set theory - Help with intuition on Cardinal Arithmetic Problems

It happens a lot to me that when I find an intuitive model (picture) of a mathematical entity, the proofs left as exercises in books are very easy to solve. For example when dealing with filters and ultrafilters on sets (specially $\omega$) I just need to imagine the Hasse diagram of the Poset $\langle\mathcal{P}(\omega),\subseteq\rangle$ and most proofs and definitions come naturally.

I have been trying to find a model/picture for arithmetic with cardinals that helps me solve the problems that come in some books but I don't know if cardinal numbers are too big and I cannot picture them correctly or I just have to use different models/pictures for different problems. So far the two that have been working best are parallel lines (when comparing cardinals) and my intuition with injective and surjective functions. But those approaches only work quickly with easy problems (adding or multiplying finitely many cardinals or relating the cardinality of two specific sets). However, this techniques become lengthy when new concepts are introduced (like cofinality and exponentiation of cardinals). Moreover, I have not been able to solve many problems with just these two methods. I'll quote two of those problems and try to make this ideas more clear:

1. (This one is in Andras Hajnal & Peter Hamburguer's Set Theory Book) If $\kappa$ is an infinite cardinal number and $\kappa=\sum_{\lambda
   

For this one I just changed $\kappa^{cf(\kappa)}$ for $\prod_{\lambda

2. (A friend gave me this one but I have a feeling there is a typo somewhere) Suppose that $\alpha$ is a limit ordinal and that $\langle\kappa_\xi\rangle_{\xi<\alpha}$ is a strictly increasing sequence of cardinals such that $\kappa=\sum_{\xi<\alpha}\kappa_\xi$ and if $0<\lambda
   
   

Although not stated in the problem, every cardinal $\kappa_\xi$ must be strictly less than $\kappa$ (otherwise the sum is greater than $\kappa$). And given that for every infinite cardinal $cf(\kappa)=\min\{\lambda\in CN_\infty\mid\forall\xi\in\lambda(\kappa_\xi<\kappa)\wedge\kappa=\sum_{\xi<\lambda}\kappa_\xi\}$ we must have that $cf(\kappa)\leq\alpha$. Under the same principle one just have to prove that $cf(\kappa^\lambda)\leq\alpha$. There is also a trivial inequality in this problem: $\kappa^\lambda=(\sum_{\xi<\alpha}\kappa_\xi)^\lambda\geq\sum_{\xi

So the questions are: Do you use a single model/picture to help you solve this kind of problems about infinite sums, products and exponentiation of cardinals? Which one? and could you provide a hint on how to use such a model/picture to solve problems 1. and 2.?

Answer

I just realized that you are asking for other methods than constructing explicit functions, however, I figure direct attacks might be the most intuitive for problems like this.

In order to show that $\kappa^{cf(\kappa)}\leq\prod_{\lambda

Given $f\in \kappa^{cf(\kappa)}$, recursively build a sequence $g$ such that $g(\lambda)=\begin{cases}
f(i)+1 && \text{if $i$ is the least such that f(i) is in }[0,\kappa_\lambda) \text{ and i is not used previously} \\
0 && otherwise
\end{cases} $.

Note we eventually exhaust all f-values as $cf(\kappa)$ is regular.

If $f\neq h$, then let $i\in cf(\kappa)$ be the least point where they differ. There are two cases to consider one is there exists a least undefined $\lambda$ such that $f(i), h(i)\in [0,\kappa_\lambda)$ and another case being there exists a least undefined $\lambda$ such that $\kappa_\lambda$ separates $f(i)$ and $h(i)$. In either case, the sequences constructed from two functions are different. So you have an injection.