Georg Cantor

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Georg Cantor
Cantor, early 1900s
Born	Georg Ferdinand Ludwig Philipp Cantor March 3, 1845 Saint Petersburg, Russian Empire
Died	January 6, 1918 (aged 72) Halle, Province of Saxony, German Empire
Nationality	German
Alma mater	Swiss Federal Polytechnic University of Berlin
Known for	Set theory
Spouse(s)	Vally Guttmann ​ (m. 1874)​
Awards	Sylvester Medal (1904)
Scientific career
Fields	Mathematics
Institutions	University of Halle
Thesis	De aequationibus secundi gradus indeterminatis (1867)
Doctoral advisor	Ernst Kummer Karl Weierstrass

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor, German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. It is important to note that not all theologians were against Cantor's theory, prominent neo-scholastic philosopher Constantin Gutberlet was in favor and Cardinal Johann Baptist Franzelin accepted as a valid theory (after Cantor made some important clarifications).

The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder.

The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."

Life of Georg Cantor

Youth and studies

Cantor, around 1870

Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg, Russia, and brought up in the city until he was eleven. Cantor, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden, then to Frankfurt, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. In 1862, Cantor entered the Swiss Federal Polytechnic. After receiving a substantial inheritance upon his father's death in June 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctorate degree in 1867.

Teacher and researcher

Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle University.

In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and it usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.

In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta. He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica."

Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:

... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.

This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature thinking there might be evidence that Francis Bacon wrote the plays attributed to William Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on December 29, 1891. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.

In 1889, Cantor was instrumental in founding the German Mathematical Society and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians, which was held in Zürich, Switzerland, in 1897.

Later years and death

After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on December 16 (Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.

Mathematical work

Cantor's work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.

In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).

Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875, is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers.

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter ${\displaystyle \aleph }$ (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.

Number theory, trigonometric series and ordinals

Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set S_n of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S₁ as its set of zeros, where S₁ is the set of limit points of S. If S_k+1 is the set of limit points of S_k, then he could construct a trigonometric series whose zeros are S_k+1. Because the sets S_k were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets S, S₁, S₂, S₃,... formed a limit set, which we would now call S_ω, and then he noticed that S_ω would also have to have a set of limit points S_ω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω, ω + 1, ω + 2, ...

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.

Set theory

An illustration of Cantor's diagonal argument for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs from diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.

Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers as a sequence a₁, a₂, a₃, .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers. Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also introduced the Cantor set during this period.

The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem. Cantor wrote on the Goldbach conjecture in 1894.

Passage of Georg Cantor's article with his set definition

In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory. The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem.

One-to-one correspondence

Main article: Bijection

 

A bijective function

Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of dimension.

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rⁿ has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

Continuum hypothesis

Main article: Continuum hypothesis

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can be neither proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").

Absolute infinite, well-ordering theorem, and paradoxes

In 1883, Cantor divided the infinite into the transfinite and the absolute.

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought".

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph. First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof.

Cantor avoided paradoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is inconsistent. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the Burali-Forti paradox (which was just mentioned), Cantor's paradox, and Russell's paradox. Russell named paradoxes after Cesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes.

In 1908, Zermelo published his axiom system for set theory. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem. Zermelo had proved this theorem in 1904 using the axiom of choice, but his proof was criticized for a variety of reasons. His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of 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. Von Neumann stated that a class is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined models of set theory that satisfy von Neumann's axiom.

Philosophy, religion, literature and Cantor's mathematics

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. Joseph Dauben has traced the effect Cantor's Christian convictions had on the development of transfinite set theory.

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind. Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as L. E. J. Brouwer and especially Henri Poincaré adopted an intuitionist stance against Cantor's work. Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake: "... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.". It is to note that prominent neo-scholastic german philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.

Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.

It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 Grundlagen, he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz. In making these claims, Cantor may have been influenced by FA Trendelenburg, whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's Ethica. FA Trendelenburg was also the examiner of Cantor's Habilitationsschrift.

