Justified True Belief

(●) The mathematician who proved that some infinities are larger than others, and who believed God had shown him how. + For two thousand years the settled answer about infinity was that there is no such thing as a finished one. You can keep counting, and you can keep dividing, and nobody ever reaches the end, but the endlessness is a promise about what you could do next rather than a thing sitting there complete. Aristotle said so, the medieval schoolmen said so after him, and the great mathematicians of the early nineteenth century said so too. Then a professor at a small German university sat down and treated the completed infinite as an ordinary object of study, gave it a number, and did arithmetic with it. Within a generation the whole of mathematics was being rebuilt on what he found. Georg Cantor matters on a Christian site for two reasons that pull in different directions, which is part of what makes him worth the shelf. The first is that he was a believing Lutheran who thought his mathematics was theology’s business as much as arithmetic’s, who wrote to Catholic theologians and to a cardinal of the Roman Curia asking whether his infinite numbers endangered the doctrine of God, and who kept the cardinal’s favourable answer close for the rest of his life. The second is that the mathematics he built is now the standing objection to one of the oldest arguments for God’s existence, the case that the universe cannot have an endless past. Defenders of that case accept every line of his arithmetic and deny that it can be laid over the furniture of the world. Cantor would have denied the denial. That argument is still running. One fact about him is worth carrying away before anything else. The theory that made him famous also made him a target, and at thirty-nine he broke down for the first time. From 1899 he went back again and again to the nerve clinic at Halle, sometimes for whole winters, once for the last seven months of his life. He did the work anyway, between the returns.

(●) 1845 AD: Born in Saint Petersburg, and handed a letter at his confirmation that he carried for the rest of his life. + Georg Ferdinand Ludwig Philipp Cantor was born in Saint Petersburg on 3 March 1845, the eldest of six children. His father, Georg Waldemar Cantor, was a merchant who had been educated at the Lutheran mission in the city and traded on its stock exchange, a serious and pious man whose letters to his son read like a pastor’s. His mother, Maria Anna Böhm, was Roman Catholic and came from a family of musicians. The household was full of music, and Georg grew up a good enough violinist that people remarked on it. In 1856, when the boy was eleven, his father’s health failed in the Russian winters and the family moved to Germany: first Wiesbaden, then Frankfurt, with school for Georg at Darmstadt. Four years later, when he was fifteen and about to be confirmed, his father wrote him a letter. Many gifted men, the letter said, are broken by the people who resist their ideas, and only an unshakable religious conviction would keep his son from ending up as one more so-called ruined genius. What his father wanted for him was that he should become a shining star on the horizon of science. Cantor kept that letter. Read now, knowing what came after, it is hard not to read it as a forecast: the gift, the resistance, and the faith he held onto through all of it.

(●) 1863 AD: He gave up engineering for mathematics, and at Berlin sat under Leopold Kronecker, who would fight his work to the end of his own life. + His father had picked a career for him, as fathers did: engineering, a respectable profession with a future, and a technical school at Darmstadt to prepare for it. In the spring of 1862, with that schooling nearly done, he wrote home asking to be let off. He wanted mathematics. His father said yes, and the son, overjoyed, went that autumn to the Polytechnic at Zürich to study it; he remembered the permission with gratitude for decades. In June 1863 his father died, and Cantor moved to the University of Berlin, which was then the best place in the world to learn mathematics. He heard Karl Weierstrass, who was teaching a generation how to make the calculus rigorous, and Ernst Kummer, and Leopold Kronecker. The last of those names matters. Kronecker was brilliant, combative, and convinced that mathematics should deal only with objects you can construct in a finite number of steps. His student would end up building the most spectacular violation of that rule anyone had attempted, and Kronecker would spend the rest of his life saying so. Cantor took his doctorate at Berlin in 1867 with a dissertation in number theory, De aequationibus secundi gradus indeterminatis, on whole-number solutions to a certain kind of quadratic equation. Appended to it, in the custom of the day, were three short theses in Latin that he had to defend in the examination hall that December, and the third of them reads like a mission statement for everything he did afterwards: in mathematics, the art of proposing a question is to be held of higher value than solving it.

