[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,

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         | [EMAIL PROTECTED]

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