Tuesday, August 2, 2011

Wow! The Continuum Hypothesis Solved???...

...well, I doubt it, but what do I know: Apparently a major Berkeley mathematician/set theorist, Hugh Woodin, using a "radically stronger logical structure" known as "ultimate L," believes he has accomplished what no one has been able to do in well over a century, and demonstrate that Gödel's Cantor's Continuum Hypothesis is true (essentially, that there exist no infinite sets lying between the set of integers and the set of real numbers -- of course it still all hinges on the initial axiomatic system one adapts):

http://www.newscientist.com/article/mg21128231.400-ultimate-logic-to-infinity-and-beyond.html?full=true

(Coincidentally, this fascinating article is from Richard Elwes who I was just highlighting a few days back.)

If you're not interested in infinity or sets, skip this article; otherwise, dive in!

3 comments:

Sam Alexander said...

In any particular model of ZFC, CH is either true or false. Examples of both are known (Godel found an example where CH is true, Cohen found one where CH is false). This is no more deep nor mystical than saying that some groups are abelian, and some are not: "abelianness is independent of the group axioms".

Neither is it particular surprising that CH be a consequence of certain stronger systems than ZFC. (One such stronger system-- trivially-- being ZFC+CH). Whether such extended systems are more "right" than ZFC or not is subjective and philosophical, not mathematics. Of course, kudos to Woodin anyway. The man's a living god when it comes to large cardinal axioms!

Anonymous said...

CH is much better attributed to Cantor, not Godel.

Shecky R. said...

Of course!! Ughhh, another senior moment (corrected above)...