# Thomson's lamp: a useless paradox?

Thomson's lamp was mentioned at How to understand numbers that become really large? (as well as a number of posts elsewhere on SE). I have mentioned elsewhere that in addition to Cantorian infinite numbers (ordinals and cardinals), there are infinite numbers of NSA such as nonstandard integers, which could be called ringinals as they are elements of a ring. From this point of view, Thomson's lamp paradox appears to be due merely to a confusion of cardinals and ringinals.

What creates the paradox is that an infinite cardinal is used where an infinite ringinal should be used, instead. If a ringinal is used, the paradox disappears: the lamp is on or off in accordance with the parity of the ordinal.

Is there any depth to this famous paradox beyond such a conflation of kinds of infinite number?

• What is "ringial"? Commented 2 days ago
• My understanding of the problem is village-bumpkin simple. We're switching the lamp on and off an infinite number of times, which the original author, if memory serves, interpreted as how infinity is usually interpreted: there's one more i.e. if you say it's on, there's an off following it and if you say it's off, there's an on following it. I suppose the question is meaningless, which is another word for useless. Commented 2 days ago
• Not really. You constructed a scenario that calls for the end of an infinite sequence; you need relativistic physics to consider an answer. For a variant that has an interesting answer, I provided one here stats.stackexchange.com/a/316227/107903 Commented 2 days ago
• "If a ringinal is used, the paradox disappears: the lamp is on or off in accordance with the parity of the ordinal." So what is the parity of an infinite ringinal? What is the ordinal of the infinite sequence of switches? Commented yesterday
• @Hudjefa is exactly right, the question of Thomson's lamp is ill-posed: If you observe a state of the light bulb after an infinite number of switches have been performed, it requires that a last switch has been performed. And that last switch can never happen, because infinity means that there must be a following switch. The disappointing answer is that the limit of the infinitely switching light bulb is simply not defined. Commented yesterday

The resolution to the paradox is that it violates the laws of physics. It would take an infinite amount of energy to move the lamp switch so fast. "Would the lamp be on?" is a question about physics, which in principle must be answered by following the differential equations of physics. If you introduce singularities (such as infinite kinetic energy in the moving switch), it's not surprising that the equations can't give a meaningful result.

Perhaps you object that this is an imaginary lamp, not a physical one. In that case, calling it a "lamp" does nothing but invoke inappropriate intuitions about how a lamp works - intuitions derived from physical lamps following the laws of physics. The intuition that lamps are on or off depending on how many times the switch was flipped is inappropriate here, for example, since the switch is not flipped an integer number of times. We should call it something else, perhaps a "blonker," and then the paradox can only be answered by following the axioms of "blonkers," which you'll have to define yourself, and which have no relation to reality as we know it.

Thomson introduced this paradox as a way of arguing for the impossibility of "supertasks," tasks where an infinite number of "steps" are completed in a finite amount of time. His argument was that since we have no way to say whether the lamp is on or off at the end, the task must be impossible. Of course, that's not the only reason the task is impossible!

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented 22 hours ago

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The paradox arises from confusion between an open set and the closure of that set. It also makes the common error of confusing infinity with a large integer.

The total time it takes to switch on and off is an open set. As the time interval between on and off gets shorter, the time only approaches 2 minutes. It never reaches it.

The total time that the switch is being moved is the sum of the elements of an open set. 2 minutes is the closure of that set but it is not an element of the set.

It's exactly like telling somebody to write down every positive integer twice as fast as the previous integer, and after 2 minutes, when he's reached infinity, ask him if the last number he wrote was was even or odd.

There IS no "last number he wrote." It's an illusion. It's stage magic.

FREE BONUS FACT!

This is also a common mistake in understanding black hole event horizons. The edge of spacetime there is an open set. The event horizon is the mathematical closure, but it doesn't actually exist.

And because relativity makes you get shorter and slower as you get closer, you can never reach it.

• Is infinity odd or even? Commented 2 days ago
• @Hudjefa That's the fallacy of treating infinity like it was just a big integer. Commented 2 days ago
• @Hudjefa That's the fallacy of treating infinity like it was just a big integer. Commented 2 days ago
• @Hudjefa: For posterity: The set of natural numbers N and the set of extended natural numbers N∞ are distinct. "All elements are even or odd" is a property of N but not N∞. As this answer notes, it might not be the case that N + 1 is equivalent to N∞, so we can't extend even-and-odd reasoning up to N∞ either by declaring that infinity is even or odd. The given answer here is topological, which is honestly a much better way to think about N∞ than to imagine that infinity is its own element. Commented yesterday
• "The event horizon is the mathematical closure, but it doesn't actually exist. And because relativity makes you get shorter and slower as you get closer, you can never reach it." By my understanding, this is false. Relativity tells us that an outside observer won't see you reach the event horizon, but if you yourself are travelling to the event horizon, then in your own frame of reference, you will reach it after a finite time. Commented yesterday

It's similar to the Achilles-and-tortoise paradox: The situation is described as an infinite series of points in time, which converge on a given time t-limit; since the description does not define what happens beyond t-limit, asking what happens after that time given just the description is meaningless.

For Achilles and the tortoise, the situation can be easily extrapolated beyond t-limit: Achilles will overtake the tortoise, the description was artificially phrased in a way that ends at t-limit but the situation is completely linear.

For Thomson's lamp, there is no such extrapolation: The Grandi series 1-1+1-1+1... diverges, i.e. fluctuates between 0 and 1 without ever stabilizing.
One could state that at an arbitrary interval between 0 and t-limit, the lamp is on and off 50% of the time, respectively, so it is effectively half-on and half-off, but I think even that does not converge but alternates between 0.25 and 0.75.
So, no, there isn't a state assignable to t-limit for Thomson's lamp.

