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.