitively felt the existence of the 'paradoxes' since 1883—as he claims, but we lack sources supporting the assertion that Cantor had formally found the 'paradoxes' by then." However, Cantor's letter states: "The undoubtedly correct proposition that there are no transfinite cardinal numbers other than the alephs I intuitively recognized over 20 years ago (during the discovery of the alephs themselves). In his 1883 Grundlagen, Cantor introduced his well-ordering principle "every set can be well-ordered," which he declared to be "law of thought." This well-ordering principle, which Cantor felt was intuitively true in 1883, implies that all the transfinite cardinal numbers are alephs. Cantor's reasoning makes no reference to the paradoxes.

In this letter, Cantor states that in 1883 he had "intuitively recognized" the theorem that there are no transfinite cardinal numbers other than the alephs. In his 1883 Grundlagen, Cantor introduced his well-ordering principle "every set can be well-ordered," which he declared to be "law of thought." This well-ordering principle, which Cantor felt was intuitively true in 1883, implies that all the transfinite cardinal numbers are alephs.

** REWRITE NEXT PARAGRAPH!! 

So the 1899 date has the strongest direct evidence for Cantor knowing that a contradiction is generated by assuming the multiplicity of all ordinals is a set. Earlier dates have circumstantial evidence, the later dates accumulate more of this evidence. Unfortunately, although Cantor saved nearly everything he wrote, these most of these writings were lost when his house was taken over in World War 2 and they were used to warm the house. Without these writings, there is plenty of room for disagreement on when Cantor discovered that a contradiction occurs if certain multiplicities are assumed to be sets. [*** NEED REF about WW2 loss of Cantor's writings: Probably: Ivor Grattan-Guinness, Towards a biography of Georg Cantor, Annals of Science 27 (4), 1971, pp. 345-391.]

The set-theoretic paradoxes

Cantor avoided the set-theoretic paradoxes by recognizing that some multiplicities are not sets and by not allowing these multiplicities to be elements of a multiplicity. In letters to Hilbert, Cantor pointed out that in his 1883 Grundlagen, he defined set _____________________, and in his 1895 article, he defined set

However, in these articles, Cantor did not explicitly state that he had worded his definitions to avoid contradictions resulting from treating certain multiplicities as sets. Later mathematicians stated that his 1895 definition " " produced a naive set theory.

"Totalities that cannot be regarded as sets …, I have already many years ago called absolute infinite totalities …, which I sharply distinguish from transfinite sets."

In 1923, John von Neumann developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. These collections are now called proper classes. In von Neumann's set theory, the class of all ordinals is a proper class. The paradoxes are avoided by not allowing a proper class to belong to any class. Cantor put the same restriction on an inconsistent multiplicity, but for a different reason. Cantor does not allow an inconsistent multiplicity to belong to any multiplicity because "it cannot be understood as one whole and thus cannot be considered as one thing." In von Neumann's theory, a proper class is "one thing," but it cannot belong to any class. The difference between Cantor's and von Neumann's approaches is that Cantor does not allow non-set objects in his set theory, while von Neumann allows them and restricts the usage of proper classes. Von Neumann not only admits proper classes into his theory but also insists that "they should be as much like sets as possible."

Bibliography

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• Leibniz, Gottfried Wilhelm (1896), New essays Concerning Human Understanding (PDF), translated by Langley, Alfred Gideon, London: The Macmillan Company.
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• Thomas-Bolduc, Aaron R. (2016), "Georg Cantor, God, and Inconsistent Multiplicities" (PDF), Studies in Logic, Grammar and Rhetoric, 44 (1): 133–146, doi:10.1515/slgr-2016-0008, S2CID 55298195.
• van Heijenoort, Jean, ed. (1967), From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, ISBN 978-0-674-32449-7.
• von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
• Wang, Hao (1996), A Logical Journey: From Gödel to Philosophy, London: MIT Press, ISBN 978-0-262-23189-3.
• Welch, Philip; Horsten, Leon (2016), "Reflecting on Absolute Infinity" (PDF), The Journal of Philosophy, 113 (2): 89–111, doi:10.5840/jphil201611325. Online version differs from published version. This article uses the online version, which is numbered 1-34.

* REMOVE BEFORE PUBLISHING!!]

(1) CHECKED in Refs.

(2) Hallett, p. 165-8, 173-5 (see SUBJECT INDEX: p. 324 "absolute or inconsistent collections ... and Cantor's absolute"

EXTRA!!

all numeration being but still the adding of one unit more, and giving to the whole together, as comprehended [191] in one idea, a new or distinct name or sign, whereby to know it from those before and after, and distinguish it from every smaller or greater multitude of units.

Locke, 13. No positive idea of infinity. Though it be hard, I think, to find anyone so absurd as to say he has the positive idea of an actual infinite number; the infinity whereof lies only in a power still of adding any combination of units to any former number, and that as long and as much as one will; the like also being in the infinity of space and duration, which power leaves always to the mind room for endless additions; yet there be those who imagine they have positive ideas of infinite duration and space.

