Cartesian Products of Countable Sets Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

The cardinality of the Cartesian product of two finite sets is the product of the cardinalities of the two individual sets. To the extent that this idea generalizes to infinite sets, understanding the cardinality of a Cartesian product of countably infinite sets helps understand what infinity times infinity is (although that’s a risky analogy to push too far). This discussion helps you explore that idea, by following two lines of thought. Comment on either or both, as you wish.

First, to what extent do you think it’s meaningful to talk about “infinity times infinity” at all? Is infinity a number? Is multiplication defined for it?

Second, discuss how you might analyze the cardinality of the Cartesian product of two countably infinite sets. This is a big question that no one post is likely to answer, so start by mentioning ideas, asking questions, etc. One fruitful source of ideas might be our textbook’s proof that the set of positive rational numbers is countable.