Why does aleph-null plus aleph-null equal aleph-null?

In mathematics, the equation ${\aleph }_0+{\aleph }_0={\aleph }_0$ holds true because aleph-null (${\aleph }_0$) represents the cardinality of a countably infinite set, such as the set of natural numbers.12

When you add two countably infinite sets together, the resulting collection can still be put into a one-to-one correspondence with the original set of natural numbers. A classic illustration of this is the Infinite Hotel Paradox, where a hotel with infinitely many rooms can accommodate an additional infinite busload of guests by moving existing occupants to new rooms (for example, by doubling their room numbers), thereby freeing up an infinite number of rooms for the new arrivals.132

Because the combined set of guests can be listed or "counted" one by one without leaving any out, the total size remains ${\aleph }_0$. This demonstrates that, unlike finite arithmetic, adding infinity to infinity does not result in a larger quantity; it remains the same transfinite number.21