I'd say the lamp is simultaneously on and off at 2 minutes. Take the two sequences independently, (a) "time at which the lamp turns on" and (b) "time at which the lamp turns off". (a) goes: 0, 1+1/2 min, 1+7/8 min, 1+31/32 min, 1+127/128 min, ... 1+(2^i-1)/(2^i) min where i is all the odd integers. (b) goes: 1, 1+3/4 min, 1+15/16 min, 1+63/64 min, ... 1+(2^i-1)/(2^i) min where i is all the even integers. You can see that the limit in both cases as i approaches infinity is 2 minutes. This means that at 2 minutes the lamp will turn on according to sequence (a) and off according to sequence (b). The problem is equivalent to asking whether infinity is an odd or an even integer, the usual answer to which is (I think) that infinity is neither - in fact, is not an integer at all. You may be disatisfied with this and say that "2 minutes" is a definite, measurable interval in the real world, and that there must be an actual, observable, on-or-off state of the lamp at this time. The problem is, of course, that no such ideal lamp could exist in the real world. If it did, aside from any other considerations, instantaneous switching on would mean infinite current, and hence infinite power in a finite volume, which would probably cause a big explosion putting an end to your experiment, and possibly putting an end to your universe as well! Big explosions, infinite power densities, end-of the-universe... where have I heard all this before?
Stathis Papaioannou Melbourne, Australia
-----Original Message----- From: Norman Samish [mailto:[EMAIL PROTECTED] Sent: Monday, 20 October 2003 5:07 PM To: [EMAIL PROTECTED] Subject: Thompson's Lamp
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 ----- Original Message ----- From: incarn81 To: [EMAIL PROTECTED] Sent: Saturday, October 18, 2003 11:36 PM Subject: Joining
Hello
I'm mainly an idoit, sometimes a savant. I get most of the references that I've read so far, but don't really have a deep technical background in any one area. Can't wait to catch up on the archives!
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