We can, I think, have no positive idea of any space or duration which is not made up of, and commensurate to, repeated numbers of feet or yards, or days and years; which are the common measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up of infinite parts, it can have no other infinity than that of number capable still of further addition; but not an actual positive idea of a number infinite. For, I think it is evident, that the addition of finite things together (as are all lengths whereof we have the positive ideas) can never otherwise produce the idea of infinite than as number does; which, consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind; without coming one jot nearer the end of such progression.

Finite, transfinite, and the absolute

The concept of the absolute predates Cantor's work. In 1883, Cantor introduced the absolute into his mathematical work in an article that is now called "Cantor's Grundlagen." This article is a mathematical and philosophical investigation of the infinite. In it, Cantor referenced philosophical works by Gottfried Wilhelm Leibniz, Baruch Spinoza, René Descartes, and John Locke. Then he stated:

However different the theories of these writers may be, in their judgment of the finite and infinite they essentially agree that finiteness is part of the concept of number and that the true infinite or absolute, which is in God, permits no determination whatsoever. As to the latter point I fully agree, and cannot do otherwise; …"

Cantor not only accepted the concept of the absolute but also adopted Leibniz's terminology: "The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts." By stating that the absolute is in God, Cantor agreed with Leibniz who had stated that the absolute is an attribute of God. Spinoza, Descartes, and Locke also had spoken of the infinite as characteristic of God. Cantor's statement that the absolute "permits no determination whatsoever" means that it is beyond human comprehension, so it cannot be understood using mathematics.

On the other hand, Cantor disagreed that finiteness was part of the number concept and that there was only the finite and the absolute:

I can find no justification for the assumption that besides the absolute (which is not attainable by any determination) and the finite there should be no modifications which, although not finite, nevertheless are determinable by numbers and are therefore what I call proper-infinite."

Cantor introduced transfinite (infinite) numbers using infinite well-ordered sets: each of these sets has a corresponding ordinal number. He used well-ordered sets to define addition and multiplication of ordinal numbers, explored the properties of these numbers, and discovered that some of their properties differ from those of finite numbers. For example, addition and multiplication are not commutative: n + ω = ω ≠ ω + n for 1 ≤ n < ω and nω = ω ≠ ωn for 2 ≤ n < ω. This resolved an objection of Aristotle to the infinite: "if the infinite existed then it would absorb the finite and destroy it." If ω is added to n, the result n + ω equals ω. However, if n is added to ω, the result ω + n is greater than ω. So the infinite absorbs the finite only if it is placed after the finite. Cantor said that these facts were "entirely unrecognized by Aristotle".

Cantor's discovery that some multiplicities cannot be sets

There is disagreement on when Cantor discovered that a contradiction occurs when certain multiplicities are assumed to be sets. The historian Alejandro Garciadiego wrote:

Historians have suggested that Cantor discovered the paradoxes in one of the following years: 1883, 1890, 1892, 1895, 1896 and 1899. There is evidence both for and against all of these dates. The clues contained in Cantor's public works and private correspondence are brief and scarce.

Garciadiego gives evidence only for the 1899 date in his article. The evidence for this date is in Cantor's 1899 letter to Dedekind, which is discussed above. This letter shows that by 1899 Cantor had discovered that assuming the multiplicity of all ordinals is a set leads to a contradiction. It also shows that Cantor had found a way to prevent this contradiction from implying his set theory is inconsistent–namely, by classifying this multiplicity as "an inconsistent multiplicity" and not allowing an inconsistent multiplicity to be an element of a multiplicity.

As for the earlier dates, only evidence for the 1883 date will be considered here–this is the earliest date in Garciadiego's list and the date that Cantor's Grundlagen appeared. The evidence for this date comes from this article, from later letters in which Cantor mentions the article, from simple mathematical arguments that Cantor supposedly would not miss, and from one of the works cited by Cantor in his article.

In his article, Cantor states that the transfinite follows the finite and that it "like the finite can be determined by well-defined and distinguishable numbers. Following the transfinite is "the absolute (which is not attainable by any determination)." This implies that the absolute cannot be determined by numbers like the finite and transfinite. In his endnotes, Cantor defines "set": "In general, by a 'manifold' or 'set' [*** LOOKUP! Does it have a comma after 'set'?] I understand every multiplicity which can be thought of as one, i.e. every aggregate of determinate elements which can be united into a whole by some law." Concerning the transfinite numbers and the absolute, Cantor states of the transfinite that "we shall never reach a boundary that cannot be crossed;" and states of the absolute: "that we shall also never achieve even an approximate conception of the absolute." He also states "The absolutely infinite sequence of [ordinal] numbers thus seems to me to be an appropriate symbol of the absolute."

So in 1883, Cantor made a clear distinction between the transfinite and the absolute. He also gave two examples of absolutely infinite multiplicities: the absolutely infinite sequence of all ordinals and the absolutely infinite sequence of all number classes, which are sets of ordinals, each set having the next greater cardinality than the cardinalities of the previous number classes. Cantor used number classes in his theory of cardinal numbers (their cardinalities are the aleph numbers). To prove that the number classes form an absolutely infinite sequence, Cantor states, but does not prove, that there is a number class associated with each ordinal. Hence, there is a one-to-one correspondence between the two multiplicities.

