ATAR Notes: Forum

Uni Stuff => Science => Faculties => Mathematics => Topic started by: humph on May 17, 2008, 10:01:21 pm

Title: Lethal Questions
Post by: humph on May 17, 2008, 10:01:21 pm: Post your überhard maths questions here for all to compare!

I may be a bit bitter about that kinda thing right now. A question on my Analysis 2 assignment:

Prove there is an ordering $\prec$ of $\mathbb{R}$ with the property that for each $y \in \mathbb{R}$ the set $\{ x \in \mathbb{R} : x \prec y \}$ is at most countable.

... which you need to assume the continuum hypothesis to prove. Crazy stuff ???
Title: Re: Lethal Questions
Post by: kamil9876 on September 04, 2010, 11:34:41 pm: Doesn't the following ordering suffice?

$x \prec y$ iff $y-x$ is a non-negative rational number. Easy to check for transitivity and that other property whose name i keep forgetting. The countability follows from the countability of $\mathbb{Q}$ .

edit: just found out today it is called "anti-symmetry".
Title: Re: Lethal Questions
Post by: Ahmad on October 09, 2010, 10:09:35 am: I don't think that's a total order though

One idea is that there exists a minimal uncountable well-ordered set (by AC). Every section (the subsets in the initial post) is countable by minimality. We can find a bijection of this set with R assuming CH and the result follows.
Title: Re: Lethal Questions
Post by: kamil9876 on October 09, 2010, 09:38:37 pm: yeah it is not, he didn't say it has to be total though.