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In taking a philosophy of maths course I have been very curious about the notion of infinity, and whether or not it is paradoxical. One thing I have frequently thought is that "infinity" as a concept is, basically, lots of concepts that we bundle under the same title. To this end I'm considering Hilbert's famous hotel paradox. Could it not be argued that the reason this is so "unintuitive" is that we conflate ordinal and cardinal numbers?

If we have the hotel with $\aleph_0$ rooms, and we put the standard order on the hotel rooms that we do on natural numbers we can say we have $\omega$ rooms. Then we take the case where we move every person to the next room on and stick someone new in room 1, we still have $\aleph_0$ rooms, but by moving everyone "one room on" the hotel now has an ordinality of $\omega +1$ right? So the ordinality increased but the cardinality stays the same, making this not as unintuitive as one might first think.

I anticipate that I may have misunderstood something in ordinal and cardinal numbers (it has been a while since I looked at my notes for my set theory module), so if there is some serious flaw with my reasoning I would love to know.

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No, moving every room up gives you an ordinality of $1+\omega$, not $\omega+1$. And $1+\omega=\omega$, as any fule kno.

Updated to add: But yes, you seem to have misunderstood something. There are "more" ordinals than cardinals; although $\omega+1\ne\omega$, it is still the case that each of these ordinals has cardinality $\aleph_0$.

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