We are all familiar with the standard nomenclature for the smallish natural numbers, such as
one, two, three, ..., one hundred, one hundred one, ..., fifteen thousand two hundred forty-nine.
I have in mind the simple American number naming conventions, together with the names for large numbers. (Update Names of large numbers seems to be more thorough. Note to Wikipedians: should probably merge those two pages somehow.)
Preliminary question. Is there a sensible naming system that provides a canonical name for every natural number?
That is, I want a naming system that extends the current naming system sensibly in such a way that every number gets a unique name. Please provide a system and explain why it is sensible.
For example, if there were some natural way to extend the Latin naming convention indefinitely, that would be great.
Let me assume that some of you will be able to provide such a naming system.
Main Question. What is the order-type of the set of natural numbers, when written in alphabetical order?
For example, the order will not be the same as the order $\omega$ of the natural number themselves, since presumably there will be infinitely many numbers starting with "o", as in one hundred, one million, one thousand, and so on, and these will all be alphabetically preceding two hundred, two million, two thousand and so on.
So the order type will probably be related naturally $L\times 26$ for some order $L$, or actually, less than $26$, since probably not every letter will be a legitimate first letter of a number name.
It is conceivable that the order type will depend on syntactic features of the naming convention.
Here is a part of the order, for numbers up to 100: (from hervé graumann 1988)
1) eight
2) eighteen
3) eighty
4) eighty-eight
5) eighty-five
6) eighty-four
7) eighty-nine
8) eighty-one
9) eighty-seven
10) eighty-six
11) eighty-three
12) eighty-two
13) eleven
14) fifteen
15) fifty
16) fifty-eight
17) fifty-five
18) fifty-four
19) fifty-nine
20) fifty-one
21) fifty-seven
22) fifty-six
23) fifty-three
24) fifty-two
25) five
26) forty
27) forty-eight
28) forty-five
29) forty-four
30) forty-nine
31) forty-one
32) forty-seven
33) forty-six
34) forty-three
35) forty-two
36) four
37) fourteen
38) hundred
39) nine
40) nineteen
41) ninety
42) ninety-eight
43) ninety-five
44) ninety-four
45) ninety-nine
46) ninety-one
47) ninety-seven
48) ninety-six
49) ninety-three
50) ninety-two
51) one
52) seven
53) seventeen
54) seventy
55) seventy-eight
56) seventy-five
57) seventy-four
58) seventy-nine
59) seventy-one
60) seventy-seven
61) seventy-six
62) seventy-three
63) seventy-two
64) six
65) sixteen
66) sixty
67) sixty-eight
68) sixty-five
69) sixty-four
70) sixty-nine
71) sixty-one
72) sixty-seven
73) sixty-six
74) sixty-three
75) sixty-two
76) ten
77) thirteen
78) thirty
79) thirty-eight
80) thirty-five
81) thirty-four
82) thirty-nine
83) thirty-one
84) thirty-seven
85) thirty-six
86) thirty-three
87) thirty-two
88) three
89) twelve
90) twenty
91) twenty-eight
92) twenty-five
93) twenty-four
94) twenty-nine
95) twenty-one
96) twenty-seven
97) twenty-six
98) twenty-three
99) twenty-two
100) two
101) zero
Let me add that I don't necessarily expect that the order is a well-order. For example, if we have a naming convention whereby $10^k$ is represented for large $k$ simply by repeating "penpenpenpen$\cdots$pen", then we could make a descending sequence via penpenpenpen$\cdots$pen twelve, which would descend as the number of pen's increased, since we would be replacing t with p.