Can someone explain Hilbert's hotel paradox simply?

Great question! The Hilberts Hotel paradox shows exactly why infinity is NOT a number. **If infinity were a number, then $\infty + 1 > \infty$.** But with infinite sets, adding one element doesnt change the "size" — theres a bijection between the set and itself plus one element. For Hilberts Hotel: Map guest $n$ to room $n+1$. This is a bijection between $\{1,2,3,\ldots\}$ and $\{2,3,4,\ldots\}$, proving they have the same cardinality ($\aleph_0$). **For the infinite buses:** Use the Cantor pairing function. Map bus $b$, seat $s$ to room number: $$f(b,s) = \frac{(b+s-1)(b+s-2)}{2} + s$$ This gives a 1-to-1 mapping between $\mathbb{N} \times \mathbb{N}$ and $\mathbb{N}$, proving countability. The lesson: Infinite sets have proper subsets of the same size, which never happens for finite sets. This is the defining property of infinite sets.