So, the answer is: it is not a paradox but a divergent series, and any attempt at assigning a limit value to it is useless.
It would be useful for teaching, and more or less useless depending on what kinds of axioms you use to define the situation.

• It all boils down to, I suppose, there's no final step and so the question of whether the lamp is on/off is moot. Commented 2 days ago
• @Hudjefa That, in combination with the switch alternating between a 0 and a 1 position. E.g. if you have a "more physical" lamp with the usual inductivity of electrical currents, a 50% current intensity will result; mathematically, the series will become more like an integral over the last switching cycles and converge to a 50% value, while an alternating 0-1 series does not converge. I.e. you need both, an infinite series and nonconvergence. Commented 2 days ago
• Now that you brought 0 and 1 to my attention, I recall reading the switch's state is 1/2 or 0.5 ... somewhere between off and on, kinda like quantum superposition, hovering between 2 mutually exclusive states. You can't say it's on and you can't say it's off. Is it neither or is it both? RIP Sanjaya Belathiputta. Commented yesterday
• @Anixx The Grandi series here represents to the sequence of states; the intervals do not matter for this. The states On, Off, On, Off, ... can be perfectly represented as +1, -1, +1, -1, ... which is exactly the Grandi series. Commented yesterday
• @Anixx The intervals matter only insofar as their diminishing length places the series' "infinity" at a finite point in time, so we need to the series' limit to assign a meaningful state to that time. Turns out the series does not have a limit. Commented yesterday

Is there any depth to this famous paradox beyond such a conflation of kinds of infinite number?

Yes. It plays a role in metaphilosophical discourse in speaking to the power and limits of intuition (SEP) in the philosophical method. The point for a philosopher of mathematics, physics, or computation, for instance, might be to attempt to resolve the paradox. For a philosopher interested in exploring the philosophical method itself, it raises questions about the reliability of philosophy itself. In An Introduction to Metaphilosophy, for instance, Overgaard et al. explore the use of intuition in Chapter 4: The Data of Philosophical Arguments.

On the face of it, it's a question that examines reasoning about computation in physical systems (SEP). If one considers computers in virtue of their physical implementation, then one must tackle certain philosophical challenges regarding logic truths versus nomological truths. Supertasks are a class of thought experiments that push the limits of intuition to explore how what is logically possible might violate what is nomologically possible. In the case of Thompson's lamp, for instance, although one can accelerate the activity of switching mathematically indefinitely, the same cannot be said of building physical machines to do so.

Consider that the flow of electricity cannot occur infinitely fast physically because of the constraints of the speed of light, and yet the switching of a lamp without taking this nomological constraint in effect knows no limit. Thus, Thompson's paradox and other supertasks are philosophically not mathematical questions, but questions of metaphysical grounding (SEP) since they impinge on questions about how explanations among various classes of theories and claims work.

A selection of the various answers others provide demonstrate the value of challenging intuition. Intuition skepticism in metaphilosophical literature is a position in the epistemological function of intuition (SEP) that rejects the use of intuition in a philosophical method. One might argue Thompson's paradox supports intuition skepticism because it shows that our intuitions can lead astray without careful consideration of the mathematics of nomological constraints. Of course, a philosopher who defends the use of intuition can point to Einstein's thought experiments as a support for the use of intuition. But either way, the attempt to dissolve the paradox and the questions that the paradox raise on method are important to understanding the nature of philosophy.

I am with Causative on this.

This is a hilarious example of creating faux profundification by mixing immiscible categories:

• empirical (physics) with rational/Platonic (math)
• more importantly, pure math with math modelling

In about 34 steps we are in picosecond flips — 'nuf said?

If one insists, it's a couple more steps to Planck time.

34 < ∞ methinks!

At the 2 minute mark the lamp is being turned on and off simultaneously and so it's in a superposition of being both on and off. At the first instant after the 2 minute mark the superposition collapses and the state is observed.

You'd probably need a few robust lamps and a willing superbeing to say more about the probability distribution of the final state and it seems reasonable that the state would depend on the specific lamp. But supposing a perfect platonic lamp then it seems hard and pointless to argue that it would be anything other than 50/50.

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• Downvote the new guy with no explanation of why. Wonderful way to stop people from contributing. I stopped posting on stack overflow for the same reason. Shame to see this site is no better .This seems a better solution to the paradox than any of the others I've read. Commented yesterday
• Not 100% I follow this part (“At the first instant after the 2 minute mark the superposition collapses and the state is observed”) but this is generally how I view this problem and I think it agrees with the Grandi series summing, in one view, to 1/2.
– bob
Commented yesterday
• I have not downvoted, but the answer is just wrong: The question is not about quantum-mechanical things, superpositions are not probabilities, and there is no final state by construction so the answer talks about things that do not happen. BTW that's the very purpose of Thomson's Lamp construction: That there is no way to define the lamp's state. Commented yesterday
• Time continues to flow and so at 2 minutes the superbeing is simultaneously switching the button both on and off. Thus the lamp is in multiple states (on and off) at the same time. That's the definition of a superposition. The "I must have switched it on and then off and vice versa" argument is specious because the unlimited being can and must perform both actions simultaneously. And by construction there must be a final state because time continues and the switch pressing stops. The only question we have left is which state the superposition collapses to. Commented 13 hours ago
• Sorry I didn't read that it was at the 2 minute mark. At the two minute mark it's both on and off at the same time, it's in a superposition. Commented 13 hours ago