Many levels of infinity (Image: Emm.A/Marie Emmermann)
The mysteries of infinity could lead us to a fantastic structure above and beyond mathematics as we know it
WHEN David Hilbert left the podium at the Sorbonne in Paris, France, on 8 August 1900, few of the assembled delegates seemed overly impressed. According to one contemporary report, the discussion following his address to the second International Congress of Mathematicians was "rather desultory". Passions seem to have been more inflamed by a subsequent debate on whether Esperanto should be adopted as mathematics' working language.
Yet Hilbert's address set the mathematical agenda for the 20th century. It crystallised into a list of 23 crucial unanswered questions, including how to pack spheres to make best use of the available space, and whether the Riemann hypothesis, which concerns how the prime numbers are distributed, is true.
Today many of these problems have been resolved, sphere-packing among them. Others, such as the Riemann hypothesis, have seen little or no progress. But the first item on Hilbert's list stands out for the sheer oddness of the answer supplied by generations of mathematicians since: that mathematics is simply not equipped to provide an answer.
This curiously intractable riddle is known as the continuum hypothesis, and it concerns that most enigmatic quantity, infinity. Now, 140 years after the problem was formulated, a respected US mathematician believes he has cracked it. What's more, he claims to have arrived at the solution not by using mathematics as we know it, but by building a new, radically stronger logical structure: a structure he dubs "ultimate L".
The journey to this point began in the early 1870s, when the German Georg Cantor was laying the foundations of set theory. Set theory deals with the counting and manipulation of collections of objects, and provides the crucial logical underpinnings of mathematics: because numbers can be associated with the size of sets, the rules for manipulating sets also determine the logic of arithmetic and everything that builds on it.
These dry, slightly insipid logical considerations gained a new tang when Cantor asked a critical question: how big can sets get? The obvious answer - infinitely big - turned out to have a shocking twist: infinity is not one entity, but comes in many levels.
How so? You can get a flavour of why by counting up the set of whole numbers: 1, 2, 3, 4, 5... How far can you go? Why, infinitely far, of course - there is no biggest whole number. This is one sort of infinity, the smallest, "countable" level, where the action of arithmetic takes place.
Now consider the question "how many points are there on a line?" A line is perfectly straight and smooth, with no holes or gaps; it contains infinitely many points. But this is not the countable infinity of the whole numbers, where you bound upwards in a series of defined, well-separated steps. This is a smooth, continuous infinity that describes geometrical objects. It is characterised not by the whole numbers, but by the real numbers: the whole numbers plus all the numbers in between that have as many decimal places as you please - 0.1, 0.01, √2, π and so on.
Cantor showed that this "continuum" infinity is in fact infinitely bigger than the countable, whole-number variety. What's more, it is merely a step in a staircase leading to ever-higher levels of infinities stretching up as far as, well, infinity.
While the precise structure of these higher infinities remained nebulous, a more immediate question frustrated Cantor. Was there an intermediate level between the countable infinity and the continuum? He suspected not, but was unable to prove it. His hunch about the non-existence of this mathematical mezzanine became known as the continuum hypothesis.
Attempts to prove or disprove the continuum hypothesis depend on analysing all possible infinite subsets of the real numbers. If every one is either countable or has the same size as the full continuum, then it is correct. Conversely, even one subset of intermediate size would render it false.
A similar technique using subsets of the whole numbers shows that there is no level of infinity below the countable. Tempting as it might be to think that there are half as many even numbers as there are whole numbers in total, the two collections can in fact be paired off exactly. Indeed, every set of whole numbers is either finite or countably infinite.
Applied to the real numbers, though, this approach bore little fruit, for reasons that soon became clear. In 1885, the Swedish mathematician Gösta Mittag-Leffler had blocked publication of one of Cantor's papers on the basis that it was "about 100 years too soon". And as the British mathematician and philosopher Bertrand Russell showed in 1901, Cantor had indeed jumped the gun. Although his conclusions about infinity were sound, the logical basis of his set theory was flawed, resting on an informal and ultimately paradoxical conception of what sets are.
