Cz: "Peegee" is a valid number

[Dr Unne]

Actually now that I think about it, since the set {a,z} is greater than the set {0,9}, using a 1:1 association, we quickly find out that the {a,z} set of 'words' is greater than the {0,9} set of numbers (this presumes that 'words' we come up with doesn't follow any schema).

The set {a..z} is bigger than the set {0..9}. But the set of all possible strings using arbitrary amounts of elements of {a..z} and the set of all possible strings using arbitrary amounts of elements of {0..9} are equally infinite. They have the same cardinality i.e. same size. Both are countable sets.

Consider this: which set is bigger, the set of all integers when you write them in binary, or the set of all integers when you write them in decimal? It's clear that they are the same set, written down using different representations.

The alphabet, a-z, could be considered a way of representing numbers in base 26. There's a 1:1 mapping from the set of all integers written in binary to the set of all integers written in decimal, base 26, or any other base. They're all equally infinite.

Note that this question is analogous to asking "Which is bigger: the set of all integers, or the set of all real numbers?". The integers are a subset of the real numbers, but they have no bounds themselves, so both sets are infinite.

Both are infinite, but the set of real numbers is bigger than the set of integers, in the sense that there is no 1:1 mapping from real numbers to integers. In any mapping of integers to real numbers, there will always be at least one real number that has no corresponding integer. Per Cantor's diagonal argument.

The fact that integers are a subset of real numbers doesn't matter. Cardinality of infinite sets isn't a straightforward topic. Which set has more elements, the set of odd integers, or the set of all integers? Turns out they have the same cardinality, even though the set of odd numbers is a subset of the set of integers.