Cz: "Peegee" is a valid number

[o_O]

If we applied constraints that we typically give to english words, you are right in that there are finite "possible" words in the english language. I don't think english words can lack vowels and words like millionaaaa/millionaaaaa/millionbbbbbbbbbbbbb lack any meaning and are indistinguishable in speech; I was just playing with the notion that since we can define a word as a n (any) lengthed string using characters from the set {a,z}, both sets should be equal in length.

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). Thus not only is peegee and face and ... oO all possible numbers, but that every conceivable number does have an associated, albeit eventually nonsensical name.

I agree with this point, but it raises the question of which set is bigger: the set of all possible numbers or the set of all possible strings?
Take some length <i>n</i>, define the set N to be all possible <i>n</i>-digit numbers and the set W to be all possible <i>n</i>-character strings.
Observe that for <i>n = 1</i>, we have <i>|N| = 10¹ = 10</i>, but <i>|W| = 26¹ = 26</i>.
Similarly, <i>|N| = 10² = 100</i>, but <i>|W| = 26² = 676</i> (Where |A| is the size of the set A).
This implies that:
- We can define the size of the set of all possible numbers of length <i>n</i> to be <i>|N| = 10ⁿ</i> (simply 1 with <i>n</i> zeros).
- We can define the size of the set of all possible words of length <i>n</i> to be <i>|W| = 26ⁿ</i>.
- Therefore, for <i>any</i> value of <i>n</i>, the size of the set of all numbers that can be formed from any length up to <i>n</i> is <i>∑(10ⁱ)</i> for <i>i = 0 : n</i>.
- Similarly, for <i>any</i> value of <i>n</i>, the size of the set of all word that can be formed from any length up to <i>n</i> is <i>∑(26ⁱ)</i> for <i>i = 0 : n</i>.

If you perform any of these calculations (you shouldn't even need to though ), you'll see that <i>|W|</i> for some finite <i>n > 0</i> is always larger than <i>|N|</i>. So what happens as <i>n</i> tends towards infinity? There is clearly no limit on the number of numbers you can have, and no limit on the number of words, but the set of words always appears to be larger.
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.

edit: also, and for some reason I forgot to say, that we could, if words were just like character strings (in programming), we could name every conceivable number using only 10 characters.

This is true of the concept whereby we assign a value to each character in the subset of the alphabet that is equal in size to the set of all digits with a 1:1 correspondence in values (mapping {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => {a, b, c, d, e, f, g, h, i, j}). If we're talking about simply <i>naming</i> each number, then without constraint, we could do it with an alphabet of only one character (naming {0, 1, 2, 3,...} as {a, aa, aaa, aaaa, aaaaa,...}).