It was not until 1922 that two German mathematicians, Ernst Zermelo and Abraham Fraenkel, devised a series of rules for manipulating sets that was seemingly robust enough to support Cantor's tower of infinities and stabilise the foundations of mathematics. Unfortunately, though, these rules delivered no clear answer to the continuum hypothesis. In fact, they seemed strongly to suggest there might even not be an answer.
Agony of choice
The immediate stumbling block was a rule known as the "axiom of choice". It was not part of Zermelo and Fraenkel's original rules, but was soon bolted on when it became clear that some essential mathematics, such as the ability to compare different sizes of infinity, would be impossible without it.
The axiom of choice states that if you have a collection of sets, you can always form a new set by choosing one object from each of them. That sounds anodyne, but it comes with a sting: you can dream up some twisted initial sets that produce even stranger sets when you choose one element from each. The Polish mathematicians Stefan Banach and Alfred Tarski soon showed how the axiom could be used to divide the set of points defining a spherical ball into six subsets which could then be slid around to produce two balls of the same size as the original. That was a symptom of a fundamental problem: the axiom allowed peculiarly perverse sets of real numbers to exist whose properties could never be determined. If so, this was a grim portent for ever proving the continuum hypothesis.
This news came at a time when the concept of "unprovability" was just coming into vogue. In 1931, the Austrian logician Kurt Gödel proved his notorious "incompleteness theorem". It shows that even with the most tightly knit basic rules, there will always be statements about sets or numbers that mathematics can neither verify nor disprove.
At the same time, though, Gödel had a crazy-sounding hunch about how you might fill in most of these cracks in mathematics' underlying logical structure: you simply build more levels of infinity on top of it. That goes against anything we might think of as a sound building code, yet Gödel's guess turned out to be inspired. He proved his point in 1938. By starting from a simple conception of sets compatible with Zermelo and Fraenkel's rules and then carefully tailoring its infinite superstructure, he created a mathematical environment in which both the axiom of choice and the continuum hypothesis are simultaneously true. He dubbed his new world the "constructible universe" - or simply "L".
L was an attractive environment in which to do mathematics, but there were soon reasons to doubt it was the "right" one. For a start, its infinite staircase did not extend high enough to fill in all the gaps known to exist in the underlying structure. In 1963 Paul Cohen of Stanford University in California put things into context when he developed a method for producing a multitude of mathematical universes to order, all of them compatible with Zermelo and Fraenkel's rules.
This was the beginning of a construction boom. "Over the past half-century, set theorists have discovered a vast diversity of models of set theory, a chaotic jumble of set-theoretic possibilities," says Joel Hamkins at the City University of New York. Some are "L-type worlds" with superstructures like Gödel's L, differing only in the range of extra levels of infinity they contain; others have wildly varying architectural styles with completely different levels and infinite staircases leading in all sorts of directions.
For most purposes, life within these structures is the same: most everyday mathematics does not differ between them, and nor do the laws of physics. But the existence of this mathematical "multiverse" also seemed to dash any notion of ever getting to grips with the continuum hypothesis. As Cohen was able to show, in some logically possible worlds the hypothesis is true and there is no intermediate level of infinity between the countable and the continuum; in others, there is one; in still others, there are infinitely many. With mathematical logic as we know it, there is simply no way of finding out which sort of world we occupy.
That's where Hugh Woodin of the University of California, Berkeley, has a suggestion. The answer, he says, can be found by stepping outside our conventional mathematical world and moving on to a higher plane.
Woodin is no "turn on, tune in" guru. A highly respected set theorist, he has already achieved his subject's ultimate accolade: a level on the infinite staircase named after him. This level, which lies far higher than anything envisaged in Gödel's L, is inhabited by gigantic entities known as Woodin cardinals.
