Flashing light bulb problem

[Mobius]

I have asked this question before but have never really had a satisfactory answer to it, so I will throw it out here and see what happens.

 

the problem is based on a faulty lightbulb that flashs. However it's flashing is based on the infamous infinite sequence 1 + 1/2 + 1/4 + 1/8 .....

 

i.e. the light is on for 1 minute and off for 1/2 a minute, goes on for 1/4 of a minute and off for 1/8th of a minute. Now it is well known that this sequence never reaches 2. Therefore at two minutes is the bulb on or off????

 

Now I'm well aware that this could not physically happen due to the nature of a light bulb, but I want a theoretical answer! i.e. the ideal light bulb, or an electron flipping from state spin up to spin down.

[DQW]

At 3 minutes, is the bulb on or off ?

[BigMoosie]

By the time 2 minutes is reached the theoretical light bulb will have changed state an infinite number of times which is impossible in a finite time period therefore the question makes no sense to begin with.

 

Here, this explains it nicely: http://www.random.abrahamjoffe.com.au/public/scrap/light.htm

[YT2095]

since a light bulb has a finite On/Off life cycle, and it`s flashed infinate times in that 2 or 3 mins, the answer would most definately be Off, as the bulb would fail.

[mezarashi]

Now I'm well aware that this could not physically happen due to the nature of a light bulb' date=' but I want a theoretical answer! i.e. the ideal light bulb, or an electron flipping from state spin up to spin down.[/quote']

 

I think you've stated the solution to your own question. Physically, the bulb's build would be the limiting factor. On the theoretical side, the light bulb's state is also not defined at 2 minutes and beyond. Let's extend the question further. What is its state at 3 minutes? The series does not conclude that if it were on at 2.00 minutes, it should be on at 2.01. You have a function defining its state for t=0 to t<2. In anycase, I would probably consider this more of a mathematical question, and there should be room for interesting comments there.

[matt grime]

Therefore at two minutes is the bulb on or off????

 

 

who knows? you've only told us about its state at time t when t is greater than 0 and strictly less than 2 minutes, and that is the theoretical answer. practically it is not a possible situation.

[YT2095]

another point, that needs mentioning.

if we ignore the actual Dynamics of this setup, the On/Off rate will be at a frequency impossible to attain at even a quantum level. it`s on a par with ideas about "what happens when we travel facter than C" or "irresistable force meets an imovable object" and the likes. Simply the situation cannot Occur.

 

Any answers to these, would be pure speculation.

[matt grime]

i thought practical considerations were to be ignored.

 

the "theoretical" reason is perhaps this:

replace on, off with the fucntion of time f(t)=0 when off, 1 when on. for 0<=t<2 as above. then removing all physical considerations, what is f(2)? or f(3)?

 

well, we've not defined it.

 

BUT

 

there are ways of extending functions ot larger domains, but this usually depends on some structure of the function, and some natural "best" kind of extension.

 

for instance, factorial thatm ost people think is defined oddly on 0 is simply a natural extension of it for the positives (useful and consistent and fits into the recusive definition of n!=n.(n-1)!.

 

most naturally is the idea of continuous extension, or extension via continuity, that appears in say the arithmetic of the extended complex plane. or perhaps we want to take a continous function on the rationals and extend it to a continuous function on the reals. if possible. perhaps we want to extend to end points of an interval. note this isn't always possible.

 

finally, there is analytic continuation where we try to extend a taylor series beyond its radius of convergence where possible. eg

 

1+z+z^2+z^3+...

 

converges only for |z|<1, yet we all know that it is (1-z)^{-1} so it extends to a function on all of C except 1.

 

however, youir fucntion has no nice natural properties that we may wish to extend in a natural way.

[YT2095]

i thought practical considerations were to be ignored.

 

I did ignore them, I brought it right down to Frequency (oscillation) not of any particular thing. I still get the same answer, The situation cannot occur.

no matter HOW I try, the answer`s the same.

[YT2095]

you`de never get past a few Ghz at best, and that`s using a Computer (for the /2 decrement) AND assuming that you`de get a port chip with that kind of slew rate.

you`de only increase this Max osc rate marginaly if you used an analogue VCO fed by a cap discharge.

 

it cannot be built to perform this with entirity.

 

edit: AHA! just had a Brainwave LOL

lets re-work this problem around the to the Inverse!

you have an everlasting bulb, set to a timer (of the same standard) the bulb lights up for one sec then off for 2 secs then on for 4 secs and off for 8 secs and so on...

after an Infinate amount of time, will the bulb be On or Off?

 

see it doesn`t really mater whether it`s X2 or /2 for infinity, there`s never going to be an end, like C and Alef Null.

[matt grime]

I did ignore them' date=' I brought it right down to Frequency (oscillation) not of any particular thing. I still get the same answer, The situation cannot occur.

no matter HOW I try, the answer`s the same.[/quote']

 

 

but your reason why this frequency cannot occur is a physical property (assumption really) of your choice of model for physics at a small scale.

 

that it is of no particular thing doesn't mean it isn't physical. by saying the situation cannot occyur in this way you are giving a practical reason why it cannot occur.

[Mobius]

I think the problems come into the case when the time intervals between the flashes becomes very short. Perhaps when the time interval reaches Planck time, 10^(-43) seconds, the light bulb will move into one of the states, meaning that it did not have to flip states an infinite amount of times.

