Xias wrote:I could have defined it to include the endpoints. Then at midnight, there would be exactly one element in the set, and that element would be zero.

It would have one element because set theory coalesced an infinite number of zero values into one. That is an indication that set theory is not the right framework for that particular problem - it's thrown away information. If you modeled the numbers as a series then you would indeed end up with 0,0,0,... Or if you modeled each element as a pair of values including the n that generated it, you would end up with an infinite set {{0,1},{0,2},{0,3},...}

poker wrote:What happens when x is infinite? First, x cannot be part of Bj for any finite j, since since each finite Bj only contains finite values.

phlip wrote:So now, we need to do the same here... in order for L_inf to be non-empty there needs to be some specific I_j which is non-empty.

Xias wrote:What stops me from arguing the same way for {n+1,...10n} at the limit? Nothing! I am arguing the same way for {n+1,...10n} at the limit!

Well, if we're considering looking at the limit (which we must if we are to get past midnight):

Defining B

_{n}={n+1,n+2,...10n}

L

_{sup}= ∩

_{n≥1}∪

_{j≥n}B

_{j}

Every ∪

_{j}will contain all members of B

_{∞}, therefore the intersection of all ∪

_{j}also contains all members of B

_{∞}

L

_{inf}= ∪

_{n≥1}∩

_{j≥n}B

_{j}

∩

_{j=∞}B

_{j}= B

_{∞}, therefore the union of all ∩

_{j≥n}B

_{j}= B

_{∞}

So the limit of B is B

_{∞}and I can't see any way to claim that it's empty.

Xias wrote:It is especially ironic for you to argue this, since the entire point of the puzzle is to reference a particular solution to the Ross-Littlewood problem. If we add ten balls and remove one ball at every step, the jug will intuitively be full at midnight. However, if we remove balls in a particular and systematic way, we end up with an empty jug, leading to the Ross-Littlewood Paradox. This is the essence of the puzzle.

The paradox is that there are a number of rational arguments for there being infinite infinitely-numbered balls in the jug. These arguments seem to be contradicted by the statement S: "For every number that enters the jug I can give you a time when it leaves the jug". That is what seems at first glance to be paradoxical - but the statement S is only true for finite numbers, so it does not in fact lead to a paradox. Every finite number leaves the jug but actually at midnight an infinite number are added and an infinite number leave - and that does not imply that nothing is left, since we know that infinity minus infinity cannot be said to be equal to zero.

The set-theory argument using strictly finite numbers is simply a mathematical formulation of the initial statement S. By ignoring infinite numbers you make it seem as though nothing is left, whereas in fact all you can say is that no finite numbers are left. If you take the limit actually to infinity, not just approaching it, then the set B

_{∞}(an infinite set of infinite numbers) remains and the paradox is resolved.