- (Mathematical Realism or Platonist) There IS a model of the reals that is the RIGHT one.In that model CH is either true of false. ZFC just isn't up to the task of figuring it out.Paul Cohen thought that there were an INFINITE number of cardinalities between N and R.I've heard rumors that Kurt Godel thought there was exactly ONE cardinality between N and R.Hugh Woodin has some mathematical reasons to think there is exactly ONE:CHone CHtwo. Many people prefer the simplicity of having NONE---the infinity after N is R. Some people think that we need to add new axioms to ZFC such as Large Cardinals or the Axiom of Determinacy to settle the question. Are these really candidates for axioms?That may be a later post.
- (Not sure what these people are called.) Since ZFC settles virtually everything else in mathbut not this question, CH has no answer. There is No `correct' copy of the reals.The weakness in this response may be the virtually. Are there questions in math that need it? Are there such questions outside of Set Theory? That may be a later post.
Monday, April 02, 2007
What to make of the Ind of CH ?
Dave Barrington suggested I blog about Paul Cohen since he just died. Scotts Blog already reported on Paul Cohen's death, and there were many comments on C* algebras and PAC learning (none of which Paul Cohen worked on). Paul Cohen's most important result was that CH is independent of ZFC. What does this mean and what do we make of it? CH is the statement there is no cardinality strictly between N and R ZFC is Zermelo-Frankl Set Theory (with the Axiom of Choice). Virtually all of Math can be derived from these axioms. (There are quibbles about this which might be a latter blog.) Kurt Godel showed that there is a model of ZFC where CH is TRUE. Paul Cohen showed that there is a model of ZFC where CH is FALSE. Together we have that CH is INDEPENDENT OF ZFC. What to make of this? Here are opinions I have heard over the years: