*Aleph-null bottles of beer on the wall,*

*Aleph-null bottles of beer!*

*Take one down, pass it around,*

*Aleph-null bottles of beer on the wall!*

*— Math nerd drinking song*

You may already know the standard story about infinite sets like {1,2,3,…} and {2,3,4,…}. Even though the second set seems to be *smaller *(it’s missing one of the elements in the first set), Cantor taught us that the two sets are the *same size *(in the sense that there’s a one-to-one correspondence between them). The two sets have the same “number” of elements (namely aleph-null), and aleph-null minus one equals aleph-null. For many students, that anomaly takes some getting used to.

But there’s a perfectly respectable mathematical sense in which the two sets do *not* have the same number of elements. With a suitable notion of what it means to “count” the elements of an infinite set of numbers, different from Cantor’s, the size of {2,3,4,…} *is* smaller than the size of {1,2,3,…}; in fact, it has *one fewer element*. Likewise, in this alternative way of measuring how big sets of numbers are, the set {1,3,5,…} is slightly bigger than the set {2,4,6,…}. How much bigger? Half an element! (Though see Endnote #2.) Continue reading →