Can you really fit ALL the counting numbers into the spaces between 0 and 1?
After you watchCan you really fit ALL the counting numbers into the spaces between 0 and 1?
The short answer
Yes — some infinities are bigger than others. The counting numbers 1, 2, 3… and the even numbers are the same size because you can pair them one-to-one forever, but the decimals between 0 and 1 are a bigger infinity that can never be listed.
Try this next
- What if you slip the brand-new decimal back onto your list as a fresh row and run the machine again? Predict whether the machine is now stuck. Then add the new number to the list and re-run the diagonal — watch it build yet another decimal that dodges your patched list too.
- What if you only changed the diagonal digit on every OTHER row instead of every row? Guess whether the new number still has to differ from every row. Then trace the rows you skipped — does the new decimal still avoid matching them?
Now you — bend it
- What if Invent your own digit-flip rule for the diagonal machine instead of the built-in one.Keep your rule away from 0 and 9 so you dodge the 0.4999…=0.5 trap, then check the new number still differs on every diagonal.
- What if Try to design a list of decimals so clever that the diagonal machine finally fails to find a missing number.Whatever order you pick, the machine just walks down your diagonal — predict where the dodge happens before you run it.
- What if Push the idea up a level: can you imagine a collection so big that not even ITS subsets can be listed against it?Every pile's set of subsets is always strictly bigger, which is why the sizes of infinity never stop climbing.
Can you prove it?No numbered list can ever hold every decimal between 0 and 1. — Take any list, run the diagonal machine to build a decimal that differs from row 1 in digit 1, row 2 in digit 2, and so on, then check it can't equal any row — so the list was always missing one.
Design your own test:Before you run it, predict whether reshuffling the rows lets you finally cover every decimal — or whether the diagonal always finds a fresh escapee.
Explain it to a 6-year-old: Some forever-piles can be matched up one-for-one, but the in-between numbers are so crowded that any list always leaves one out.
The whole story
How it works
The fair way to compare two endless piles is to try matching them one-to-one with nobody left over. You can pair every counting number to an even number (1→2, 2→4, 3→6…) forever, so those two infinities are the same size. The decimals between 0 and 1 are different: take any numbered list of them and build a new decimal whose 1st digit differs from row 1's 1st digit, whose 2nd digit differs from row 2's 2nd digit, and so on. That new decimal can't equal any row, so it's missing from the list. No list can ever hold them all, which makes the decimals a strictly bigger infinity.
What people get wrong
People often think infinity is just one thing, so every endless collection must be the same size. It isn't. The even numbers feel like 'half' of the counting numbers but are actually the same size, while the decimals between 0 and 1 are genuinely bigger — you can pair up and list the counting numbers, but you can never list all the decimals.
The catch
The even and counting numbers count as the 'smallest' endless size precisely because they can be lined up in a list, even though your gut says there are fewer evens. The decimals are a bigger infinity because no list can hold them all — but 'bigger infinity' does not mean a bigger number; both still go on forever, one just can't be lined up at all.
Questions kids ask
How can the even numbers be the same size as all the counting numbers?
Because you can match them one-to-one with nobody left out: 1→2, 2→4, 3→6, and so on forever. Every counting number gets exactly one even partner and every even number is used, so the two endless piles are the same size — even though it feels like there should be fewer evens.
Why can't you list all the decimals between 0 and 1?
Take any numbered list of them. Build a new decimal that differs from row 1 in its 1st digit, row 2 in its 2nd digit, row 3 in its 3rd digit, and so on. That new decimal can't be the same as any row, so it was left off. This works on every list, so no list is ever complete.
Does 'a bigger infinity' mean a bigger number?
No. Both the counting numbers and the decimals go on forever, so neither is a finite number. 'Bigger infinity' means one pile can't be paired one-to-one with the other no matter how you try — the decimals can't be lined up against the counting numbers, so there are strictly 'more' of them.
Is there a biggest infinity of all?
No. For any collection you can always build a strictly bigger one (the set of all its subsets is always bigger). So the sizes of infinity go on forever too — there is no largest infinity.
Talk about it
- Guess first: which feels like more, all the counting numbers or just the even ones? Then say why your gut might be wrong.
- If I handed you a never-ending list of decimals and swore it had every one, how could you catch me in the lie?
- What do you think it means for something endless to still be 'bigger' than something else endless?
For grown-ups
Two infinite sets have the same cardinality if there is a one-to-one matching (bijection) between them. The even numbers and the counting numbers ℕ are both countably infinite (ℵ₀). The reals in (0,1) are uncountable, proven by Cantor's diagonal argument: assume any list of them, build a number whose nth digit differs from the nth listed number's nth digit, and it is absent from the list, so no list is complete. To avoid the 0.4999…=0.5 edge case, choose replacement digits that avoid 0 and 9. By Cantor's theorem there is no largest infinity — the power set of any set is always strictly bigger.
Keep going
What else makes you wonder?
- Is there an infinity that sits between the counting numbers and the decimals, or do you jump straight from one to the next?
- Are there more fractions than counting numbers, or do those two piles still pair up one-to-one?
- If the decimals are bigger than the counting numbers, what could possibly be bigger than the decimals?