Why can the reals be well-ordered but not countable?
June 12, 2012 10:58 AM Subscribe
Math-Filter - Set theory edition:
I am struggling with the difference between countable infinite sets and well-ordering. Longer explanation inside.
posted by heybearica to Science & Nature (16 answers total) 4 users marked this as a favorite
Ok the longer version:
I get that there are countably infinite sets (like natural numbers).
I get that some sets were proven to be countable by changing their order so as to set up a 1-to-1 equivalence to the natural numbers (like the rational numbers).
I get Cantor's diagonal argument for proving that the real numbers are uncountable.
I think that I understand what it means for a set to be well-ordered. (From wikipedia:"In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set."
Here's where the confusion lies:
Apparantly it has been proven that there exists a well-ordering of the reals and some mathematicians think that it is likely that all sets may have a well-ordering. But if you can put a set into a well-order can't you then just count the elements in that order? Wouldn't they be in a 1-to1 relationship with the natural numbers at this point? I am definitely stuck with why something could be well-ordered but not countable.
I have googled and read wikipedia for long enough that I think I am getting more confused not less.
Thanks to anyone who can help my brain hurt less.