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 

>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."

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