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim, as well as theologians such as Cardinal Johann Baptist Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's. Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom." These ideas parallel those of Edmund Husserl, whom Cantor had met in Halle.

Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics".

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering: "... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, René Descartes, George Berkeley, Gottfried Leibniz, and Bernard Bolzano on infinity. Instead, he always strongly rejected Kant's philosophy, in the realms of both the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophistical Philistine who knew so little mathematics."

Cantor's ancestry

The title on the memorial plaque (in Russian): "In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor", Vasilievsky Island, Saint-Petersburg.

Cantor's paternal grandparents were from Copenhagen and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on them. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about Georg Waldemar's origin or education. Cantor's mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:

Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ...

("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...") which could be read to imply that she was of Jewish ancestry.

According to biographers Eric Temple Bell, Cantor was of Jewish descent, although both parents were baptized. In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).

In a letter written to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde...." ("He was born in Copenhagen of Jewish (lit: 'Israelite') parents from the local Portuguese-Jewish community.") In addition, Cantor's maternal great uncle, a Hungarian violinist Josef Böhm, has been described as Jewish, which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.

In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows:

Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a regular just Germain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany.

There were documented statements, during the 1930s, that called this Jewish ancestry into question:

More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew ..."

Biographies

Until the 1970s, the chief academic publications on Cantor were two short monographs by Arthur Moritz Schönflies (1927) – largely the correspondence with Mittag-Leffler – and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst". Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell – including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents. A critique of Bell's book is contained in Joseph Dauben's biography. Writes Dauben:

Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible.

Hilbert's paradox of the Grand Hotel

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel 

 

Hilbert's Hotel

Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in (Hilbert 2013, p.730), and was popularized through George Gamow's 1947 book One Two Three... Infinity.

The paradox

Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.

Finitely many new guests

Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. In general, assume that k guests seek a room. We can apply the same procedure and move every guest from room n to room n + k. In a similar manner, if k guests wished to leave the hotel, every guest moves from room n to room n − k.

Infinitely many new guests

It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n (2 times n), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.

Infinitely many coaches with infinitely many guests each

Further information: Pairing function

It is possible to accommodate countably infinitely many coachloads of countably infinite passengers each, by several different methods. Most methods depend on the seats in the coaches being already numbered (or use the axiom of countable choice). In general any pairing function can be used to solve this problem. For each of these methods, consider a passenger's seat number on a coach to be ${\displaystyle n}$, and their coach number to be ${\displaystyle c}$, and the numbers ${\displaystyle n}$ and ${\displaystyle c}$ are then fed into the two arguments of the pairing function.

Prime powers method

Empty the odd numbered rooms by sending the guest in room ${\displaystyle i}$ to room ${\displaystyle 2^{i}}$, then put the first coach's load in rooms ${\displaystyle 3^{n}}$, the second coach's load in rooms ${\displaystyle 5^{n}}$; for coach number ${\displaystyle c}$ we use the rooms ${\displaystyle p^{n}}$ where ${\displaystyle p}$ is the ${\displaystyle c}$th odd prime number. This solution leaves certain rooms empty (which may or may not be useful to the hotel); specifically, all odd numbers that are not prime powers, such as 15 or 847, will no longer be occupied. (So, strictly speaking, this shows that the number of arrivals is less than or equal to the number of vacancies created. It is easier to show, by an independent means, that the number of arrivals is also greater than or equal to the number of vacancies, and thus that they are equal, than to modify the algorithm to an exact fit.) (The algorithm works equally well if one interchanges ${\displaystyle n}$ and ${\displaystyle c}$, but whichever choice is made, it must be applied uniformly throughout.)

Prime factorization method

You can put each person of a certain seat ${\displaystyle s}$ and coach ${\displaystyle c}$ into room ${\displaystyle 2^{s}3^{c}}$ (presuming c=0 for the people already in the hotel, 1 for the first coach, etc. ...). Because every number has a unique prime factorization, it's easy to see all people will have a room, while no two people will end up in the same room. For example, the person in room 2592 (${\displaystyle 2^{5}3^{4}}$) was sitting in on the 4th coach, on the 5th seat. Like the prime powers method, this solution leaves certain rooms empty.