(●) 1874 AD: From a post at provincial Halle he proved that the points on a line cannot be counted. + In 1869 Cantor took a junior teaching post at the University of Halle, a modest place compared with Berlin or Göttingen, and he never left it. He climbed the German ladder there, made extraordinary professor in 1872, which is the salaried rank below a full chair rather than a compliment, and full professor in 1879. What he wanted was a call from one of the great universities, and he waited for it for the rest of his career and never received it. He became convinced Kronecker was blocking him. Whether that was true in the form he believed it is another matter, but he believed it, and it ate at him. Halle gave him one thing Berlin might not have: a problem. Eduard Heine, a senior colleague there, was working on trigonometric series, the sums of sine and cosine waves used to represent a function, and he put to Cantor the question of whether such a representation is unique. Could the same curve be built out of those waves in two different ways? Cantor solved it in 1870, showing that it could not, and then kept pushing, allowing more and more exceptional points where the representation might misbehave. The pushing forced two pieces of groundwork, and the second of them is where the infinite got in. First he had to say exactly what a real number is, a real number being any number that marks a point on a line: the whole numbers and the fractions, together with the ones that are neither, like the square root of two. He gave the definition in 1872 by taking sequences of fractions that close in on a value and letting the sequence itself stand for the value it closes in on. Second, he had to work with an operation that takes a set of points and hands back another set of points: the limit points of the first set, meaning the spots that have members of the set crowding in on them however closely you look. Take the set of exceptional points, take its limit points, then take the limit points of that, and keep going. Nothing in the counting numbers tells you when to stop. After the first stage, the second, the third and every stage you can count to, there can still be points left over, and what Cantor wanted to talk about was the stage that comes after all of them, and then the stage after that one. There is no ordinary number to label those stages with. He had to count past every ordinary number to get one. The infinite arrived in his work as a tool he needed, not as a doctrine he went looking for. Then came the year everything changed. In late 1873 he wrote to Richard Dedekind, the friend he had met on holiday in Switzerland the year before and with whom he would trade letters for a quarter of a century, with a proof that the real numbers cannot be arranged in a single endless list. The fractions can. The whole numbers can. The points on a line cannot, and no cleverness will make them. He published it in 1874. In August of that same year he married Vally Guttmann, a friend of his sister’s, and spent much of the honeymoon at Interlaken talking mathematics with Dedekind, who happened to be there; they went on to have six children. Three years later he proved something stranger still, that every point of a square can be matched one for one with a point of a single line segment, so that a two-dimensional patch has exactly as many points as a one-dimensional strip. He sent it to Dedekind on 29 June 1877 with a line in French that has followed him ever since: I see it, but I do not believe it.

(●) 1883 AD: His book the Grundlagen defended the completed infinite, and Kronecker turned a disagreement into a campaign. + Between 1879 and 1884 Cantor published a six-part series in the Mathematische Annalen called "On infinite linear point-manifolds." The fifth part, which appeared in 1883, was so much more ambitious than the rest that he had it reprinted the same year as a book of its own: Grundlagen einer allgemeinen Mannigfaltigkeitslehre, foundations of a general theory of manifolds, with the subtitle "a mathematical-philosophical essay in the doctrine of the infinite." That subtitle is not decoration. Roughly a third of the book argues with Aristotle, Augustine, Thomas Aquinas, Spinoza, Leibniz and Kant about whether a finished infinite is so much as possible. The mathematics in it was the transfinite numbers: a first number after all the counting numbers, which he wrote as ω, then ω plus one, and on up a ladder that never ends. The philosophy in it was his claim that mathematics does not need permission from metaphysics to introduce a new number, only consistency and clear definition. And the theology in it, tucked into the endnotes, was the distinction that governed everything else he wrote about God. Beyond the whole endless ladder stands what he called the Absolute, which is not the top rung and not a number at all, but God, who is not on the ladder in the first place (T3). Kronecker was unmoved and unrelenting. He had already delayed one of Cantor’s papers at Crelle’s Journal in 1878, which reached print only after Dedekind intervened. He is remembered for a remark Heinrich Weber reported in a memorial notice two years after his death, that the whole numbers were made by God and everything else is the work of man, which is a fair summary of why he could not accept a word of Cantor’s programme. Cantor complained to friends that Kronecker was calling his work humbug. The sharper epithets often repeated in later accounts, that Kronecker called him a corrupter of youth and a charlatan, come to us through report rather than through anything in Kronecker’s own hand, and are worth holding loosely. The professional hostility needs no exaggeration. It was real, it was sustained, and it cost Cantor the career he wanted.

