lordatog wrote:My thinking on this problem as a whole is that there are compelling arguments for both sides. "Every ball that is added is later removed, so there cannot be anything in the urn" - check. "The number of balls in the urn strictly increases in each step" - also check. The fact that the two arguments come to opposing conclusions doesn't necessarily mean that either of them is wrong. It could instead mean that the problem as a whole is inconsistent, which is the answer I favor.

Alright, let's try yet another variation. You have an infinite number of balls, lined up in a nice row. They are numbered in order, 1, 2, 3, and so on. At each step, you put the next ten balls into the urn, remove the lowest-numbered ball from the urn, and, using any kind of method you want, choose an integer larger than the largest number in the urn. Relabel the removed ball with that number, put it in the appropriate spot in the row of balls, and add one to the number of every ball above it, including the one that previously held that number. Then, repeat. Now, every ball that is added to the urn is later removed - and every ball that is removed from the urn is later added again. How many balls are in the urn at the end of the process?

You add and remove each ball an infinite number of times, so, yeah, it looks undefined to me as well. The problem here is, that from the urn's mouth point of view, the original case and this modified one are identical. Then why are the outcomes different? Perhaps from the urn's POV the results will be identical, but from the POV of someone who knows how the selection of the balls works the result might be different. Are there any other situations where the amount of known information changes the outcome?

mike-l wrote:With the lamp setup, there's no reason limits should commute in this case, so even if the behaviour is identical to Thompson's lamp, I don't see any contradiction.

The argument I was making was the following:

In case a), where you remove the lowest ball, one can argue that because each ball has been removed at a certain step, then the urn will end up being empty. However, consider the case where the algorithm consists of adding a ball with n written on it on the (2n-1)th step and removing it on the (2n)th step. Again, if there is a ball n at the end in the jar, then it should've been removed on the 2nth step, so the jar is empty. If you connect the jar to Thompson's Lamp, then the lamp should end up being switched off, but that isn't the case, since the Lamp doesn't really stop in one state.

Now, skeptical has argued that the information needed at each step is different in the two cases (Thompson's case and the "2n-1, 2n" case). In the former, you require to know the lamp's last position to toggle the switch, while, by connecting the lamp to the urn, it becomes irrelevant, since you can just determine the state of the urn, and thus, that of the lamp's. While skeptical is right, he did not disprove my point. I only connected the lamp to the jug to try and show the fact that the two cases are analogous.

The state of the lamp can be determined by the current step's number, if it is an odd step, the lamp is off, otherwise it's on. Similarly, the state of the jug (empty or non-empty) can be determined by the current step's number, if it is an odd step, the jug is non-empty, otherwise it's empty. Or, if you only want to know the state of the lamp without numbering the steps(if the lamp is off, turn it on, otherwise turn it off), you can also have an analog in the jug's case: if the urn is non-empty, remove the ball from it and write n on a piece of paper, otherwise add a ball with n+1 on it and erase the paper).

Also, I'm considering the case where each ball added is a different one because jestingrabbit, I believe, argued that having different balls at each step changes the outcome, something I'm not convinced of.

jestingrabbit wrote:Let us consider the case of a device turning the lamp on and off as you suggest. If the only rules for the device are that it alternates its position in the usual way then we have no reason to believe that the device has just turned the lamp on or just turned the lamp off. If there was some hard coded instruction like "when you get to the end of your lamp switching you must make sure that the lamp is off" then we would agree, I expect, that after the switching had been and gone, the lamp was definitely off. I claim that the fact that we can know something about the device that is doing the switching takes the place of the hard coded instruction at the end of the lamp flickering on and off.

I'm not sure I'm understanding what you're saying, but I believe that, basically, you're arguing that by connecting the lamp to the jug the original Thompson's paradox is no longer valid, because the switching of states is done on different grounds, namely the state of the jug. In that case, the above applies. Otherwise, I'm afraid I might need you to explain your point again.

I'm starting to believe that the jug problem doesn't have a solution because it is analogous with the Lamp one. You can argue the jug is empty because all balls are removed, but you can argue it is non-empty because balls are always added. I'm not saying that either argument is wrong. They both make sense, but neither trumps the other. Rearranging the terms of the 1-1+1-1.. series changes the apparent result, but, in reality, no result is correct because there exists no result.