This method can also easily be expanded for infinite nights, infinite entrances, etc. ... ( ${\displaystyle 2^{s}3^{c}5^{n}7^{e}}$ )

Interleaving method

For each passenger, compare the lengths of ${\displaystyle n}$ and ${\displaystyle c}$ as written in any positional numeral system, such as decimal. (Treat each hotel resident as being in coach #0.) If either number is shorter, add leading zeroes to it until both values have the same number of digits. Interleave the digits to produce a room number: its digits will be [first digit of coach number]-[first digit of seat number]-[second digit of coach number]-[second digit of seat number]-etc. The hotel (coach #0) guest in room number 1729 moves to room 01070209 (i.e., room 1,070,209). The passenger on seat 1234 of coach 789 goes to room 01728394 (i.e., room 1,728,394).

Unlike the prime powers solution, this one fills the hotel completely, and we can reconstruct a guest's original coach and seat by reversing the interleaving process. First add a leading zero if the room has an odd number of digits. Then de-interleave the number into two numbers: the coach number consists of the odd-numbered digits and the seat number is the even-numbered ones. Of course, the original encoding is arbitrary, and the roles of the two numbers can be reversed (seat-odd and coach-even), so long as it is applied consistently.

Triangular number method

Those already in the hotel will be moved to room ${\displaystyle (n^{2}+n)/2}$, or the ${\displaystyle n}$th triangular number. Those in a coach will be in room ${\displaystyle ((c+n-1)^{2}+c+n-1)/2+n}$, or the ${\displaystyle (c+n-1)}$ triangular number plus ${\displaystyle n}$. In this way all the rooms will be filled by one, and only one, guest.

This pairing function can be demonstrated visually by structuring the hotel as a one-room-deep, infinitely tall pyramid. The pyramid's topmost row is a single room: room 1; its second row is rooms 2 and 3; and so on. The column formed by the set of rightmost rooms will correspond to the triangular numbers. Once they are filled (by the hotel's redistributed occupants), the remaining empty rooms form the shape of a pyramid exactly identical to the original shape. Thus, the process can be repeated for each infinite set. Doing this one at a time for each coach would require an infinite number of steps, but by using the prior formulas, a guest can determine what his room "will be" once his coach has been reached in the process, and can simply go there immediately.

Arbitrary enumeration method

Let ${\displaystyle S:=\{(a,b)\mid a,b\in \mathbb {N} \}}$. ${\displaystyle S}$ is countable since ${\displaystyle \mathbb {N} }$ is countable, hence we may enumerate its elements ${\displaystyle s_{1},s_{2},\dots }$. Now if ${\displaystyle s_{n}=(a,b)}$, assign the ${\displaystyle b}$th guest of the ${\displaystyle a}$th coach to the ${\displaystyle n}$th room (consider the guests already in the hotel as guests of the ${\displaystyle 0}$th coach). Thus we have a function assigning each person to a room; furthermore, this assignment does not skip over any rooms.

Further layers of infinity

Suppose the hotel is next to an ocean, and an infinite number of car ferries arrive, each bearing an infinite number of coaches, each with an infinite number of passengers. This is a situation involving three "levels" of infinity, and it can be solved by extensions of any of the previous solutions.

The prime factorization method can be applied by adding a new prime number for every additional layer of infinity ( ${\displaystyle 2^{s}3^{c}5^{f}}$, with ${\displaystyle f}$ the ferry).

The prime power solution can be applied with further exponentiation of prime numbers, resulting in very large room numbers even given small inputs. For example, the passenger in the second seat of the third bus on the second ferry (address 2-3-2) would raise the 2nd odd prime (5) to 49, which is the result of the 3rd odd prime (7) being raised to the power of his seat number (2). This room number would have over thirty decimal digits.

