[tlocke]

I know the light switch idea as Thomson's lamp:

https://en.m.wikipedia.org/wiki/Thomson%27s_lamp

I used it as an interview question many years ago. I wasn't very rigorous about it, basically any plausible answer was good enough for me. Answers fell into two categories, the theorists (it's an infinite series that doesn't converge) and the pragmatists (you couldn't physically do it).

[nom]

> .. they say that its sum is 1/2. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on.

Maybe a lamp is a bad example, it is clearly half-on: PWM modulated with 50% duty cycle and infinite frequency. I don't see any Problem ;)

[pixl97]

Darn quantum lamps

[taneq]

As anyone who's played light switch disco will know, the light bulb will blow long before you reach the theoretical limit of the series. :D

(Actually thinking about it, that's probably not true because the switching accelerates so fast... the limiting factor is probably actually friction from moving the switch, and/or the tensile strength of the switch housing. I wonder if it would just disintegrate or if it would build up enough heat to properly explode...?)

[mitthrowaway2]

I think the limiting factor will be the inductance of the wire and lightbulb filament, which sustains a current even when the switch attempts to break the circuit!

[jerf]

I'll give you a good in between. The problem is underspecified because the universe in question is underspecified. You can get pretty much any result you want out of such a strange thing by specifying a more fully-fleshed out universe in which that is the answer. You can create one in which the answer is half-on, half-off, even though you initially specified your cells as only containing integers and not realized that your axioms entailed additional values, which themselves could be limited to rationals with the right axioms, or the computables, or the reals, or several other things.

In this, your universe may admit of a natural "averaging" operation, where you take a "half on" thing and a "quarter on thing" and produce a "three quarters on thing" in some natural manner. Playing around in my head, I find you need to be careful about your definition of "fully on" and "fully off" here; you may or may not want to permit an "infinite sequence" that contains zero changes in it. But you'll get a weirder number system if full-on and full-off are not permitted, where you can arbitrarily close to them but not quite achieve them.

Or you may find your axioms force you to admit a new "undefined" value that, again, you didn't realize when you started that you were adding but it turns out you were. Or you could create a scenario where this actually freezes your entire universe because there simply is no way to proceed past the singularity, because your "take next step" function simply stops working and you get the mathematical equivalent of a crashed or hung program. (Just because something is "mathematical" does not imply totality; there's many, many, many cases where the answer is just "That function doesn't work there". Sometimes that leads to an exploration of "well what if we force it anyhow", e.g., "square root of negative numbers", but sometimes it really is just "this function stops here".)

An intriguing exercise for those who may want to experience this sort of "I didn't realize these axioms did that!" first hand is to look up the "surreal numbers" if you've never heard of them before. Read the definition of surreal and try to guess what numbers will emerge from it. Then watch as your mind is blown about what you admit when you admit those axioms. Unfortunately, I could not find a presentation of surreal numbers that started with the bare definition; for obvious reasons everyone leads with where you're going because who reads unmotivated mathematical definitions for fun, right? Still, try to imagine what you would have expected such axioms to produce, what someone might have "intended" them to produce, versus what the actually do.