Shelah’s PCF theory
I have recently started to read a bit about Shelah’s theory of possible cofinalities. It is a quite interesting topic in Set Theory that I have wanted to study for a long time, but I did not really find time until now. I am going to give an overview in this blog post for people who are interested but also mostly to help me organizing my ideas and understanding of this subject.
First, this theory originated from (infinite) cardinal arithmetics, which is itself the Cantor’s invention that marked the birth of Set Theory. The addition and multiplication of infinite cardinals are trivial, since we have
$\forall{κ,λ>=ℵ_{0}},\,κ+λ=κ.λ=\max{\{κ,λ\}}$ 
The exponentiation is much more difficult (and interesting!). Here is Theorem 5.20 from Thomas Jech’s book "Set Theory":
Theorem 0.1
Let $λ$ be an infinite cardinal. Then for all infinite cardinal $κ$, the value of $κ^{λ}$ is computed as follows, by induction on $κ$:

1.
If $κ\leq λ$ then $κ^{λ}=2^{λ}$.

2.
If there exists some $\mu<κ$ such that $\mu^{λ}\geq κ$, then $κ^{λ}=\mu^{λ}$

3.
If $κ>λ$ and $\mu^{λ}<κ$ for all $\mu<κ$, then

(a)
if $\operatorname{cf}(κ)>λ$, then $κ^{λ}=κ$,

(b)
if $\operatorname{cf}(κ)\leq λ$, then $κ^{λ}=κ^{\operatorname{cf}κ}$

(a)
This theorem reduces the problem of exponentiation to the determination of the continuum function $κ\mapsto 2^{κ}$ and, on singular cardinals $κ$, of the gimel function $κ\mapsto\operatorname{\gimel}(κ)=κ^{\operatorname{cf}κ}$ . There are simular reductions of infinite sums and products to $\sup$ and exponentation operations, so computing these two functions is the crucial point of cardinal arithmetics.
Set theorists proved by forcing that we can not really say more about the value of the continuum for regular cardinals. Indeed, the only constraints are that the function must be increasing and must satisfy some consequences of König’s theorem (namely $2^{κ}>κ$ and $\operatorname{cf}(2^{κ})>κ$).
At the opposite it turns out that more properties can be proved for the singular case. For instance, we have the striking bound of $\operatorname{\gimel}(ℵ_{ω})$ mentioned on Shelah’s Web Archive:
Theorem 0.2
$\operatorname{\gimel}(ℵ_{ω})=ℵ_{ω}^{ℵ_{0}}\leq 2^{ℵ_{0}}+ℵ_{ω_{4}}$ 
Shelah developed his PCF theory in order to obtain these kinds of results. The basic idea is to consider an "interval" of regular cardinals $A$ i.e. with the property that any regular cardinal between two elements of $A$ is itself an element of $A$. Then we define the set of possible cofinalities of ultraproducts of $A$:
$\operatorname{pcf}(A)=\left\{\operatorname{cf}{\left(\prod A/U\right)}U\text{ is an ultrafilter on }A\right\}$ 
where $\prod A/U$ is a totally ordered set: an ordinal function is smaller than another if it is smaller "almost everywhere" (i.e. on an element of the ultrafilter $U$). The cofinality of such a totally ordered set is defined in a similar way as for cardinals: it is shortest length of an unbounded sequence.
Now, $\prod A/U$ is a topological space (a basis is naturally given by the open intervals with respect to the total order) and so we can use topological methods to get information on it. In particular, this provides information on $\prod A$, which in turn provides infomation on $\left\prod A\right$, the infinite product of the (regular) cardinals in $A$.
For example, if $A={\left\{\aleph_{n}\right\}}_{n=0}^{\infty}$ and if we assume that $\aleph_{\omega}$ is strong limit then one can show that $\operatorname{pcf}(A)$ is itself an interval of regular cardinals and has a maximum element which is $2^{\aleph_{\omega}}$. So its elements are in the sequence of cardinals
$\min{\operatorname{pcf}(A)}=\aleph_{\alpha},\aleph_{\alpha+1},\aleph_{\alpha+2 },\ldots,\aleph_{\alpha+\lambda}=\max{\operatorname{pcf}(A)}=2^{\aleph_{\omega}}$ 
and obviouly, $\alpha+\lambda\leq{\left\operatorname{pcf}A\right}$. But the size of $\operatorname{pcf}A$ is not greater than the number of ultrafilters of $A$, which are sets of subsets of the countable set $A$ i.e. there are at most $2^{2^{\aleph_{0}}}$ many of them. Since we assume that $\aleph_{\omega}$ is a strong limit cardinal, we have $\alpha+\lambda\leq 2^{2^{\aleph_{0}}}<\aleph_{\omega}$ and finally
Theorem 0.3
If $\aleph_{\omega}$ is a strong limit cardinal then
$2^{\aleph_{\omega}}<\aleph_{\aleph_{\omega}}$ 
These are of course just two examples of what we can say about the values of the continuum and gimel functions at a singular cardinal (here for $\aleph_{\omega}$, the smallest infinite singular cardinal). Several other results are known and the PCF theory also applies to other areas than cardinal arithmetics. Many beautiful topics that I would like to learn about…