Are some infinities bigger than others?
One partner for each
Pair every child with one chair. Then check whether anyone is left alone.
Counting numbers meet even partners
show pairs3 pairs · rule keeps going
1 pair6 pairs, then more
The rule n → 2n gives every counting number one even partner.
Number the claimed decimals
If every decimal had a counting partner, this list would hold them all. Test row 1 at digit 1, row 2 at digit 2, and on.
Always make the digit different
input digit1 changes to 2
digit 1digit 9
The whole rule: 1 becomes 2. Every other digit becomes 1.
Its diagonal is 1, 8, 1, 4, 9, 1
The rows and digits continue beyond the screen. What will the rule build?
Use 1 → 2, and every other digit → 1.
Which decimal will the machine build?
Sweep down the diagonal. Watch each changed digit land.
diagonal sweepready at row 1
row 1compare all rows
The screen is small. The rule is not.
Only six rows fit here +
The real argument keeps going: row n supplies digit n, and the new digit differs there. The screen is only a readable window.
Why use only 1 and 2? +
Some decimals have two names: 0.4999... equals 0.5. Building with only 1s and 2s avoids that naming trap.
Is infinity a giant number? +
No. “Bigger” here means one endless set cannot be paired one-to-one with the other—not that infinity sits at the end of counting.
Some forever-piles fit one numbered line. The decimals between 0 and 1 do not: every claimed list leaves a diagonal escape.
Psst, grown-ups — the precise idea
A one-to-one matching is a bijection. The counting numbers and evens are countably infinite. Cantor's diagonal argument shows the real numbers in (0,1) are uncountable. Using the replacement rule 1→2 and every other digit→1 avoids tails of repeating 9s.