 

Alternatively it could be "on" in our universe and "off" in an alternative universe (satisfying the condition of being both on and off at the same time) after flashing an infinte amount of time (according to it) but a finite amount of time (according to us) ;-)

 

I was wondering whether to post this in the maths or physics section, glad I posted it in the maths one:)

[YT2095]

perhaps it`s a mental flaw on my part then, if it cannot occur, The question is irrelevant (to my way of thinking).

 

the question as outlined in the OP is invalid.

[Mobius]

Just because it cannot happen physically, that does not mean we can't speculate theoretically what would happen.

 

I feel the question poses a challenge and is worth considering. That is the reason I posted it on a maths thread as physics (my background) generally dismiss this question as invalid....

[matt grime]

we're not talking about the physical possibility of this situation at all, simply what happens when we pass to the limit

 

this is a very common mathematical sitaution. can we pass from some case defeind for all finite number of steps to say anything about an infinfite case?

 

firstly we must deicide if the limit of the index exists, and then if the property passes through to the limit.

 

example:

 

let S_1, S_2,.. be the nested sets

 

{1} {1,2} {1,2,3} etc

 

then the limit of the sets is clealry the natural numbers in any reasonable sense,

 

 

 

now consider some properties of the sets.

 

eg S_n is a set of natural numbers. this property is passed on to the limit

 

S_n is a finite set, this is not a property preseved in the limit

 

in this case we have on or off (0 or 1) at each finite stage, but no canonical way of passing it thruogh to the limit (at time t=2)

 

 

of course the situation so given is physically impossible; but apparenlty that isn't the point of the question (though this is purely my interpretation of "no practical considerations")

[YT2095]

your question (simply put) asks how many times can something be divided by 2 until it cannot be anymore.

 

such a question is unacceptable. (at least when a reasonable answer is expected).

 

My opinion.

[matt grime]

again that is a physical interpretation of the question and not a mathematical one. it is akin to xeno's paradox, and not a mathematical problem at all.

[Mobius]

There are lots of questions in physics where the maths breaks down. The beginning of the universe at the big bang, the singularities in black holes, travelling at the speed of light (your time passed effectively become 0) and so on...

 

These questions deserve a physical description. The flashing light bulb problem is loosely based on zeno's paradox but it is based on time rather than distance and motion. Whenever infinity raises its ugly head in our finite world it always casues problems. It seems that the physical world solves this problem (maybe by not allowing the situation to happen in the first place).

 

When we discuss the start of the universe we generally say it started at t=10^(-43) seconds as the physics before this is not understood, maybe as I stated already that if we had such a light bulb than as the interval between the flashes became this short than it may jump to a final state.

 

As for after the two minutes it stands to reason that it will remain in the same state that it was at 2 minutes as the interval between the flashes is 0.

 

The answer is not altogether important, but the question is!!!!

[YT2095]

again that is a physical interpretation of the question and not a mathematical one. it is akin to xeno's paradox, and not a mathematical problem at all.

 

I did my utmost to elliminate ANY ref to Physicality, and to keep it as Maths(ish) as I could.

 

the way I see the problem as boiled down to the Raw basics as possible is that which I stated before, it`s division by 2 until it`s no longer divisible (that point being the 2 minute marker).

 

it`s like presenting 10 / 3 =

and asking for a finite answer.

 

there is non. at least Not without specifying a resolution (in decimal places).

[matt grime]

As for after the two minutes it stands to reason that it will remain in the same state that it was at 2 minutes as the interval between the flashes is 0.

 

 

nonsense; you've given no information at all that allows you to discuss its state at time t=2 or greater.

[Mobius]

Fair enough, but surely as the limit is reached, there would be no more flashing?

[YT2095]

from a Maths standpoint I see it as an exponential curve that never quite reaches its end, but curves off to almost a flat line.

 

from a Physical standpoint, since it`s spending as much time On as Off then you`de get a bulb that would "Appear" at half brightness, simply due to latent heat and retina image retention (but that`s just a Factual/Material phenomenon).

 

btw, I`m unfamiliar with Xenos paradox, I think I`ll google that and have a read

[Mobius]

From a physical point of view it would not appear at half brightness. you would (after a short amount of time) see it on fully. this is due to 'persitance of vision'. We cannot differentiate between flashes of about 20 per second.

 

A real light bulb flashes 50 times a second due to the ac current running through it. We just see it on...

[matt grime]

are you talking physically or purely from the mathematical vewi point of the 0,1 function interpretation I gave? It is *your* assumption that when the input is 2 the fucntion is fixed at 0. in your words, that is fair enough if that is how you wish to model the situation. there is no mathematical reason to suppose that is what happens, and no physical one either since the situation is unphysical.

 

consider the following example.

 

Suppose that a quantity x satisfies x(t)=1/t for all strictly positive t. what is x(0)?

 

teh question makes no mathematical sense, obviously, but if i were to ask you to state some reasonable assumptions and then make a conclusion, what would you think the answer is?

 

to me, as i've not defined x(0) i can declare it to be anything I wish and it is still a function of t for positive t (inckluding zero), and there is no way to make it continuous on the extended domain. that is all you can say about it.

 

if you want to give some more physical meaning to it, do so, but i thought you weren't interested in the physics of it merely the mathematics of it.

[YT2095]

Matt I`m interested in the maths side, I`m yet to be convinced (in a way I can understand), as for the physics of it, I think that`s already been (inadvertantly) covered here.

 

btw, I did a seach for "xenos paradox" it`s all garbage that pops up, do you have a link that explains it concisely? pref something without online shopping and philosophy