We know the approximate age of the cosmos because we can observe light and we know the speed of light, and using that information we can calculate how long it took the light we can see to get where it is now, and that tells us where it came from and where it was 13 billion years ago.
It has nothing to do with "carbon dating".
How do you know how far away the stars are?
By observing them.
I disagree.
How old do you believe the cosmos is?