You’d also need an infinitely durable switch and lightbulb, not to mention that at some point the circuit would probably be unable to register the switch being switched because it was happening too fast for it, so add in an infitely responsive circuit and there you go.
And a special universe in which you can move infinitely fast with no weird side effects.
I have a feeling that what you’re asking is how does the maths work, and a discussion of the physics isn’t what you’re after. SO the real question, I suspect, is this:
How can you occupy infinite positions on a timeline which is finite in length?
Well, I’m saying that calculus/analysis (which is what we’re doing here) is abstract math…perhaps pure math is a better description? Anyway I think abstract in the sense of ‘not having to obey physical limitations’ is accurate enough.
Oh, and a reply to my post just above your last one would be much appreciated as I want to be sure we’re talking the same language here.
Well, think about two distinct places on the timeline, say time=1min58secs and time=1min59secs…how many times is it possible to occupy between those two times?
You just answered the lamp question! Every time you switch the switch you just move to the next position on the timeline which is halfway further along than you were before. There is a position for you to move to, and so the question of how you can do inifinite things in a finite time has been answered.
We **do not know ** what happens at two minutes because the rule describing his actions only gives us information about what happens within two minutes, not at or after two minutes.
Mathematically, we are looking at the inifinite sum of {1+(1/2)^n}
A little calculus reveals that this sum is equal to 2 BUT never reaches 2. 2 is ‘the least upper bound (supremum) of the sum’
I’m not quite sure where you’re misunderstanding us so maybe you could go into a little more detail as to where you’re getting stuck?
It’s an indefinite number and that’s the hardest concept to grasp. Try to wrap your mind around this though experiment. I have two people each pace off the length of an infininte line; Vern “Mini Me” Troyer and NBA tall guy Manute Bol. We know that it takes Vern more steps to pace off the length of a basketbal court than Manute. Does that mean it takes Vern more steps to pace off infinity? There is no meaningful answer, it’s undefined in our concept of numbers. If you give any numeric answer then by definition they have not paced off infinity.
Time would have to stop relative to the person (or whatever) performing the action. Without time stopping, there is no way even in theory or thought that an infinite amount of action could be performed in a finite amount of time.
At the two minute mark, relative to the being performing the action, time stops and the number of actions reaches infinity. The light is neither on nor off…time has stopped for this being and his light. They have ceased to exist.
In general, it’s very difficult, if not impossible, to deduce what happens in the infinite case from the known behaviors of the finite case. That’s the problem here.
You wish to know the end result of an infinite number of discrete steps, is that right?
I know infinity is tough to wrap your mind around, but it exists in mathematics, especially in the Calculus. It’s so tough, we’ve come up with numerous gedanken experiments to help us understand it. The best known are variations on Zeno’s Paradox: You know the endpoint, you know the steps to reach the endpoint, but you also know you’ll never get there. The Tortoise will never reach Achilles, and the light will never stop flicking on and off. In the exact same sense, the graph will never reach its asymptote. No matter how infinitely close it comes, there’s still an infinite number of steps between where it is now and its goal.
Weirdly enough, false. Explaining why is going to give me a bit of a headache though…ah, what the heck, I’ll give it a go:
There are different kinds of infinity. The first is the “infinity of the whole numbers” i.e. start at 1 and count upwards 2,3,4,5…jumping up one whole number each time. The set of all whole numbers is, of course, infinite. This infinity is called countable because there is a rule which describes how you get every number in it. The rule is simply start at 1 and add 1 each time.
Many other infinite sets (such as all the negative whole numbers, all the fractions and so on) are also countable because you can find a rule to get you every number in the set.
The next biggest infinity is the ‘continuum of the reals’ and is just the size of the set of all real numbers. It is uncountable because you can’t define a rule to give you every number in the set. It is the infinity that describes all the real numbers, or the length of a line or, the number of positions you can occupy on a continuous timeline.
There are infinite positions you can occupy between 0 and 2 minutes.
Scaling this interval down by any factor you like, you can see that there are infinite positions you can occupy between any smaller time interval
And so in a way, it is obvious that someone could perform infinite operations in this time interval.
So from the point of view of the guy switching the lamp, there’s always an infinite amount of switching to do before he gets to the 2 minute mark. You CAN’T count the number of times he has switched the switch. He will carry on switching the lamp faster and faster as the two minute mark approaches, and his acceleration becomes infinite, so his speed becomes infinite.
So I guess the answer to your question of ‘how is it possible to do an inifinite amount of things in a finite time’ is simply this:
You can, if you move at infinite speed to do them. That’s what is happening here.
Please, if you have any more questions on this problem, make sure you’re specific. Thankyou.
The last thing he does is turn the switch on, and off, and and on , and off…
alright, it is impossible to tell which, so this is why the whole question is unanswerable.
We know we cannot model this physically so let us have a go at modelling it abstractly. Define events ‘turn on’ and ‘turn off’ at times described. Define ‘light on’ if the most recent event was ‘turn on’ and ‘light off’ if ‘turn off.’ We would hope this is sufficient, but in fact we have not defined the light state at 2, since there is no most recent event. We would have to extend our model - there is no natural answer.
This doesn’t occur in a real world example, because things have to be continuous, so events involving discrete movement can’t happen dense in time (mostly)
However, that can only happen at the very ‘last’ interval, the one he never gets to, that brings him to the two minute mark. If the demon moves at infinite speed at any interval before the ‘infinite’ interval, he gets ahead of his task. As long a miniscule gap remains to be cut in half with a switch of the light, the demon’s speed must remain below infinity. The demon’s speed only reaches ‘infinite’ at the two minute mark, which is of course never unless you include the theoretical moment when time has stopped completely relative to the demon.
I like the fact that this is a really thought provoking question, but they need to point out that "Hit the switch once, it turns it on. Hit it again, it turns it off. " Really? See I can’t grasp that because everytime I try it it turns into a light switch rave and I get really high.