The interleaving method can be used with three interleaved "strands" instead of two. The passenger with the address 2-3-2 would go to room 232, while the one with the address 4935-198-82217 would go to room #008,402,912,391,587 (the leading zeroes can be removed).

Anticipating the possibility of any number of layers of infinite guests, the hotel may wish to assign rooms such that no guest will need to move, no matter how many guests arrive afterward. One solution is to convert each arrival's address into a binary number in which ones are used as separators at the start of each layer, while a number within a given layer (such as a guest's coach number) is represented with that many zeroes. Thus, a guest with the prior address 2-5-1-3-1 (five infinite layers) would go to room 10010000010100010 (decimal 295458).

As an added step in this process, one zero can be removed from each section of the number; in this example, the guest's new room is 101000011001 (decimal 2585). This ensures that every room could be filled by a hypothetical guest. If no infinite sets of guests arrive, then only rooms that are a power of two will be occupied.

Infinite layers of nesting

Although a room can be found for any finite number of nested infinities of people, the same is not always true for an infinite number of layers, even if a finite number of elements exists at each layer.

Analysis

Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.

Initially, this state of affairs might seem to be counter-intuitive. The properties of "infinite collections of things" are quite different from those of "finite collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. Thus, in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is ${\displaystyle \aleph _{0}}$.

Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.

References in fiction

• BBC Learning Zone repeatedly screened a 1996 one-off educational docudrama Hotel Hilbert set in the hotel as seen through the eyes of a young female guest Fiona Knight, her name a pun on finite. The programme was designed to educate viewers about the concept of infinity.
• The novel White Light by mathematician/science fiction writer Rudy Rucker includes a hotel based on Hilbert's paradox, and where the protagonist of the story meets Georg Cantor.
• Stephen Baxter's science fiction novel Transcendent has a brief discussion on the nature of infinity, with an explanation based on the paradox, modified to use soldiers rather than hotels.
• Geoffrey A. Landis' Nebula Award-winning short story "Ripples in the Dirac Sea" uses the Hilbert hotel as an explanation of why an infinitely-full Dirac sea can nevertheless still accept particles.
• In Peter Høeg's novel Miss Smilla's Feeling for Snow, the titular heroine reflects that it is admirable for the hotel's manager and guests to go to all that trouble so that the latecomer can have his own room and some privacy.
• In Ivar Ekeland's novel for children, The Cat in Numberland, a "Mr. Hilbert" and his wife run an infinite hotel for all the integers. The story progresses through the triangular method for the rationals.
• In Will Wiles's novel The Way Inn, about an infinitely large motel, the villain's name is Hilbert.
• In Reginald Hill's novel "The Stranger House" the character Sam refers to the Hilbert Hotel paradox.
• The short story by Naum Ya. Vilenkin The Extraordinary Hotel (often erroneously attributed to Stanislaw Lem) shows the way in which Hilbert's Grand Hotel may be reshuffled when infinite new hosts arrive.
• The comic book saga The Tempest from the League of Extraordinary Gentlemen series by Alan Moore and Kevin O'Neill shows a villain called Infinity. In the story it is suggested that the villain goes to the hotel based on Hilbert's paradox. Georg Cantor is mentioned as well.

Kalam cosmological argument

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Kalam_cosmological_argument 

The Kalam cosmological argument is a modern formulation of the cosmological argument for the existence of God. It is named after the kalam (medieval Islamic scholasticism) from which its key ideas originated. William Lane Craig was principally responsible for giving new life to the argument, due to his The Kalām Cosmological Argument (1979), among other writings. The kalam cosmological argument's premises surrounding causation and the beginning of the universe were discussed by various philosophers, the philosophical view of causation being a subject of David Hume's An Enquiry Concerning Human Understanding and the metaphysical arguments for a beginning of the universe being the subject of Kant's first antinomy.