(●) 1884 AD: A collapse at the end of May began an illness that kept coming back for the rest of his life. + It came without warning. At the end of May 1884 Cantor broke down, and by late June he was writing to his Swedish friend Gösta Mittag-Leffler that he did not know when he would return to scientific work, that at the moment he could do absolutely nothing with it. He recovered within weeks that first time. He did not recover permanently. What was pressing on him was partly the opposition and partly a question he could not answer. He had proved that the points on a line are more numerous than the counting numbers, and he had conjectured that there is nothing in between, no size of infinity larger than the counting numbers and smaller than the line. That conjecture is the continuum hypothesis, and he attacked it for the rest of his life, some seasons convinced he had proved it, other seasons convinced he had refuted it, and he settled it neither way. In 1885 Mittag-Leffler asked him to withdraw a paper already set in type, saying it was about a hundred years too soon; Cantor answered with a joke about being asked to wait until 1984, and the friendship cooled. From 1899 the illness had him for good. He took medical leave in the winters of 1899 to 1900, 1902 to 1903, 1904 to 1905 and 1907 to 1908, and spent long stretches in the nerve clinic at Halle. His mother died in 1896, a younger brother in 1899, and on 16 December of that same year his youngest son Rudolph died suddenly. From the first illness onwards the enforced idle months went into Elizabethan literature, and he became convinced that Francis Bacon had written the plays published under Shakespeare’s name, printing pamphlets on it in 1896 and 1897. Modern writers are careful not to say the mathematics caused the illness; what the record actually shows is a recurring depressive illness, an unusually hostile professional life, and a man who kept working in the gaps.

(●) 1895 AD: His last mathematical papers, the Beiträge, set out the finished theory, and within two years the first contradictions inside it surfaced. + The last two mathematical papers he published are the summary of everything: Beiträge zur Begründung der transfiniten Mengenlehre, contributions to the founding of the theory of transfinite numbers, in two parts in 1895 and 1897. They open with the definition of a set that students still learn, and then sort the infinite in two directions at once, a distinction worth carrying away. A cardinal number answers how many, and two sets have the same cardinal when their members pair off one for one. An order type answers in what arrangement. The counting numbers in their usual order, and the counting numbers with 1 picked up and put after all the rest, have exactly the same cardinal and different order types, because rearranging a set does not change how many things are in it. Order types of well-ordered sets, meaning sets arranged so that any part you pick out of them has a first member, are the ordinal numbers, and ω is the first infinite one. On top of that the two papers lay out the arithmetic of the alephs, the letter ℵ from the Hebrew alphabet that Cantor chose for the infinite sizes. This is the pair of papers Philip Jourdain translated into English in 1915, which is how most English readers meet him. Two things had gone right in the meantime. Cantor had spent years trying to build a forum where new mathematics could get a hearing without the permission of a small circle in Berlin, and in 1890 the Deutsche Mathematiker-Vereinigung, the German Mathematical Society, was founded with him as its first president. At its first meeting, at Halle in 1891, he read a three-page proof of his old result about the real numbers by a completely new method: line up any list of them, then build a new number that differs from the first in its first digit, from the second in its second, and so on down the diagonal. The new number cannot be on the list, and the list was any list. That argument, printed in the society’s first yearbook, is now taught in first-year courses everywhere, and the same trick shows there is no largest infinity at all. Then the ground shifted. By 1896 and 1897 Cantor had noticed that certain collections break the theory, and it is worth seeing how quickly they break it. Gather every ordinal number into one set. That set is well-ordered like any other, so it has an ordinal of its own, and that ordinal has to be larger than every ordinal in the set while also being one of them. The same trap springs on the collection of all the cardinal numbers. Treat either of them as a set and the contradiction arrives within a line or two. He wrote to David Hilbert about it; Cesare Burali-Forti published one version in 1897; Bertrand Russell found another in 1901, and it was Russell’s that caused the crisis. Cantor’s own answer, worked out in letters to Dedekind in the summer of 1899, was to divide multiplicities in two. Some can be thought of as one finished thing, and those are sets. Some cannot, because the assumption that all their members are together contradicts itself, and those he called absolutely infinite. The Absolute of his theology had turned into a working distinction inside his mathematics.

