Wrong Proof: Infinite cartesian product of countable sets is countable

We know that if we have two sets $$A,B$$, $$A \times B$$ is countable. So basically, the cartesian product of two sets is countable. Then, we know $$(A \times B) \times C$$ is countable, which means $$((A \times B) \times C) \times D$$ is countable, and so on and so forth for infinity. Why does this fail?

• By doing this you can inductively prove that any finite product of countable sets is countable however you never actually consider a countable (or uncountable) cartesian product of countable sets. Aug 30 '21 at 16:24
• The product $\{0,1\} \times \{0,1\} \times \cdots$ is basically the binary representation of the numbers on $[0,1]$ which is uncountable. Aug 30 '21 at 16:24
• Ahem... Which of these sets are countable? It is obviously not true that the product of any two sets is countable.
– user562983
Aug 30 '21 at 16:25
• "and so on and so forth for infinity" is hiding all the important details Aug 30 '21 at 16:26
• Showing something is true for all finite values of, say $n$, does not mean it is true for an infinite number. Aug 30 '21 at 16:26

$$\{0\}$$ is a finite set
$$\{0,1\}$$ is a finite set
$$\{0,1,2\}$$ is finite set