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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.

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up vote 4 down vote accepted

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.

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