The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals. I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is:
Viewed as a product of ordinals, $2\cdot 3\cdot 4\cdot 5\cdot \cdots = \omega$  ($\omega$ being the first limit ordinal)
and same question but viewed as a product of cardinals.
I do not even know where to begin, and I'm not sure how being a cardinal changes the game.
 A: Where to begin? Well, perhaps with the definitions.
What is the definition of the infinite multiplication the ordinals? What is the definition of the infinite product of cardinals?
Recall that $\omega$ is the least infinite ordinal, so if you have a sequence of ordinals which is increasing, and all its elements are finite the its supremum has to be $\omega$.
For the cardinals, your claim is wrong. The infinite product of the finite cardinals is not $\omega$, but rather $2^{\aleph_0}$. You can use the fact that if $\lambda_i\leq\kappa_i$ for all $i\in I$ then $\prod_{i\in I}\lambda_i\leq\prod_{i\in I}\kappa_i$. Now take $\lambda_i=2,\kappa_i=i$ and $\mu_i=\omega$ for $i\in\omega$. Now we have that $\lambda_i\leq\kappa_i<\mu_i$. Show that all the products have the same cardinality.

The most important tip I can give you is this:
Ordinals are not cardinals. Even if the finite ordinals behave the same in their arithmetics when treated as ordinals, cardinals, integers, rationals, real numbers, complex numbers, and so on; when doing infinitary operations all the rules change, and each system can be discerned from the others.
