A Supertask Paradox
Thomson's Lamp:
The Impossible Light
Flip a switch ON at 1 min, OFF at 1.5 min, ON at 1.75 min... After 2 minutes, is the lamp ON or OFF?
A lamp starts OFF. You turn it ON after 1 minute.
Then OFF after another 30 seconds. Then ON after 15 seconds. Then OFF after 7.5 seconds.
Each toggle takes half as long as the previous one. At exactly 2 minutes, infinitely many toggles have occurred.
Is the lamp ON or OFF at t = 2?
Mathematics cannot answer this question.
The sequence ON, OFF, ON, OFF... has no limit.
Watch the Lamp Toggle
James F. Thomson proposed this thought experiment in 1954 to challenge the coherence of supertasks—infinite sequences of actions completed in finite time.
Toggle Count
0
Time
0:00.000
Notice how the toggles accelerate. Each one takes half as long as the previous. The total time: 1 + 1/2 + 1/4 + 1/8 + ... = exactly 2 minutes.
The infinite series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2.
This is mathematically uncontroversial—it is a geometric series.
The controversy is what happens at t = 2.
Time Compression: Zeno Style
The toggle points form a sequence converging to t = 2:
Toggle 1
t = 1.0
Lamp ON
Toggle 2
t = 1.5
Lamp OFF
Toggle 3
t = 1.75
Lamp ON
Toggle 4
t = 1.875
Lamp OFF
Toggle Points Converging to t = 2
As you add more toggles, they cluster closer to t = 2.
Time remaining after toggle 5: 1.875000 seconds
Like Zeno's paradox of motion, we fit infinite events into finite time. But unlike Zeno, we are not describing continuous motion—we are describing discrete state changes.
The Sequence Has No Limit
Here is the crux of the paradox. Consider the sequence of lamp states:
The Sequence Has No Limit
Odd toggles (1, 3, 5, ...)
State: ON
There are infinitely many
Even toggles (2, 4, 6, ...)
State: OFF
There are infinitely many
ON, OFF, ON, OFF, ON, OFF...
This sequence oscillates forever. It does not approach any single value.
Mathematics gives no answer for the final state.
In calculus, we say a sequence a_n has limit L if the terms get arbitrarily close to L as n increases. But the sequence 1, 0, 1, 0, 1, 0... does not approach any single value.
The limit does not exist.
There is no mathematical answer to "What is the lamp state at t = 2?"
The function S(t) is simply undefined at t = 2.
But Some State Must Obtain
Here is what makes Thomson's Lamp philosophically troubling:
A physical lamp must be either ON or OFF.
There is no third option. The lamp cannot be "undefined."
At t = 2, the lamp exists. It has some state.
But mathematics cannot tell us which.
This reveals a gap between mathematical models and physical reality. The supertask is mathematically well-defined (the times converge to 2), but the state function is not.
Explore Different Scales
| Toggle # | Time | State | Time Since Prev |
|---|---|---|---|
| 1 | 1.5000000000 | ON | 1.50e+0 min |
| 2 | 1.7500000000 | OFF | 2.50e-1 min |
| 3 | 1.8750000000 | ON | 1.25e-1 min |
| 4 | 1.9375000000 | OFF | 6.25e-2 min |
| 5 | 1.9687500000 | ON | 3.13e-2 min |
| 6 | 1.9843750000 | OFF | 1.56e-2 min |
| 7 | 1.9921875000 | ON | 7.81e-3 min |
| 8 | 1.9960937500 | OFF | 3.91e-3 min |
| 9 | 1.9980468750 | ON | 1.95e-3 min |
| 10 | 1.9990234375 | OFF | 9.77e-4 min |
| ... and so on ... | |||
Time remaining at toggle 10:
9.7656e-4 minutes
Gap between toggles 9 and 10:
9.7656e-4 minutes
No matter how many toggles you examine, infinitely more remain.
The gaps shrink exponentially, but the pattern ON-OFF-ON-OFF never settles.
The Mathematical View
The state function S(t) is not defined at t = 2. We simply cannot extend the function to that point. The question is meaningless in the same way asking "What is 0/0?" is meaningless.
The Physical View
A real lamp must have a state. Either our mathematical model is incomplete, or supertasks reveal that some infinite processes cannot be physically realized.
What is Your Guess?
At exactly t = 2 minutes, the lamp is...
What Does This Mean?
Thomson's Lamp connects to deep questions about infinity, physical reality, and the limits of mathematical modeling.
Actual vs. Potential Infinity
Can infinity be "completed"? Aristotle distinguished between potential infinity (always more) and actual infinity (a completed totality). Supertasks assume actual infinities can exist.
Does Math Describe Reality?
Mathematical models often lack answers (division by zero, undefined limits). Does this mean reality also lacks answers, or that our models are incomplete?
Physical Possibility
Real lamps have switching delays. Real time may be quantized (Planck time). The thought experiment may assume something physically impossible.
The Meta-Question
Perhaps the paradox shows that some questions are malformed. "Is the lamp on or off?" assumes a binary state, but supertasks may transcend binary logic.
Other Supertasks to Explore
Zeno's Dichotomy
To walk across a room, first go half, then half of the rest...
Infinite steps but continuous motion. Position is well-defined.
Hilbert's Hotel
A hotel with infinite rooms can always accommodate more guests.
Demonstrates infinity + 1 = infinity. No supertask needed.
Benardete's Paradox
Infinite gods each plan to block you before you reach a point.
Shows problems with "backward" supertasks.
Thomson's Lamp reveals the edge of mathematical reasoning.
Some supertasks (like Ross-Littlewood) have definite answers.
Others (like Thomson's Lamp) do not.
The difference is whether the sequence converges.
Explore More Paradoxes
Thomson's Lamp is one of many thought experiments that challenge our intuitions about infinity.
Reference: Thomson (1954), "Tasks and Super-Tasks"