A Supertask Paradox
How to Empty an
Infinitely Full Vase
Add 10 balls, remove 1. Repeat infinitely. How many balls remain? The answer is zero.
At step 1, put balls 1-10 in a vase, remove ball 1.
At step 2, put balls 11-20 in, remove ball 2.
Continue infinitely, each step in half the previous time.
After exactly 2 minutes, all infinite steps are complete. How many balls are in the vase?
Zero.
You added infinitely many balls. Net +9 every step.
After infinite steps: no balls remain.
The Supertask Setup
A supertask is an infinite sequence of operations completed in finite time. The trick: each operation takes half as long as the previous one.
Step 1
t = 1:00
Takes 1 minute
Step 2
t = 1:30
Takes 30 seconds
Step 3
t = 1:45
Takes 15 seconds
The total time: 1 + 1/2 + 1/4 + 1/8 + ... = 2 minutes exactly.
At t = 2 minutes, infinitely many operations have completed. This is mathematically well-defined—it is the geometric series converging to 2.
The question is not whether we can perform infinite operations.
The question is: what is the state at t = 2?
Watch the Vase Fill
Let's watch what happens step by step. At each step n:
- Add balls (n-1)×10 + 1 through n×10 (ten balls)
- Remove ball n (one ball)
- Net gain: +9 balls
Current Step
0
Time (supertask)
0:00
In Vase
0
= 9 x 0 = 0
Removed
0
After n steps: 9n balls in vase
As n → infinity, ball count → infinity
But at t = 2 min... zero balls?
Notice: after n steps, you have exactly 9n balls. As n approaches infinity, the ball count approaches infinity.
And yet...
The Disappearing Act
Here's the key insight. Ask yourself: which specific ball is in the vase at t = 2 minutes?
Pick any ball. Ball 1? Removed at step 1. Ball 42? Removed at step 42. Ball 1,000,000? Removed at step 1,000,000.
Pick Any Ball Number
Every ball gets removed at a finite step.
Ball n enters at step ⌈n/10⌉ and is removed at step n.
No ball survives to t = 2.
This is the paradox. The number of balls at step n is 9n → infinity. But the set of balls at t = 2 is empty.
The Paradox Visualized
Ball count grows without bound... but reaches zero at the limit
At any finite step n:
9n balls in vase
This goes to infinity as n increases
At t = 2 minutes (completion):
0 balls in vase
Every specific ball has been removed
The mathematical resolution:
The limit of the cardinality (count) is not the same as the cardinality of the limit (set).
lim(n→∞) |S_n| = ∞, but |lim(n→∞) S_n| = |∅| = 0
Labeling Matters
Here's what makes this truly strange. The result depends entirely on how you label the balls.
Change the removal rule slightly, and you get completely different answers—even though you're always adding 10 and removing 1.
The Removal Rule Changes Everything
Rule Description
Original Rule
Remove ball n at step n
Result at t = 2 min
Zero balls remain
Every ball N is removed at step N
Balls removed (first 10 steps):
Survivors at limit: None
Same operation (add 10, remove 1), same count (9n balls)
But different labeling → completely different limits
With the original rule, every natural number eventually gets removed. With the modified rule, 90% of natural numbers are never touched.
Same physical operation. Different mathematical result. The labeling is doing the heavy lifting.
The Philosophy of Infinity
The Ross-Littlewood paradox reveals something deep about infinity and limits.
Limits vs. Limits
The limit of counts (cardinalities) and the count of limits (set membership) are fundamentally different operations. They do not commute.
Physical Impossibility
In the real world, supertasks are impossible. Matter has minimum scales, time has limits, and nothing can move infinitely fast. This is purely mathematical.
Well-Defined Weirdness
The paradox is not a contradiction—it is a counterintuitive but logically consistent result. Every step is valid. The conclusion follows.
Information vs. Quantity
The labeling (information) matters as much as the quantity. 9n balls does not tell you which balls remain—and that turns out to be everything.
Infinity is not just "a very large number."
It has its own rules that break our finite intuitions. The Ross-Littlewood paradox is a perfect example: 9 × ∞ = 0, when the right ball is removed at each step.
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Reference: Ross (1988), Littlewood (1953)