The Farmers and Birds Infinity Paradox: Why Infinite Sets Behave Unexpectedly

A Classic Infinity Paradox in Disguise

With the short workweek behind us, it’s the perfect time to talk about the paradoxes of infinity. Infinity is a tricky thing — much like division by zero. Most people have heard of Hilbert’s infinite hotel, but here’s another paradox worth your attention.

The Setup: Two Farmers and Very Persistent Birds

Imagine two farmers: Alice and Bob. Both are about to plant an infinite row of seeds. But they share one big problem — pesky birds keep interfering.

For Alice, the birds steal every fifth seed she plants. For Bob, after every five plantings, the birds take the first remaining seed from his row.

The question: when Alice and Bob “finish” planting, will any seeds remain in their rows?

Surprising Answers to the Infinity Paradox

Alice will have seeds left. The birds take every fifth seed from an infinite supply — so infinitely many seeds still remain in her row.

Bob will have nothing. The birds take the first remaining seed each time, and by the end, his entire row is empty.

Why the Same Rate Produces Opposite Results

Here’s where the infinity paradox truly kicks in: Alice and Bob work simultaneously, and the birds steal seeds at exactly the same rate. So why does Alice end up with infinity while Bob ends up with nothing?

The paradox lies in the counterintuitive properties of infinite sets. An infinite set can be put into a one-to-one correspondence (a bijection) with one of its own proper subsets — this is actually one of the defining characteristics of infinity. So even though the number of seeds is infinite, the set the birds took from Alice appears smaller.

The Key Insight About Infinite Set Cardinality

In other words, Alice’s birds and Bob’s birds take away sets of seeds with equal cardinality — but one of those sets is a proper subset of the other.

The order in which elements are removed from an infinite set completely changes what remains. Remove every fifth element, and the structure of the remainder is preserved. Remove the first remaining element each time, and you systematically eliminate everything — even from an infinite supply.

That’s the heart of the infinity paradox: two processes that look equivalent at every finite step can lead to completely different outcomes when carried to infinity.