I have been researching the countability of the Cartesian product of three integer sets (which are countably infinite sets). Although I find information on the infinite countability of the cartesian product of two integer sets and the uncountability of the Cartesian product of infinite integer sets, I have yet to see anything on three sets.
Would the Cartesian product of three integer sets, therefore, fall under infinitely countable, like that of two integer sets, or would it be uncountable, like that of infinite sets?
Thanks for the insight, in advance! I am interested in hearing what you think!
It is countably infinite.
Suppose that $A,B$, and $C$ are countably infinite sets. You already know that $A\times B$ is countably infinite. Depending on exactly how $A\times B\times C$ is defined, either it is $(A\times B)\times C$ by definition, in which case you already know that it is countably infinite as the Cartesian product of two countably infinite sets, or there is a trivial bijection between it and $(A\times B)\times C$, in which case you also know that it is countably infinite.