(●) 1918 AD: He died of a heart attack in the Halle clinic, in the last winter of the war. + The honours came late and awkwardly. In 1911 the University of St Andrews invited him to its five hundredth anniversary as a distinguished foreign scholar. He went, talked at length to anyone who would listen about Bacon and Shakespeare, and travelled on to London hoping to meet Bertrand Russell and argue about the Principia Mathematica; word that his son had fallen ill turned him back to Germany before the meeting came off. St Andrews gave him an honorary doctorate in law the next year, which he was too unwell to collect in person. He retired from teaching in 1913. The war brought shortages to Halle, and Cantor, an old man in poor health, went hungry with everyone else; the celebration planned for his seventieth birthday in 1915 was cut down to a small gathering at home. In June 1917 he entered the sanatorium for the last time and asked repeatedly to be allowed to go home. He died there of a heart attack on 6 January 1918. Seven years later, in a lecture called "On the infinite" that went into print the following year, David Hilbert, the most commanding mathematician of the next generation, said a sentence about him that has been quoted ever since: no one shall drive us out of the paradise which Cantor has created for us. The verdict of the profession had gone the other way from the verdict of Cantor’s own lifetime, and it has stayed there.

What He Taught

(T1) Some infinities are larger than others. + Walk into a concert hall where every seat is taken and nobody is standing. Without counting anybody, and without counting the seats, you know there are exactly as many people as chairs, because each person is sitting in one chair and each chair has one person in it. That pairing is the whole idea. Cantor took it and made it the definition of same size, and then did the daring thing, which was to keep using it after the collections stopped being finite. - A set (his German word was Menge) is any collection of definite, distinguishable things gathered into a whole, whether the things are numbers, points, or chairs. - Two sets are the same size when they can be matched one for one with nothing left over on either side. Cantor called the size of a set its power, and later its cardinal number. - A set is countable when it can be matched against the counting numbers 1, 2, 3 and so on, which is the same as saying its members can be written out in a single endless list. Now the results, and they come fast. The even numbers can be matched against all the counting numbers: pair 1 with 2, 2 with 4, 3 with 6, forever. So a set can be the same size as a part of itself. Galileo had noticed this in 1638 and concluded that size simply does not apply to infinite collections; Cantor concluded instead that this is what infinite means, and kept going. The fractions are countable too, which is startling, since between any two of them lie infinitely many more. Even the algebraic numbers can be put in one list. An algebraic number is one that solves a polynomial equation with whole-number coefficients, which is to say an equation built out of nothing but whole numbers and powers of x: the square root of two solves x squared minus 2 equals 0, so the square root of two is algebraic. Then the real numbers, which is to say the points on a line, and there the run of successes stops. Cantor proved in 1873 that no list can hold them all, and in 1891 he gave the argument that is now famous for its simplicity. Suppose somebody hands you a list claiming to contain every decimal between 0 and 1. Build a new decimal by making its first digit different from the first digit of the first number on the list, its second digit different from the second digit of the second number, and so on down the diagonal. The number you have built differs from every number on the list in at least one place, so it is not on the list. But the list was supposed to contain everything. So no such list exists, and the points on a line are a strictly larger infinity than the counting numbers. The same argument, generalised, shows something with no floor and no ceiling: the collection of all subsets of a set is always larger than the set itself. Take any infinity you like, however large, and there is a larger one. The ladder goes up forever.

(Q) "We say that two aggregates M and N are 'equivalent' ... if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other." + Source: Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, section 1 (Jourdain translation, 1915). Jourdain’s "aggregate" is Cantor’s Menge, which everyone now calls a set. The sentence is doing more work than it looks: everything Cantor built about the sizes of infinite collections rests on taking this pairing test seriously past the point where counting gives out (T1).

