How to evaluate $\prod_{i\in \omega}\aleph_i$ I would appreciate help regarding several aspects regarding evaluating $\prod_{i\in \omega}\aleph_i$.
-- Multiplication of cardinals is defined as the cardinality of their Cartesian product. How is this used when it is the Cartesian product of an infinite number of cardinals. And, in particular, when all the cardinals are infinite.
-- There is a theorem stating that if $\kappa$ and $\lambda$ are cardinals with $\kappa\leq\lambda$ and $\lambda$ is infinite, then $\kappa\lambda=\lambda$. How is this applicable to evaluating the product. And especially since in the context of the infinite product, there is no explicit maximum multiplicand - only a least upper bound $\aleph_0$.
Thanks
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 A: This is $\aleph_\omega^{\aleph_0}$. First of all, this cardinal is an obvious upper bound. Second, if $A\subseteq\omega$ is infinite, $\prod_{i\in A}\aleph_i$ is clearly at least $\aleph_\omega$. The result follows, by splitting $\omega$ into countably many infinite sets.

In general, the rules governing infinite products and exponentials are far from being well-understood. By König's theorem, $\aleph_\omega^{\aleph_0}>\aleph_\omega$, but how large it is is very much open: It could be $2^{\aleph_0}$, but assuming (say) that $\aleph_\omega$ is strong limit, all we know is that it is smaller than $\aleph_{\omega_4}$, see

MR1318912 (96e:03001). Shelah, Saharon. Cardinal arithmetic.
  Oxford Logic Guides, 29. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1994. xxxii+481 pp. ISBN: 0-19-853785-9. 

(That result is fairly involved, certainly beyond the level of simple cardinal arithmetic of products of finitely many cardinals.)
One other thing that can be said quickly is that if $\aleph_\omega$ is strong limit, then $\aleph_\omega^{\aleph_0}=2^{\aleph_\omega}$, see for instance this answer. 

You ask a more general question, essentially regarding the definition of infinite products. For sets $A_i$, $i\in I$, by $\prod_{i\in I}A_i$ we understand the set of all functions $f$ with domain $I$ such that $f(i)\in A_i$ for all $i\in I$. 
An infinite product of cardinals is understood as the cardinality of this Cartesian product, and in general $\bigl|\prod_{i\in I} A_i\bigr|=\prod_{i\in I}|A_i|$. 
Note that this generalizes the case of finite products, where $A\times B$, for example, can be identified with the set of functions $f$ defined on $\{0,1\}$ with $f(0)\in A$ and $f(1)\in B$. Indeed, each ordered pair $(a,b)$ can be seen as such a function $f$ with $f(0)=a$ and $f(1)=b$.
We do not have all the rules, but some are understood. Many generalize familiar rules for finite products. For instance, for nonempty sets $A_i$, $|\prod_{i\in I}A_i|\ge|A_j|$ for each $j\in I$, so the product is at least the supremum of the $|A_i|$. Also, if $I$ is the disjoint union of sets $J,K$, then there is an obvious bijection between $\prod_{i\in I}A_i$ and $\bigl(\prod_{j\in J}A_j\bigr)\times\bigl(\prod_{k\in K}A_k\bigr)$, and similarly if $I$ is the infinite disjoint union of sets $I_c$, $c\in C$, then once can easily find a bijection between $\prod_{i\in I}A_i$ and $\prod_{c\in C}\bigl(\prod_{i\in I_c}A_i\bigr)$, which is what I used in the answer above.
