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limit superior of $A_n$: $\cap_{n=1}^{\infty} \cup_{m=n}^{\infty}A_m$

limit interior of $A_n$: $\cup_{n=1}^{\infty} \cap_{m=n}^{\infty}A_m$

Let $A_n$ be defined to be the set of n positive integers greater than or equal to n and less than 2n. Then since no positive integer belongs to infinitely many of the sets of $A_n$, the limit superior and limit inferior are both empty. Yeah it is simple, I get it.

The proceeding example can be visualized in terms of placing pairs of billard balls. which bear numbers $0,1,2...$, into a bag while repeatedly withdrawing one. For example at one minute before noon balls numbered $0$ and $1$ are placed in the bag and ball number $0$ is removed. At $\frac{1}{2}$ minute before noon balls numbered $2$ and $3$ are added and ball number 1 is taken out. At $\frac{1}{3}$ minute before noon balls $4$ and $5$ are added and ball number $2$ is removed. This process is continued, and the question is asked: "How many balls are in the bag at noon?" Answer: "None."

Why none? Isn't this question about the cardinality which goes to infinity? Yeah I know that you can't point any ball that is in the bag but still I am confused.

Asaf Karagila
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romperextremeabuser
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    The issue is that cardinality is not continuous in the appropriate sense: the limit of the cardinalities of the sets involved need not be, and indeed is not here, the cardinality of the limit (in the appropriate sense) of the sets involved. I'm posting this as a comment as opposed to an answer since I'm quite confident this question has been asked here before, but FWIW it's a reasonable and common confusion to have. – Noah Schweber Aug 11 '22 at 14:17
  • I don't undertand your comment :(, how is cardinality not continuous in the appropriate sense? What does it even mean?

    So the answer "none" is correct?

    Could you give me a link to the post where this question was asked?

     – romperextremeabuser Aug 11 '22 at 15:10
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    "how is cardinality not continuous in the appropriate sense? What does it even mean?" It means exactly that we cannot conflate "cardinality of limit" (the thing we're interested in here, which is $0$) and "limit of cardinalities" (which is $\infty$). You're implicitly assuming that we can perform this sort of swap, but what the example you're looking at shows is exactly that we can't. – Noah Schweber Aug 11 '22 at 15:52
  • Okay, I think that I don't understand why we are interested in cardinality of limit instead of limit of cardinality in question "how many balls are in the bag at noon". – romperextremeabuser Aug 11 '22 at 16:01
  • "How many balls are in the bag at noon?" Answer: "None." --- This reminds me of a (class term) paper I wrote for a philosophy class in the late 1970s on infinity machines. I basically argued that in an example like this (e.g. Thompson's lamp) we have only defined a certain function for certain input values, and then asked what the function's output value is for a different input value without providing any stipulation as to how the extension is to be made. No paradox, because there are no grounds for arguing there even is an extension. – Dave L. Renfro Aug 11 '22 at 19:08
  • https://math.stackexchange.com/questions/869916/a-strange-puzzle-having-two-possible-solutions?rq=1 okay i found the answer – romperextremeabuser Aug 11 '22 at 20:15

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