http://friam.383.s1.nabble.com/no-one-shall-expel-us-from-the-paradise-that-Cantor-has-created-Hugh-Woodin-s-ultimate-L-Richard-Elw8-tp6699752p6701367.html
Wow. Thanks for passing on such a refreshing and informative article.
> "no one shall expel us from the paradise that Cantor has created",
> Hugh Woodin's "ultimate L": Richard Elwes: Rich Murray 2011.08.18
>
>
http://www.newscientist.com/article/mg21128231.400-ultimate-logic-to-infinity-and-beyond.html?full=true>
> Ultimate logic: To infinity and beyond
>
> 01 August 2011 by Richard Elwes
> Magazine issue 2823.
>
> 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)
>
>
> within mutual service, Rich Murray
>
[hidden email] 505-819-7388 Skype audio, video rich.murray11
>
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