[Warning: maths ahead :-P and I've just stumbled into this thread, so I
might be waaay off; my apologies if that's the case]
I don't think there's necessarily an answer to this question. The
history of the lamp is a series of points in the set {ON, OFF}X R_[0,2],
but you have only defined it for {ON, OFF} X R_[0,2), yet the question
asks wheter the history includes (ON, 2) or (OFF, 2). As you defined the
history as a function F:R_[0,2) -> {ON, OFF}, one is tempted (and I
guess that's what you were thinking about) to exted the function in a
continuous way. With a suitable topology for {ON, OFF}, the domain of F
is dense, so if F were continous such an extension would exist and be
unique. Sadly, F isn't continuous unless you give {ON, OFF} the
indiscrete topology, but then you lose the power to distinguish between
ON and OFF in any case [the lamp's history converges in the limit to
both (yep, in non-Hausdorff topologies [ie, "too coarse], limits are not
necessarily unique)].
So, ignoring the physics of the issue [of which I understand nothing,
I'm afraid :-(], mathematically my guess off-the-cuff is that the
problem has no solution [as the setting doesn't give explicit
information about t=2, and either you can't extend in a natural way the
lamp's function, or if you can't it's at the price of saying anything at
all with that extension].
open to comments --- I'm fairly new at this in any case,
Marcelo
PS: A fascinating problem, really.
> Welcome,
> I've been looking for an idiot savant to answer this question: Perhaps you've heard of Thompson's Lamp. This is an ideal lamp,
> capable of infinite switching speed and using electricity that travels at infinite speed. At time zero it is on. After one minute
> it is turned off. After 1/2 minute it is turned back on. After 1/4 minute it is turned off. And so on, with each interval one-half
> the preceding interval. Question: What is the status of the lamp at two minutes, on or off? (I know the answer can't be calculated
> by conventional arithmetic. Yet the clock runs, so there must be an answer. Is there any way of calculating the answer?)
> Norman
--
Marcelo Rinesi | mrinesi.domain.name.hidden
People were stupid, sometimes. They thought the Library was
a dangerous place because of all the magical books, which
was true enough, but what made it really one of the most
dangerous places there could ever be was the simple fact that
it was a library.
-- Terry Pratchett, "Guards! Guards!"
Received on Thu Oct 23 2003 - 02:55:46 PDT