Hey, this is BIG NEWS. Stephen writes
> also must exist, thus we have the example of the Cantor Hierarchy. > > http://www.phschool.com/science/science_news/articles/infinite_wisdom.html GOOD GRIEF. Woodin may soon be up for *sainthood* among us mathematical platonists. Thanks, Stephen. Get a load of the following: To Platonists, mathematical objects such as sets exist in an ideal mathematical world, and axiomatic systems are merely useful tools for illuminating which statements about those objects are true in that world. To Platonists, the continuum hypothesis feels like a concrete statement that should be true or false. To them, if the standard axioms can't settle the continuum hypothesis, it's not that the hypothesis is a meaningless question, but rather that the axioms are insufficient. >From this point of view, Cohen's result indicates that mathematicians need to >add to their roster of axioms about infinite sets. There is a problem, however. An axiom should be so intuitively obvious that everyone agrees immediately that it's true. Yet intuition quickly evaporates when confronted with questions about infinity. Infinite Elegance In the decades that followed Cohen's 1963 result, mathematicians trying to settle the continuum hypothesis ran into a roadblock: While some people proposed new axioms indicating the continuum hypothesis was true, others proposed what seemed like equally good axioms indicating the it false, Woodin says. Woodin decided to try a different tack. Instead of looking for the missing axiom, he gathered circumstantial evidence about what the implications of that axiom would be. To do this without knowing what the axiom was, Woodin tried to figure out whether some axioms are somehow better than others. A good axiom, he felt, should help mathematicians settle not only the continuum hypothesis but also many other questions about Cantor's hierarchy of infinite sets. Mathematicians have long known that there is no all-powerful axiom that can answer every question about Cantor's hierarchy. However, Woodin suspected a compromise is possible: There might be axioms that answer all questions up to the level of the hierarchy that the continuum hypothesis concerns—the realm of the smallest uncountably infinite sets. Woodin called such an axiom "elegant." In a book-length mathematical argument that has been percolating through the set theory community for the last few years, Woodin has proved—apart from one missing piece that must still be filled in—that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false. "If there's a simple solution to the continuum hypothesis, it must be that it is false," Woodin says. And if it is false, then there are indeed infinite sets bigger than the counting numbers and smaller than the real numbers. Woodin's novel approach of sidestepping the search for the right axiom doesn't conform to the way mathematicians thought the continuum hypothesis would be settled, says Joel Hamkins of the City University of New York and Georgia State University in Atlanta. Mathematicians haven't yet absorbed the ramifications of Woodin's work fully enough to decide whether it settles the matter of the continuum hypothesis, says Akihiro Kanamori of Boston University (Mass.). "[It's] considered a very impressive achievement, but very few people understand the higher reaches [of Woodin's framework]," he says. Does Woodin himself believe that the continuum hypothesis is false? "If anyone should have an opinion on this, I should, but even I'm not sure," he answers. "What I can say is that 10 years ago, I wouldn't have believed there was a chance the continuum hypothesis was solvable. Now, I really think it has an answer."