The argument's key underpinning idea is the metaphysical impossibility of actual infinities and of a temporally past-infinite universe, traced by Craig to 11th-century Persian Muslim scholastic philosopher Al-Ghazali. This feature distinguishes it from other cosmological arguments, such as that of Thomas Aquinas, which rests on the impossibility of a causally ordered infinite regress, and those of Leibniz and Samuel Clarke, which refer to the Principle of Sufficient Reason.

Since Craig's original publication, the Kalam cosmological argument has elicited public debate between Craig and Graham Oppy, Adolf Grünbaum, J. L. Mackie and Quentin Smith, and has been used in Christian apologetics. According to Michael Martin, the cosmological arguments presented by Craig, Bruce Reichenbach, and Richard Swinburne are "among the most sophisticated and well argued in contemporary theological philosophy".

Form of the argument

The most prominent form of the argument, as defended by William Lane Craig, states the Kalam cosmological argument as the following syllogism:

1. Everything that begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.

Given the conclusion, Craig appends a further premise and conclusion based upon a philosophical analysis of the properties of the cause of the universe:

4. The universe has a cause.
5. If the universe has a cause, then an uncaused, personal Creator of the universe exists who sans (without) the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful.
6. Therefore, an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful.

Referring to the implications of Classical Theism that follow from this argument, Craig writes:

"... transcending the entire universe there exists a cause which brought the universe into being ex nihilo ... our whole universe was caused to exist by something beyond it and greater than it. For it is no secret that one of the most important conceptions of what theists mean by 'God' is Creator of heaven and earth."

Historical background

The kalam cosmological argument is based on the concept of the prime-mover, introduced by Aristotle, and entered early Christian or Neoplatonist philosophy in Late Antiquity, being developed by John Philoponus. Along with much of classical Greek philosophy, the concept was adopted into medieval Islamic tradition during the Islamic Golden Age, where it received its fullest articulation at the hands of Muslim scholars, most directly by Islamic theologians of the Sunni tradition.

Its historic proponents include Al-Kindi, Al-Ghazali, and St. Bonaventure.

One of the earliest formulations of the kalam cosmological argument in the Islamic philosophical tradition comes from Al-Ghazali, who writes:

"Every being which begins has a cause for its beginning; now the world is a being which begins; therefore, it possesses a cause for its beginning."

Between the 9th to 12th centuries, the cosmological argument developed as a concept within Islamic theology. It was refined in the 11th century by Al-Ghazali (The Incoherence of the Philosophers), and in the 12th by Ibn Rushd (Averroes). It reached medieval Christian philosophy in the 13th century and was discussed by Bonaventure, as well as Thomas Aquinas in his Summa Theologica (I, q.2, a.3) and Summa Contra Gentiles (I, 13).

Islamic perspectives may be divided into positive Aristotelian responses strongly supporting the argument, such as those by Al-Kindi, and Averroes, and negative responses critical of it, including those by Al-Ghazali and Muhammad Iqbal. Al-Ghazali was unconvinced by the first-cause arguments of Al-Kindi, arguing that only the infinite per se (that is an essentially ordered infinite series) is impossible, arguing for the possibility of the infinite per accidens (that is an accidentally ordered infinite series). He writes:

"According to the hypothesis under consideration, it has been established that all the beings in the world have a cause. Now, let the cause itself have a cause, and the cause of the cause have yet another cause, and so on ad infinitum. It does not behove you to say that an infinite regress of causes is impossible."

Muhammad Iqbal also stated:

"To finish the series at a certain point, and to elevate one member of the series to the dignity of an uncaused first cause, is to set at naught the very law of causation on which the whole argument proceeds."

Contemporary discourse

According to the atheist philosopher Quentin Smith, "a count of the articles in the philosophy journals shows that more articles have been published about Craig’s defense of the Kalam argument than have been published about any other philosopher’s contemporary formulation of an argument for God’s existence."

The Kalam cosmological argument has received criticism from philosophers such as J. L. Mackie, Graham Oppy, Michael Martin, Quentin Smith, physicists Paul Davies, Lawrence Krauss and Victor Stenger, and authors such as Dan Barker.

