The latest Modal Ontological Proof thread got me thinking of this guy again. For those in the dark (heh), apparently there’s this little demon who is tasked with flipping a light switch on and off at every halfway point up to one minute. So he flips the switch on at 30 seconds, then off at 45 seconds, then on at 52.5 seconds, continuing on and on halfway to the one minute point until he’s flippin’ like crazy because he’s cutting nanoseconds in half and then cutting them in half again ad infinitum, and the question is ‘after one minute is the light on or off?’
Many proposals were offered, some guessing either on or off, or both, or some undefined state. My original answer was that time effectively slows to a stop for the demon and his evil light, and he never makes it to one minute. If somehow the rest of us continued on to the next minute, the demon, staying true to task, couldn’t come with us, although I never fully fleshed out how we would move on without him.
In any event, I was thinking about this again and hypothesized that the demon would never flip the switch at all. He’d have to divide a minute into an infinite number of intervals, and to choose one as the first halfway point would be to put a cap on infinity. So time would effectively stop for him the moment he took on the task.
The smallest unit of time that has any meaning in our universe is Planck time. Numerically its about 5.37062E-44 seconds. For our little demon the following must apply to the light-which-is-part-of-our-physical-world.
60/2[sup]n[/sup]>=5.37062E-44
With n being the number of times the switch is changed.
This simply computes the time interval between switchings and asserts that the interval must be greater or equal to the Planck time interval.
By setting the two side equal and solving it shows that n is approximately 149.64. So after 149 switches time just doesn’t have meaning. As it’s an odd number, the state will be the opposite of how it started, as you state it was off at the start.
Well, if you want to take that tack, the problem starts earlier with the light switch itself. Each flip of the switch has the tip of the arm move about 2.8 cm. The speed of light ~300x10[sup]8[/sup] cm/s. Relativity puts our barrier at:
I think the 43rd switch will have to break the speed of light barrier, if my calculations are correct. That being the case, he can only flip it 42 times (at least, given my wall switch), and the light (if originally off) will remain off.
I’m glad to see the MOP thread has stirred up some worthwhile rational thought. And I’m glad to see that we’ve come up with two definitive and contradictory answers. Seems only fitting.
Adjusted calculations: number of flips for the speed of light 39.2, or 40. So he can flip 39 times. Don’t know what I messed up before. So now the switch is on. :rolleyes:@me
When we speak of infinite sequences, we never talk about what actually happens when we reach an infinite number of terms. Instead, we focus on limits, or what the sequence is doing as you look at more and more terms.
For a sequence of numbers, a limit is some number that the terms approach (in other words, the more terms you take, the smaller the distance between the last term and the limit). For a sequence of sets, there’s an appropriate notion.
What notion should we use to define the limit of a sequence of states?
If my fuzzy recollections of limits serve me correctly, no value is defined for a non-convergent series (which this certainly is) when it reaches its limit. Given that the switch is not allowed to have ‘no value’, we can be certain that the switch is either on, or off, afterwards (assuming the demon stopped flipping when time ran out). I would posit that this is a schroedinger’s-cat sort of situation: the switch will have a value afterwards, but that value isn’t predictable based on the demon’s previous actions. You’d just have to go and look.
