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The Ross-Littlewood Paradox

Add 10 balls, remove 1. Repeat infinitely. How many balls remain?

THE INFINITE VASE PARADOX

Before noon: add balls 1-10, remove ball 1. Then add 11-20, remove ball 2. Add 21-30, remove ball 3... At step n, NET +9 balls. After infinitely many steps... the vase is EMPTY!

Time to Noon
1/1 min

Controls

5x

Statistics

0
Step n
0
Balls Added
0
Balls Removed
0
In Vase Now

Two Arguments

INFINITE balls remain

Each step adds 10, removes 1 → net +9 balls. After ∞ steps: 9 × ∞ = ∞ balls!

ZERO balls remain

Ball #k is added at step k, removed at step k. EVERY ball has a removal step. No ball survives to noon!

Ball Fate Tracker

Start simulation to track individual balls

The Setup (Littlewood, 1953)

Imagine a vase and infinitely many numbered balls. We perform infinitely many steps, each taking half the time of the previous:

Step 1
1 min before noon
Add 1-10, Remove #1
Step 2
½ min before noon
Add 11-20, Remove #2
Step 3
¼ min before noon
Add 21-30, Remove #3
...
→ noon
∞ steps complete

The Mathematical Answer

The vase is EMPTY at noon. Despite adding infinitely more balls than we remove!

The key insight: Ask "which ball is in the vase at noon?" For ANY ball #k, we can identify exactly when it was removed (at step k). Ball 1 removed at step 1. Ball 1,000,000 removed at step 1,000,000. Every single ball has a specific removal step!

The "net +9" argument fails because infinity doesn't work like finite numbers. You can't simply multiply ∞ × 9. The cardinality of added balls equals the cardinality of removed balls: both are countably infinite (ℵ₀).

The Paradox

Our intuition about finite processes doesn't transfer to infinite ones. The vase is demonstrably empty, yet at every finite step it contained more balls than before.

Why It Matters

Supertasks: This is a "supertask" - infinitely many actions in finite time. Zeno's paradox is another example.

Set Theory: In ZFC set theory, the limit inferior/superior of the vase contents is the empty set.

Philosophy: Benacerraf argues the problem is ill-posed - we specify what happens BEFORE noon but not AT noon. The vase could contain anything, including exploding into dust!