(This is the ninth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eighth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in [...]
A nice post called “What do Christian fundamentalists have against set theory?” by Maggie Koerth-Baker Seems worth sharing.
Now that you know the basics of countable elementary submodels (CESM), you might think that you are in the clear. “Mike”, you say arrogantly, “I know all the most basic properties of CESMs, without proof I remind you, what else could I possibly want?”. I gently and patiently remind you that CESMs are worthless unless [...]
Last time, just in time for Christmas, we looked at the Secret Santa Problem. Basically the problem is to set up a secret santa type gift exchange without using any external aids like random number generators. A similar problem given to me by Sam Coskey is the following: Sam’s Problem. Is it possible for two [...]
In my ongoing love affair with compactness I am constantly revisiting a particular proof of the Heine-Borel theorem, a characterization of compactness in $\R$. There are two proofs that I know of: the standard “subdivision” proof and the “creeping along” proof. I am going to focus on the creeping along proof. Heine-Borel Theorem. A subset [...]
Last week we looked at the concepts of a collection of sets being n-linked or having the finite intersection property. The key theorem was Helly’s theorem which says: Helly’s Theorem: If a (countable) family of closed convex sets (at least one of which is bounded) in the plane are 3-linked, then they have a point [...]
I love compactness. I really do. It turns infinite things into (almost) finite things. I could gush about how great it is, but instead let me tell you about one problem where compact sets act as the delimiter. Here is one way to characterize compactness: A space X is compact if and only if any [...]