(T2) The infinite can be a finished whole, not only an endless process. + Think of a road that runs on with no end. There are two quite different things you might mean by calling it infinite. You might mean that however far you drive there is always more road, which is a statement about your driving. Or you might mean that the road, taken as one object, all of it at once, has an endless amount of pavement in it. The first is a claim about a process that never finishes. The second is a claim about a finished thing. Philosophers had names for these long before Cantor, and the tradition allowed only the first. - The potential infinite is a quantity that grows past any bound but is always finite at any moment. Cantor called it the improper infinite, and remarked that it is really just a variable finite. - The actual infinite, which he called the proper infinite, is an infinite collection considered as complete, all its members there at once. - The scholastic formula for the ban was infinitum actu non datur, there is no infinite in actuality. Aristotle had argued for it, the medieval schools inherited it, and as late as 1831 Carl Friedrich Gauss, the greatest mathematician of the age, wrote that he protested against the use of an infinite magnitude as something completed, which is never allowed in mathematics. Cantor broke the ban, and he was honest that it went against his own training to do it. His new numbers were not larger and larger finite quantities; they were the completed infinite given a name and a place in an arithmetic. After all the counting numbers comes ω, and after ω comes ω plus one, and the arithmetic up there is strange but consistent: one plus ω is ω, while ω plus one is something new. You might well want to stop him there and say that this cannot be right, that two plus three and three plus two come to the same thing and always will. So they do, for counting numbers. Up here the plus sign is not counting objects, it is laying arrangements end to end, and the order in which you lay them down is part of the answer. Put one extra item in front of an endless queue and you still have an endless queue, everybody shuffled back one place. Put it behind every member of an endless queue and you have something genuinely new: an arrangement with a last item that has no item immediately before it. Christian readers should notice both what this settles and what it does not. It settles that the actual infinite is not self-contradictory as a piece of mathematics, which removes one old objection to it. It does not settle whether an actual infinite can be realised in the world of concrete things, out among the stars and the past seconds, and that second question is where the argument with Cantor is still being had.

(Q) "I was logically forced, almost against my will, because in opposition to traditions which had become valued by me in the course of scientific researches extending over many years, to the thought of considering the infinitely great, not merely in the form of the unlimitedly increasing ... but also to fix it mathematically by numbers in the definite form of a completed infinite." + Source: Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (1883), quoted in Jourdain’s introduction to Contributions to the Founding of the Theory of Transfinite Numbers (1915). This is a man saying he did not want the result he got. The traditions he means are the ones he had been taught at Berlin and had believed, and he adds in the next breath that he does not think any objection can be brought against the step which he is unable to answer (T2).

(T3) Above every number, however large, stands the Absolute, which is God. + Walk north for a day and you are a day’s walk further north, but you are not measurably closer to the stars. Cantor thought the ladder of infinite numbers was like that. You can always climb another rung, and every rung leaves the same unclimbable distance above it. Cantor had a precise reason for saying so, and it is worth having. Climb to any transfinite number you like, and the numbers still waiting above it are at least as numerous as everything you passed on the way up. The climbing never puts a dent in what is left. Whatever stands above the entire ladder, therefore, cannot be the largest number on it. It has to be a different kind of thing. He sorted the infinite into three, and kept the three apart with some care: - The Absolute, which he named in Latin the infinitum aeternum increatum, the eternal uncreated infinite. This belongs to God and God’s attributes alone, admits of no increase and no measurement, and is not a number. - The transfinite, the infinitum creatum, the created infinite. This is what the alephs and the ω numbers measure. It is genuinely infinite and genuinely exceeded, which is why it can be counted with. - The finite, the ordinary numbers, which everybody already granted. The distinction was not a pious afterthought bolted onto the mathematics. It did work. When the paradoxes arrived in the late 1890s, Cantor’s reply was that the collection of all the ordinal numbers is an absolutely infinite multiplicity, one that cannot be gathered into a single finished thing at all, and so is not a set and does not obey the rules for sets. The theological category became the mathematical firewall. Scripture says something close to the negative half of this, and says it as worship rather than as arithmetic: "Great is our Lord, and abundant in power; his understanding is beyond measure" (Psalm 147:5), and Job’s friend asking whether anyone can find out the limit of the Almighty (Job 11:7). The psalmist trying to count God’s thoughts gives up at the sand (Psalm 139:17–18). Cantor’s contribution was to say why the giving up is not a failure of effort. Some things are not large. They are outside the scale.

(Q) "The Absolute can only be recognized, but never apprehended, even approximately." + Source: Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (1883), quoted in Jourdain’s introduction to Contributions to the Founding of the Theory of Transfinite Numbers (1915). Cantor is drawing a line between two German words, anerkennen, to acknowledge that a thing is there, and erkennen, to know what it is. He goes on to say that the endless sequence of numbers seems to him a fitting symbol of the Absolute, precisely because you can never reach it by climbing (T3).

