A perfect lamp

[BMAD]

We have a perfect machine for turning a light on and off. First, we have the light on for one minute, after which it is turned off for one half of a minute. Then it is on again for one fourth of a minute and off for one eighth of a minute. This continues with the light turned on or off after one half of the preceding time period. After two full minutes an infinite sequence of offs and ons will have occurred. At this time, will the light be on or off?

[bonanova]

@DejMar, I agree, there are noninteger too: the time intervals the lamp spends in one or the other state: the "on" times plus the "off" times sum to 2 minutes. The lamp is "on" more than it's off, but we can determine these times without ambiguity. I would add these nonintegers are limited to the rationals, however, not the reals. Still,

Spoiler

It is the integers that play the decisive role here. The state of the lamp is changed at intervals of 1/2ⁿ minutes: "on" when n is even; "off" when n is odd.  The OP asks us when these changes have been "completed" what the final state of the lamp is.

Now there are two ways to look at this. When t > 2 minutes the lamp is no longer changing between states and must be "on" or "off." Which is it? Well, it's simply the parity of the final time the lamp changed its state. That is, the parity of the final integer. That creates a paradox.

The second way is to inquire when t = 2 minutes precisely. Some persons argue this removes the paradox. The schedule for changing the lamp is to turn the lamp "on" at precisely t =0, "off" after an interval of 1/2 minutes, "on" again after another 1/4 minutes, "off" again 1/8 minutes later, and so forth. This removes the paradox because the sums of all these intervals is < 2 minutes. That leaves us free to specify things when t = 2 minutes precisely. Optimists would say the lamp is then "on." Pessimists would say it's "off." Neither answer is provably wrong.

 

[bonanova]

Ah, Mr. Thompson's lamp has been dusted off and put through its paces.

We can answer this question immediately upon discerning

Spoiler

whether the largest integer is odd or even.

 

[DejMar]

Spoiler

Whether the largest integer is odd or even is irrelevant. The set of numbers is described as the set of real numbers and not integers - the mathematical expression described is the summation of the series of inverse powers of 2, which approaches the limit of 2. Odd and even have no defined meaning in non-integers.
 
Further, the physical property of light is not discussed in the problem description. How many bundles of photons are emitted? And, though in a perfect vacuum they move at the rate of c, the question lacks the definition of what location in the space-time continuum the light is being considered. At the source, by the description of the problem, the light is indeed 'off' -- yet the photons still radiate away from that point which can allow the light, given no definitive point in space-time, to be 'on'. If to be considered, there should  be defined in this scenario whether the space-time field is warped or perfectly uniform, and whether the photons with the average statistical frequency of emission that is constant is also uniform in frequency other than in statistical averages.

In a purely mathematical analysis, the question asks whether non-integers are odd or even. The answer is undefined (though, specific numbers that are also integers may be). In this problem, the property of the light being 'on' or 'off' is the same as the property of 'even' or 'odd'. The definition that is mathematically applied does not fit within the given numerical system.

One more theoretically answer can be given with the following question: Is Schrödinger's cat 'alive' or 'dead'?

 

[dgreening]

This is very interesting, but I think there is a more concrete answer.
 

Spoiler

 

For each "On - Off" cycle, the light is ON 2/3 of the time and OFF 1/3 of the time.

After not too many cycles the additions make very little difference in the total On Time and Off time.

This would indicate that the light is on for 1:20 and off for 0:40.

 

 

 

 

[bonanova]

18 hours ago, dgreening said:

This is very interesting, but I think there is a more concrete answer.
 

  Reveal hidden contents

 

For each "On - Off" cycle, the light is ON 2/3 of the time and OFF 1/3 of the time.

After not too many cycles the additions make very little difference in the total On Time and Off time.

This would indicate that the light is on for 1:20 and off for 0:40.

 

Agree.