4 ms·
> Nat has countably many inhabitants. And one more than countably infinite is countably infinite! Two sets having the same cardinality (number of inhabitants)
by Infinity315 2y ago
> 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.
- srcreigh 2y agoThis is incorrect.. the equation represents the number of elements in Nat based on its definition. Integers could be Int = Nat | Neg Nat. So Int = 2 * Nat. Your logic would presumably interpret this as even naturals.. but ya
- Infinity315 2y ago> Your logic would presumably interpret this as even naturals.. but ya My logic comes from a Math background. Take any Abstract Algebra class and this notation is common. For example: https://math.stackexchange.com/questions/1378174/how-to-denote-an-even-number-in-mathematics#:~:text=The%20integers.,set%20of%20all%20even%20numbers https://math.stackexchange.com/questions/1378174/how-to-deno.... To say the least I take numerous issues with the notation used, but I accept it. Since the integers and the naturals have the same cardinality, we have that Int = Nat.