> Nat has countably many inhabitants. And one more than countably infinite is countably infinite!

Two sets having the same cardinality (number of inhabitants) is insufficient to prove that two sets are equal. There is an additional condition that they all share the same elements.

I read 1 + Nat as: map each element in the Naturals to its successor (i.e. Nat - {0}). 0 is in Nat, but 0 does not exist in "1 + Nat" since it got mapped to 1.

This would be true if the set we were working with were the Integers instead of the Naturals.