(T4) A new number earns its place by being consistent, not by being useful. + Every kind of number now taught to schoolchildren was once called absurd by serious people. Negative numbers were resisted for centuries: how can you have less than nothing? The square root of minus one was named "imaginary" as an insult, by mathematicians who used it while refusing to believe in it. In each case the objection was the same, that the new number does not correspond to anything out there, and in each case the objection eventually lost. Cantor turned that history into a principle. He distinguished two senses in which a mathematical concept might be called real. There is the reality it has inside thought, where it takes a determined place in our understanding, is clearly marked off from every other concept, and stands in fixed relations to the ones already accepted. And there is the reality it might have as an image of something out in the world. Applied mathematics has to care about the second. Pure mathematics, he argued, need only check the first, and a concept that passes that test has earned the name of number whether or not anyone can point at an instance of it. He gave Kummer’s ideal numbers as his example: invented to solve a problem inside number theory, corresponding to nothing anyone could point at, and now indispensable. The sentence he is best remembered for in Germany puts it flatly: the essence of mathematics lies precisely in its freedom. There is a safeguard built in, and he says so. The freedom is not licence, because a concept that is barren shows it very soon by being useless and is quietly dropped, and because the requirement of non-contradiction is severe. Still, this is the doctrine that let him introduce ω over Kronecker’s objections, and it is the doctrine that lets a Christian reader accept his arithmetic without swallowing his metaphysics. Consistency inside a system of thought is one thing. Whether the system describes what God actually made is a further question, and Cantor himself thought the answer was yes, for reasons drawn from theology rather than from mathematics.

(Q) "Mathematics is, in its development, quite free, and only subject to the self-evident condition that its conceptions are both free from contradiction in themselves and stand in fixed relations, arranged by definitions, to previously formed and tested conceptions." + Source: Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (1883), section 8, quoted in Jourdain’s introduction to Contributions to the Founding of the Theory of Transfinite Numbers (1915). Written by a man whose new numbers were being refused a hearing, this is both a philosophy of mathematics and a plea for his own work to be judged on whether it contradicts itself rather than on whether it offends anyone (T4).

(Q) "My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all, because I have followed its roots, so to speak, to the first infallible cause of all created things." + Source: Cantor to Carl Friedrich Heman, professor of theology at Basel, 21 June 1888, quoted in Joseph Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (1979), page 298. Everything that made Cantor formidable and everything that made him hard to argue with sits in these few lines. He gives a theologian three reasons for his confidence, and the last of them, the one he leans on hardest, is not mathematical at all. He has traced the transfinite back to God, and that is why he expects the objections to fail.

What Christian Thinkers Made of Him

(†) The theologians he wrote to accepted his infinite sooner than the mathematicians did. + It is one of the pleasant surprises of his story that while Berlin was calling his work humbug, Rome was reading it carefully. The timing helped. Leo XIII had issued Aeterni Patris in 1879, setting Catholic philosophy back to work on Thomas Aquinas, and one of the questions that revival reopened was exactly Cantor’s: whether an actual infinite is possible in the created order. Constantin Gutberlet, a priest at Fulda, had already published a book on the infinite in 1878 arguing for a version of it, and he took up Cantor’s transfinite numbers in his own defence. The correspondence that mattered most was with Cardinal Johann Baptist Franzelin, a Jesuit who had been among the leading theologians at the First Vatican Council. In 1886 Franzelin put to Cantor the objection that a careful theologian would put: if the created world can be actually infinite in the same sense in which God is infinite, then the difference between God and the world has been quietly erased, and what is left is pantheism, the doctrine that God and the universe are the same thing. That is not a quibble. It is the objection on which the whole question turns for a monotheist. Cantor answered with the distinction he had already drawn in the Grundlagen: the eternal uncreated infinite belongs to God alone and is beyond number, while the transfinite is created, counted, and always exceeded by something larger, so that the two are not infinite in the same sense at all (T3). Franzelin accepted it, and wrote back that the two concepts are essentially different, so that the one should be called properly infinite and the other only equivocally so, and that understood in that way he saw no danger to religious truth in the notion of the transfinite. Cantor quoted that sentence for the rest of his life. He was a Lutheran corresponding with Catholic theologians for the plain reason that they were the people then doing serious work on the question, and he also wrote to Ignatius Jeiler and Thomas Esser among others; his own published collection of these exchanges, the Mitteilungen zur Lehre vom Transfiniten, is where much of the correspondence can still be read.

