It's not being pedantic. With different assumptions you can make a system where 2 + 2 = 0 and it turns out to be extremely useful. You can also build a system where 2 + 2 = 22 like the other commenter lampoons and the free monoid that corresponds to is again useful.

If we had a radically different perspective (like Borges' Funes the memorious), you can imagine how adding wholly distinct objects might seem ridiculous and derive some other wacky system of arithmetic instead.

Of course, you could alternatively derive it from set theory, but you might also end up with something fundamentally different than what the grandparent intended like presburger or skolem arithmetic.

But “2+2=4”, in the specific formal system that is common arithmetic, is true.

Furthermore, you can translate “2+2=4” to any other formal system (your examples, “2+2=10 (base 4)”, “2+2=1 (mod 3)”, etc.), and it’s still true.

“2+2=4” is a universal truth, just expressible in different ways.