580 -Cardinal arithmetic (8)

 

4. Large cardinals and cardinal arithmetic

 

In section 3 we saw how the powers of singular cardinals (or, at least, of singulars of uncountable cofinality) satisfy strong restrictions. Here I show that similar restrictions hold at large cardinals. There is much more than one could say about this topic, and the results I present should be seen much more like an invitation than a full story. Also, for lack of time, I won’t motivate the large cardinals we will discuss. (In the ideal world, one should probably say a few words about one’s beliefs in large cardinals, since their existence and even their consistency goes beyond what can be done in the standard system I’ll however take their existence for granted, and proceed from there.)

 

1. Measurable cardinals

 

Definition 1 is a measurable cardinal iff and there is a nonprincipal -complete ultrafilter over

 

580 -Cardinal arithmetic (5)

At the end of last lecture we defined club sets and showed that the diagonal intersection of club subsets of a regular cardinal is club.

Definition 10. Let be a limit ordinal of uncountable cofinality. The set is stationary in iff for all club sets  

For example, let be a regular cardinal strictly smaller than Then is a stationary set, since it contains the -th member of the increasing enumeration of any club in This shows that whenever there are disjoint stationary subsets of Below, we show a stronger result. The notion of stationarity is central to most of set theoretic combinatorics. 

Fact 11. Let be stationary in

1. is unbounded in
2. Let be club in Then is stationary. 

Proof. 1. must meet for all and is therefore unbounded.

2. Given any club sets and and it follows that is stationary.