The historian Joseph Dauben states "… it is possible that Cantor may have been aware of the paradoxes of set theory … as early as the 1880s …" Dauben points out that in his Grundlagen, Cantor refers to "collections that are too large to be a well-defined, completed, unified entity." He also says that Cantor obscurely referred to these absolute sets in theological terms. Dauben then asks: "Was this a hint that he already understood that the collection of all transfinite ordinal numbers was inconsistent, and therefore not to be regarded as a set?" Dauben also mentions that later Cantor said that when writing the Grundlagen, he did know of the contradiction that occurs when the multiplicity of all ordinals is assumed to be a set.

Dauben is referring to letters that Cantor later wrote. In an 1897 letter to David Hilbert, Cantor wrote: "Totalities that cannot be regarded as sets …, I have already many years ago called absolute infinite totalities …, which I sharply distinguish from transfinite sets." In an 1899 letter, Cantor told Hilbert that absolute infinite totalities are fundamental for his theory: "You can find this, my fundament, in the anno 1883 published 'Grundlagen' …, actually in the notes where it appears as already completely clear, but also intentionally hidden." In a 1907 letter to Grace Chisholm Young, Cantor wrote that W [the multiplicity of all ordinals] "is not a 'set' in my sense of the word, but an 'inconsistent multiplicity.' When I wrote the 'Grundlagen' I already saw this clearly, as is evident from the remarks (1) and (2) in its conclusion, where I referred to W as the 'absolutely infinite number sequence.' In (1) I said explicitly that I designate as 'sets' only those multiplicities that can be conceived as unities, i.e., as objects." (Moore & Garciadiego 1981, p. 342.)}} NEED EVIDENCE THAT CANTOR KNEW IT WOULD GENERATE A CONTRADICTION!!

The mathematician and historian Walter Purkert believes that Cantor intentionally hid his discovery of contradictions because he was concerned that the majority of mathematicians would not accept his philosophical justifications for the absolute infinite. Without acceptable justification for this concept, the contradiction generated by the set of all transfinite numbers would be a "serious blow against his theory, which was ignored and even under attack as it was." Purkert's argument is consistent with Cantor's cautious handling of previous articles. For example, in his 1874 article, he gave a constructive proof of the existence of transcendental numbers rather than the non-constructive proof found in his earlier letters. Akihiro Kanamori states that "This presentation is suggestive of Cantor’s natural caution in overstepping mathematical sense at the time."

Purkert also believes that Cantor would have seen the following mathematical argument: If the collection of all ordinals form a set, it would be well-ordered set, which implies it has an ordinal number Ω. However, Ω would be the greatest ordinal, which contradicts the fact that each ordinal α has a successor α + 1. Purkert comments: "It is almost unthinkable that Cantor, who reflected so deeply on these matters, would have taken no notice of such a simple consequence."

The philosopher William W. Tait agrees with Purkert and states that "it would be difficult to believe that Cantor was not aware that it would be contradictory to assume that Ω [the class of all finite and transfinite numbers] is a set." However, Tait criticizes Purkert's suggested proof because it uses well-orderings and is not based on Cantor's definition of the transfinite numbers. Tait also gives his version of an argument that Cantor may have used.

The historian Alejandro Garciadiego disagrees with Purkert and Tait. He states that "It is possible that Cantor had intuitively felt the existence of the 'paradoxes' since 1883—as he claims, but we lack sources supporting the assertion that Cantor had formally found the 'paradoxes' by then." However, in the letter Garciadiego references to support his position, Cantor does not use the word "intuitively" with respect to the paradoxes. Instead, Cantor states that in 1883 he had "intuitively recognized" the theorem that there are no transfinite cardinal numbers other than the alephs. In his 1883 Grundlagen, Cantor introduced his well-ordering principle "every set can be well-ordered," which he declared to be "law of thought." This well-ordering principle, which Cantor felt was intuitively true in 1883, implies that all the transfinite cardinal numbers are alephs.

** REWRITE NEXT PARAGRAPH!! 

So the 1899 date has the strongest direct evidence for Cantor knowing that a contradiction is generated by assuming the multiplicity of all ordinals is a set. Earlier dates have circumstantial evidence, the later dates accumulate more of this evidence. Unfortunately, although Cantor saved nearly everything he wrote, these most of these writings were lost when his house was taken over in World War 2 and they were used to warm the house. Without these writings, there is plenty of room for disagreement on when Cantor discovered that a contradiction occurs if certain multiplicities are assumed to be sets. [*** NEED REF about WW2 loss of Cantor's writings: Probably: Ivor Grattan-Guinness, Towards a biography of Georg Cantor, Annals of Science 27 (4), 1971, pp. 345-391.]

Bibliography

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* REMOVE BEFORE PUBLISHING!!]

(1) CHECKED in Refs.

(2) Hallett, p. 165-8, 173-5 (see SUBJECT INDEX: p. 324 "absolute or inconsistent collections ... and Cantor's Absolute"