Here are two postulates, about the two undefined terms globs and glitches.
- Every glob has a positive, finite, integral number of glitches.
- There are more globs altogether than there are glitches on any one glob.
These two postulates yield a theorem: There exist two globs having the same number of glitches.
Convince yourself that this theorem is true, and then write a convincing argument.
Solution
Well, suppose there are, say, 10 globs. Then each glob can have 1, or 2, or 3, . . . , up to 9 glitches max. If we try to give each glob its own unique number of glitches, we can take care of 9 out of the 10 globs--but the tenth glob will have to have the same number of glitches as one of the first nine globs. This works for any number--10 was just an example.
If you frame the problem right, you can prove that there are two people in Ohio with the same number of hairs on their heads, and that there are two boxes of Cheerios in Ohio with the same number of individual cheerios in their boxes.
I am indebted to the late Donald Kreider for this problem.