Here are two posts to explain the points of view:

• In Defense Of The Axiom Of Choice, and
• A Point Against The Axiom Of Choice

When we have two finite sets, $A$ and $B$, can form the set of pairs:

• $\{ (a,b) : a\in A, b\in B \}$

When we have three finite sets we can form $(A\times B)\times C$, or we can form $A\times (B\times C)$, or we can form the collection of triples:

• $\{ (a,b,c) : a\in A, b\in B, c\in C \}$.

We can avoid this "duplication" issue by a simple technical trick. If we assume that all the sets are disjoint then instead of forming pairs or triples, we can form a set, taking an element from each contributor. Thus:

• $A\times B := \{ \{a,b\} : a\in A, b\in B \}$

We can clearly form the product of finitely many sets even if some are infinite, that's not a problem, and one could easily assume that it all works without any problems even if we have infinitely many sets.

And sometimes it does, but sometimes we get some results that might catch you by surprise.

• The Banach-Tarski Paradox;
• The Vitali set;

• In Defense Of The Axiom Of Choice, and
• A Point Against The Axiom Of Choice

Quotation from Tim Berners-Lee