Modern discourse encompasses the fields of both philosophy and science (e.g. the fields of quantum physics and cosmology), which Bruce Reichenbach summarises as:

"... whether there needs to be a cause of the first natural existent, whether something like the universe can be finite and yet not have a beginning, and the nature of infinities and their connection with reality".

Since the temporal ordering of events is central, the kalam argument also brings issues of the nature of time into the discussion.

Premise one: "Whatever begins to exist has a cause."

Craig and Sinclair have stated that the first premise is obviously true, at least more plausibly true than its negation. Craig offers three reasons why the first premise is true:

1. Rational intuition: Craig states that the first premise is self-evidently true, being based upon the metaphysical intuition that "something cannot come into being from nothing", or "Ex nihilo nihil fit", originating from Parmenidean philosophy. He states that for something to come into being without any cause is to come into being from nothing, which he says is "surely absurd."
2. Reductio ad absurdum: if false, it would be inexplicable why just anything and everything does not randomly come into existence without a cause.
3. Inductive reasoning from both common experience and scientific evidence, which constantly verifies and never falsifies the truth of the first premise.

According to Reichenbach, "the Causal Principle has been the subject of extended criticism", which can be divided into philosophical and scientific criticisms.

Philosophical objections

Graham Oppy, J. L. Mackie and Wes Morriston have objected to the intuitiveness of the first premise. Oppy states:

"Mackie, [Adolf] Grunbaum, [Quentin] Smith and I—among many others—have taken issue with the first premise: why should it be supposed that absolutely everything which begins to exist has a cause for its beginning to exist?"

Mackie affirms that there is no good reason to assume a priori that an uncaused beginning of all things is impossible. Moreover, that the Causal Principle cannot be extrapolated to the universe from inductive experience. He appeals to David Hume's thesis (An Enquiry Concerning Human Understanding) that effects without causes can be conceived in the mind, and that what is conceivable in the mind is possible in the real world. This argument has been criticised by Bruce Reichenbach and G.E.M. Anscombe, who point out the phenomenological and logical problems in inferring factual possibility from conceivability. Craig notes:

"Hume himself clearly believed in the causal principle. He presupposes throughout the Enquiry that events have causes, and in 1754 he wrote to John Stewart, 'But allow me to tell you that I never asserted so absurd a Proposition as that anything might arise without a cause'".

Morriston asserts that causal laws are physical processes for which we have intuitive knowledge in the context of events within time and space, but that such intuitions do not hold true for the beginning of time itself. He states:

"We have no experience of the origin of worlds to tell us that worlds don't come into existence like that. We don't even have experience of the coming into being of anything remotely analogous to the “initial singularity” that figures in the Big Bang theory of the origin of the universe."

In reply, Craig has maintained that causal laws are unrestricted metaphysical truths that are "not contingent upon the properties, causal powers, and dispositions of the natural kinds of substances which happen to exist", remarking:

"The history of twentieth century astrophysical cosmology belies Morriston's claim that people have no strong intuitions about the need of a causal explanation of the origin of time and the universe."

Quantum physics

A common objection to premise one appeals to the phenomenon of quantum indeterminacy, where, at the subatomic level, the causal principle; "everything that begins to exist has a cause" appears to break down. Craig replies that the phenomenon of indeterminism is specific to the Copenhagen Interpretation of Quantum Mechanics, pointing out that this is only one of a number of different interpretations, some of which he states are fully deterministic (mentioning David Bohm) and none of which are as yet known to be true. He concludes that subatomic physics is not a proven exception to the first premise.

The philosopher Quentin Smith has cited the example of virtual particles, which appear and disappear from observation, apparently at random, to assert the tenability of uncaused natural phenomena. In his book A Universe from Nothing: Why There is Something Rather Than Nothing, cosmologist Lawrence Krauss has proposed how quantum mechanics can explain how space-time and matter can emerge from 'nothing' (referring to the quantum vacuum). Philosopher Michael Martin has also referred to quantum vacuum fluctuation models to support the idea of a universe with uncaused beginnings. He writes:

"Even if the universe has a beginning in time, in the light of recently proposed cosmological theories this beginning may be uncaused. Despite Craig's claim that theories postulating that the universe 'could pop into existence uncaused' are incapable of 'sincere affirmation,' such similar theories are in fact being taken seriously by scientists."

