Thomson's Lamp | Surprising Paradoxes

OFF

Time: 0.000 min

Switches: 0

Toggle Count

Rate: 0 toggles/sec

Timeline (0 → 2 minutes)

0 min 1 min 1.5 min 1.75 min 2 min

State Sequence:

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Interval Visualization (geometric decay)

1 min 0.5 0.25 0.125 ... 0

Sum: 1 + 0.5 + 0.25 + 0.125 + ... → 2 minutes

🎬 Simulation

⚡ Speed

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🤯 The Paradox

Switch at t=1, t=1.5, t=1.75, t=1.875...
All switches complete by t=2 minutes.

At t=2, is the lamp ON or OFF?

It can't be ON – every ON was followed by OFF!
It can't be OFF – every OFF was followed by ON!

1 − 1 + 1 − 1 + 1 − 1 + ... = ???

The Setup

In 1954, British philosopher James F. Thomson devised this thought experiment to analyze supertasks—the completion of infinitely many tasks in finite time.

"Suppose I switch the lamp on at time 0. At t=1 minute, I switch it off. At t=1.5 minutes, I switch it on. At t=1.75 minutes, off again. Each switch takes half the time of the previous interval..."

The switching times form a geometric series: 1 + 0.5 + 0.25 + 0.125 + ... = 2 minutes. After exactly 2 minutes, infinitely many switches have occurred.

The Contradiction

Argument	Conclusion	Problem
Every ON is followed by OFF	Can't end ON	There's no "last" switch!
Every OFF is followed by ON	Can't end OFF	Same problem in reverse!
Lamp must be ON or OFF	Contradiction!	???

Grandi's Series: The Mathematical Heart

If we represent ON = 1 and OFF = 0, the lamp's state at any moment before t=2 is:

S = 1 − 1 + 1 − 1 + 1 − 1 + ...

This is Grandi's series, which famously has no standard sum:

Group as (1-1) + (1-1) + ... = 0 + 0 + ... = 0
Group as 1 + (-1+1) + (-1+1) + ... = 1 + 0 + 0 + ... = 1
Cesàro sum: Average of partial sums = 1/2

The sequence of partial sums is 1, 0, 1, 0, 1, 0... which oscillates forever and does not converge to any value.

Resolution: The Problem Is Underspecified

Philosopher Paul Benacerraf (1962) argued that Thomson's lamp is not truly paradoxical—it's simply underspecified.

The supertask defines the lamp's state at every time before t=2, but nothing in the setup logically determines its state at t=2. The final state is simply not entailed by the infinite sequence of switches.

"The lamp could be ON at t=2, or OFF, or transformed into a pumpkin. Nothing in the premises rules out any possibility."

Comparison with Ross-Littlewood

Unlike the Ross-Littlewood paradox (balls in a vase), Thomson's lamp has no definite mathematical answer:

Paradox	Question	Answer
Ross-Littlewood	How many balls at noon?	0 (each ball n removed at step n)
Thomson's Lamp	Is lamp on or off at t=2?	Undetermined (no convergent limit)

The difference: in Ross-Littlewood, we can trace each ball's fate. In Thomson's lamp, the ON/OFF sequence has no limit—the question is meaningless without additional assumptions about what happens at t=2.