# Hilbert's Hotel, cardinality, and equality

One of the absurdities illustrated by Hilbert's Hotel, some say, is infinity + infinity = infinity. This is absurd in the sense that "After the infinitely many new guests check in, the number of guests will remain the same as before." I think what is meant by "the number of guests is the same as before" is that the infinities in question have the same cardinality - the same number of guests before and after.

Question: Is it true that if two infinite sets have the same cardinality, they are equal? Or is it possible that two sets have the same cardinality, yet have different numbers of members?

If it is true that infinity + infinity = some greater infinity, in what sense is the claim true that "After the infinitely many new guests check in, the number of guests will remain the same as before."

Note: The context of this question is a discussion on an objection to Hilbert's Hotel on Philosofize

The term "infinity" means nothing in modern mathematics. It is just a term which represents "not finite".

In the context of the Hilbert hotel, and it seems that you discuss this in the context of infinite cardinals, rather than ordinals, or some other notion of infinity, infinities are cardinals. So when you say that "two infinities have the same cardinality they are equal" you really just say "two infinites are equal then they are equal".

On the other hand, if you mean infinities in the sense of ordinal numbers, which are another form of infinitary numbers, then addition is not idempotent. $\alpha+\alpha\neq\alpha$. But ordinals carry more structure than cardinals, and the addition of ordinals is different than the addition of cardinals because of that extra structure.

Glancing over the comments in the "objection" that you have linked, it seems to me that a particularly important point is amiss:

The notion of "number" simply means some "measurable quota" in some aspect. Cardinality of sets is a way to measure size of sets in a particular way. So cardinal numbers are numbers. It is true that those are not real numbers, but this is why we don't use the term "infinities" so wildly, and we tend to be very clear in what sort of context it is applied.

If one thinks of numbers as real numbers, and infinity as the infinity we know from calculus (a length which is longer than any finite length), then of course there is merits to this apparent objections. But this is not what Hilbert was trying to show when he opened his grand hotel to the unsuspecting public. No, he was talking about cardinal numbers which are a whole other system of numbers.

It comes down to how we can count the members of an infinite set - and as with many things, we do it by analogy to something easier to grasp. How do we count the members of a finite set $X$? We number them, starting with 1, which is equivalent to forming a bijection between $X$ and $\{1,...,n\}$. We show that this definition of "counting" is consistent in various ways - it doesn't depend on how we choose the bijection, if bijections exist between $X$ and $\{1,...,n\}$ and $X$ and $\{1,...,m\}$ then $m=n$, and if both $X$ and $Y$ can form a bijection to $\{1,...,n\}$ then you can also form a bijection between $X$ and $Y$, so you can meaningfully say that they have the same number of elements.

Having created a way to count the members of a finite set, we extend the idea to infinite ones, and we find that the definition is, at the very least, consistent for countably infinite sets - if you can put a set in bijection with $\mathbb{N}$, then we can call it countably infinite, and all such sets can be considered to have the same "number" of elements - you can still show that two sets that are both countably infinite can be put into bijection with each other. The only problem, such as it is, is that we get funny behaviour like sets having the same cardinality as some of their proper subsets, which doesn't happen for finite sets, but when was the last time everything behaved nicely at infinity?

According to the conventional definition, the number of elements in a set is defined to be its cardinality. If you want to insist that the number of elements in a set is always larger than the number of elements in a proper subset of it, we have to break the assumption that the number of elements in any set is its cardinality. I think there is a consistent system that does that.