Woodin cardinals illustrate how adding penthouse suites to the structure of mathematics can solve problems on less rarefied levels below. In 1988 the American mathematicians Donald Martin and John Steel showed that if Woodin cardinals exist, then all "projective" subsets of the real numbers have a measurable size. Almost all ordinary geometrical objects can be described in terms of this particular type of set, so this was just the buttress needed to keep uncomfortable apparitions such as Banach and Tarski's ball out of mainstream mathematics.
Such successes left Woodin unsatisfied, however. "What sense is there in a conception of the universe of sets in which very large sets exist, if you can't even figure out basic properties of small sets?" he asks. Even 90 years after Zermelo and Fraenkel had supposedly fixed the foundations of mathematics, cracks were rife. "Set theory is riddled with unsolvability. Almost any question you want to ask is unsolvable," says Woodin. And right at the heart of that lay the continuum hypothesis.
Ultimate L
Woodin and others spotted the germ of a new, more radical approach while investigating particular patterns of real numbers that pop up in various L-type worlds. The patterns, known as universally Baire sets, subtly changed the geometry possible in each of the worlds and seemed to act as a kind of identifying code for it. And the more Woodin looked, the more it became clear that relationships existed between the patterns in seemingly disparate worlds. By patching the patterns together, the boundaries that had seemed to exist between the worlds began to dissolve, and a map of a single mathematical superuniverse was slowly revealed. In tribute to Gödel's original invention, Woodin dubbed this gigantic logical structure "ultimate L".
Among other things, ultimate L provides for the first time a definitive account of the spectrum of subsets of the real numbers: for every forking point between worlds that Cohen's methods open up, only one possible route is compatible with Woodin's map. In particular it implies Cantor's hypothesis to be true, ruling out anything between countable infinity and the continuum. That would mark not only the end of a 140-year-old conundrum, but a personal turnaround for Woodin: 10 years ago, he was arguing that the continuum hypothesis should be considered false.
Ultimate L does not rest there. Its wide, airy space allows extra steps to be bolted to the top of the infinite staircase as necessary to fill in gaps below, making good on Gödel's hunch about rooting out the unsolvability that riddles mathematics. Gödel's incompleteness theorem would not be dead, but you could chase it as far as you pleased up the staircase into the infinite attic of mathematics.
The prospect of finally removing the logical incompleteness that has bedevilled even basic areas such as number theory is enough to get many mathematicians salivating. There is just one question. Is ultimate L ultimately true?
Andrés Caicedo, a logician at Boise State University in Idaho, is cautiously optimistic. "It would be reasonable to say that this is the 'correct' way of going about completing the rules of set theory," he says. "But there are still several technical issues to be clarified before saying confidently that it will succeed."
Others are less convinced. Hamkins, who is a former student of Woodin's, holds to the idea that there simply are as many legitimate logical constructions for mathematics as we have found so far. He thinks mathematicians should learn to embrace the diversity of the mathematical multiverse, with spaces where the continuum hypothesis is true and others where it is false. The choice of which space to work in would then be a matter of personal taste and convenience. "The answer consists of our detailed understanding of how the continuum hypothesis both holds and fails throughout the multiverse," he says.
Woodin's ideas need not put paid to this choice entirely, though: aspects of many of these diverse universes will survive inside ultimate L. "One goal is to show that any universe attainable by means we can currently foresee can be obtained from the theory," says Caicedo. "If so, then ultimate L is all we need."
In 2010, Woodin presented his ideas to the same forum that Hilbert had addressed over a century earlier, the International Congress of Mathematicians, this time in Hyderabad, India. Hilbert famously once defended set theory by proclaiming that "no one shall expel us from the paradise that Cantor has created". But we have been stumbling around that paradise with no clear idea of where we are. Perhaps now a guide is within our grasp - one that will take us through this century and beyond.
Richard Elwes is a teaching fellow at the University of Leeds in the UK and the author of Maths 1001: Absolutely Everything That Matters in Mathematics (Quercus, 2010) and How to Build a Brain (Quercus, 2011)
Magazine issue 2823.
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