Philosopher of science David Albert has criticised the use of the term 'nothing' in describing the quantum vacuum. In a review of Krauss's book, he states:

"Relativistic-quantum-field-theoretical vacuum states—no less than giraffes or refrigerators or solar systems—are particular arrangements of elementary physical stuff. The true relativistic-quantum-field-theoretical equivalent to there not being any physical stuff at all isn’t this or that particular arrangement of the fields—what it is (obviously, and ineluctably, and on the contrary) is the simple absence of the fields."

Likewise, Craig has argued that the quantum vacuum, in containing quantifiable, measurable energy, cannot be described as 'nothing', therefore, that phenomena originating from the quantum vacuum cannot be described as 'uncaused'. On the topic of virtual particles, he writes:

"For virtual particles do not literally come into existence spontaneously out of nothing. Rather the energy locked up in a vacuum fluctuates spontaneously in such a way as to convert into evanescent particles that return almost immediately to the vacuum."

Cosmologist Alexander Vilenkin has stated that even "the absence of space, time and matter" cannot truly be defined as 'nothing' given that the laws of physics are still present, though it would be "as close to nothing as you can get".

Premise two: "The universe began to exist."

Craig defends premise two using both physical arguments with evidence from cosmology and physics, and metaphysical arguments for the impossibility of actual infinities in reality.

Cosmology and physics

For physical evidence, Craig appeals to:

1. Scientific confirmation against a past-infinite universe in the form of the Second Law of Thermodynamics.
2. Scientific evidence that the universe began to exist a finite time ago at the Big Bang.
3. The Borde–Guth–Vilenkin theorem, a cosmological theorem which deduces that any universe that has, on average, been expanding throughout its history cannot be infinite in the past but must have a past space-time boundary.

Professor Alexander Vilenkin, one of the three authors of the Borde-Guth-Vilenkin theorem, writes:

"A remarkable thing about this theorem is its sweeping generality. We made no assumptions about the material content of the universe. We did not even assume that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires some modification, our conclusion will still hold. The only assumption that we made was that the expansion rate of the universe never gets below some nonzero value, no matter how small."

Victor J. Stenger has referred to the Aguirre-Gratton model for eternal inflation as an exemplar by which others disagree with the Borde-Guth-Vilenkin theorem. In private correspondence with Stenger, Vilenkin remarked how the Aguirre-Gratton model attempts to evade a beginning by reversing the "arrow of time" at t = 0, but that: "This makes the moment t = 0 rather special. I would say no less special than a true beginning of the universe."

At the "State of the Universe" conference at Cambridge University in January 2012, Vilenkin discussed problems with various theories that would claim to avoid the need for a cosmological beginning, alleging the untenability of eternal inflation, cyclic and cosmic egg models, eventually concluding: "All the evidence we have says that the universe had a beginning."

Actual infinities

On the metaphysical impossibility of actual infinities, Craig asserts:

1. The metaphysical impossibility of an actually infinite series of past events by citing David Hilbert's famous Hilbert's Hotel thought experiment.
2. The impossibility of forming an actual infinite by successive addition, referencing Bertrand Russell's example of Tristram Shandy.

Michael Martin disagrees with these assertions by Craig, saying:

"Craig's a priori arguments are unsound or show at most that actual infinities have odd properties. This latter fact is well known, however, and shows nothing about whether it is logically impossible to have actual infinities in the real world."

Andrew Loke has argued against the metaphysical possibility of a beginningless universe as well as that of an actual infinite existing in the real world.