(†) Mathematics kept what he built and rebuilt the foundations under it. + For roughly fifteen years his work was fought over, and then it won, so completely that the fight is now hard to imagine. Ernst Zermelo, who would later edit Cantor’s collected papers, published in 1908 a short list of axioms saying which collections may be treated as sets. An axiom is a rule you agree to start from rather than prove, the way the rules of chess are not won or lost but settled before the first move, and Zermelo’s list, extended by Abraham Fraenkel, is the foundation on which ordinary mathematics still rests. What the axioms do about the paradoxes is essentially what Cantor had proposed in his letters to Dedekind in 1899: some multiplicities are too big to be gathered into one thing, and the rules simply do not let you form them. The continuum hypothesis had a stranger end. Kurt Gödel showed in 1940 that it cannot be disproved from the standard axioms, and Paul Cohen showed in 1963 that it cannot be proved from them either. The question that Cantor worked at through every good season of his adult life is not merely unanswered; it cannot be settled by the assumptions he left behind. Somebody just needs to find the right extra rule, you might think, and mathematicians have spent decades proposing candidates and arguing about which ones deserve to be believed. The trouble is that nothing inside the system Cantor left behind picks one out. That is not a verdict against him. It is a discovery about the limits of the axioms, and it came out of the machinery he built. Hilbert’s line about the paradise Cantor created is quoted so often that its edge has worn off. He gave it in 1925 and printed it the next year as a defence, in the middle of a real dispute about whether the whole transfinite enterprise should be abandoned, and it is the considered judgement of the profession on a man most of whose contemporaries thought he had gone too far. Today no mathematics student reaches the end of a first year without meeting the diagonal argument.

(†) Where Christians part company with him: the infinite he placed inside creation, and the certainty he claimed for it. + Two things in Cantor’s own record, not in what anyone later did with his work, are worth arguing with. - He put the actual infinite into the created world, and argued for it from the nature of God. Cantor did not claim only that the transfinite is coherent as mathematics. He held that it is realised in creation, and his argument ran from God’s perfections to the world: from the highest perfection of God’s being he inferred the possibility of creating an ordered transfinite, and from God’s goodness and greatness he inferred that such a creation had actually taken place. The step from what God could do to what God has done is the weak joint, and Scripture does not license it: "Our God is in the heavens; he does all that he pleases" (Psalm 115:3). Creation answers to God’s free will, not to a requirement that he must do the most he can. Franzelin pressed exactly this point, and Cantor softened the claim in reply, saying that the necessity he had in mind was a necessity for us in inferring, not a necessity laid on God. The softened version is more defensible, and it is also much less than what he wanted. • He also tried to cash the claim out in physics. In a paper published in Acta Mathematica in 1885 he proposed, borrowing Leibniz’s word monad for the ultimate units things are made of, that matter is built of corporeal monads as numerous as the counting numbers while the ether is built of aetherial monads of the next power up, which he took to be the number of points on a line. The physics went nowhere and no one followed him into it, but it shows how literally he meant the claim. • Defenders of the kalam argument, the case that whatever begins to exist has a cause, that the universe began to exist, and that it therefore has a cause, accept Cantor’s mathematics in full and part company at this same joint. Their claim is that an actual infinite is consistent as a system of thought and cannot be instantiated in a collection of concrete things, which is why they think the past cannot be endless. Cantor would have denied the distinction, and honest accounts of the exchange do not declare a winner. - He treated his conviction as something given rather than argued. Cantor told correspondents that the transfinite numbers had been communicated to him, that he had recognised them with God’s help, and that he was only the instrument of a higher power rather than their author. A man may believe that about his own insight and be right. The difficulty is what he did with it: he made the conviction his chief reason for expecting every objection to fail, which is a way of holding a claim that no argument can reach. God gives understanding through ordinary means, through study, correction and other people, and a mathematical claim has to answer to proof whoever supplied the idea. Cantor’s own answer to the paradoxes came, in the end, not from the certainty but from going back to the mathematics and finding the distinction that saved it. None of this cancels the debt. Cantor took a question the church had argued about since Augustine, whether the infinite can be complete, and was the first to answer it with an arithmetic instead of an argument. He kept God carefully outside the number system while insisting that the number system was God’s business. And his father’s warning about what unbelief and opposition do to a gifted man turned out to be a good deal more accurate than either of them could have wanted.