Another criticism comes from Thomist philosopher Dr. Edward Feser who claims that past and future events are potential rather than actual, meaning that an infinite past could exist in a similar way to how an infinite number of potential halfway points exist between any two given points.

Conclusion: "The universe has a cause."

Given that the Kalam cosmological argument is a deductive argument, if both premises are true, the truth of the conclusion follows necessarily.

In a critique of Craig's book The Kalam Cosmological Argument, published in 1979, Michael Martin states:

"It should be obvious that Craig's conclusion that a single personal agent created the universe is a non sequitur. At most, this Kalam argument shows that some personal agent or agents created the universe. Craig cannot validly conclude that a single agent is the creator. On the contrary, for all he shows, there may have been trillions of personal agents involved in the creation."

Martin also claims that Craig has not justified his claim of creation "ex nihilo", pointing out that the universe may have been created from pre-existing material in a timeless or eternal state. Moreover, that Craig takes his argument too far beyond what his premises allow in deducing that the creating agent is greater than the universe. For this, he cites the example of a parent "creating" a child who eventually becomes greater than he or she.

In the subsequent Blackwell Companion to Natural Theology, published in 2009, Craig discusses the properties of the cause of the universe, arguing that they follow as consequences of a conceptual analysis and of the cause of the universe and by entailment from the initial syllogism of the argument:

1. A first state of the material world cannot have a material explanation and must originate ex nihilo in being without material cause, because no natural explanation can be causally prior to the very existence of the natural world (space-time and its contents). It follows necessarily that the cause is outside of space and time (timeless, spaceless), immaterial, and enormously powerful, in bringing the entirety of material reality into existence.
2. Even if positing a plurality of causes prior to the origin of the universe, the causal chain must terminate in a cause which is absolutely first and uncaused, otherwise an infinite regress of causes would arise, which Craig and Sinclair argue is impossible.
3. Occam's Razor maintains that the unicity of the First Cause should be assumed unless there are specific reasons to believe that there is more than one causeless cause.
4. Agent causation, volitional action, is the only ontological condition in which an effect can arise in the absence of prior determining conditions. Therefore, only personal, free agency can account for the origin of a first temporal effect from a changeless cause.
5. Abstract objects, the only other ontological category known to have the properties of being uncaused, spaceless, timeless and immaterial, do not sit in volitional causal relationships.

Craig concludes that the cause of the existence of the universe is an "uncaused, personal Creator ... who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful"; remarking upon the theological implications of this union of properties.

Theories of time

Craig holds to the A-theory of time, also known as the "tensed theory of time" or presentism, as opposed to its alternative, the B-theory of time, also known as the "tenseless theory of time" or eternalism. The latter would allow the universe to exist tenselessly as a four-dimensional space-time block, under which circumstances the universe would not "begin to exist": The form of the kalam he presents in his earlier work rests on this theory:

"From start to finish, the kalam cosmological argument is predicated upon the A-Theory of time. On a B-Theory of time, the universe does not in fact come into being or become actual at the Big Bang; it just exists tenselessly as a four-dimensional space-time block that is finitely extended in the earlier than direction. If time is tenseless, then the universe never really comes into being, and, therefore, the quest for a cause of its coming into being is misconceived."

Craig has defended the A-theory against objections from J. M. E. McTaggart and hybrid A–B theorists. Philosopher Yuri Balashov has criticised Craig's attempt to reconcile the A-theory with special relativity by relying on a ‘neo‐Lorentzian interpretation’ of Special Relativity. Balashov claims:

"Despite the fact that presentism has the firm backing of common sense and eternalism revolts against it, eternalism is widely regarded as almost the default view in contemporary debates, and presentism as a highly problematic view."

Craig has criticised Balashov for adopting a verificationist methodology that fails to address the metaphysical and theological foundations of the A-theory.

It has recently been argued that a defense of the Kalam cosmological argument does not have to involve such a commitment to the A-theory. Craig has since modified his view of the A-theory being necessary for the Kalam, stating that while the Kalam would need to be reformulated, "it wouldn't be fatal" on a B-theory.