What is the order-type of the set of natural numbers, when written in alphabetical order?












12












$begingroup$


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 $Ltimes 26$
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.










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    I have a method for writing the names of numbers in alphabet: For $n$ just write a string of $n+1$ times the letter "a". So $0$ is "a", and "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa" is $345$. Now the order type is again $omega$!
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 13:35






  • 1




    $begingroup$
    This exact question was asked two weeks ago on r/math subreddit.
    $endgroup$
    – Wojowu
    Dec 20 '18 at 13:48






  • 1




    $begingroup$
    @Wojowu Great! I guess the puzzle is in the air.
    $endgroup$
    – JDH
    Dec 20 '18 at 13:53






  • 1




    $begingroup$
    @Holo: It will be, the string will just be indexed with a non-standard integer.
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 20:48






  • 3




    $begingroup$
    Actually, not even all native English speakers are "familiar with" the naming scheme of your first example: in British English, 15,249 is read out as "fifteen thousand two hundred and forty nine". Aside from that the form of your question amounts to: "please invent something nice and then tell me what its properties are". This is not a good MSE question.
    $endgroup$
    – Rob Arthan
    Dec 20 '18 at 22:54
















12












$begingroup$


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 $Ltimes 26$
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.










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    I have a method for writing the names of numbers in alphabet: For $n$ just write a string of $n+1$ times the letter "a". So $0$ is "a", and "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa" is $345$. Now the order type is again $omega$!
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 13:35






  • 1




    $begingroup$
    This exact question was asked two weeks ago on r/math subreddit.
    $endgroup$
    – Wojowu
    Dec 20 '18 at 13:48






  • 1




    $begingroup$
    @Wojowu Great! I guess the puzzle is in the air.
    $endgroup$
    – JDH
    Dec 20 '18 at 13:53






  • 1




    $begingroup$
    @Holo: It will be, the string will just be indexed with a non-standard integer.
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 20:48






  • 3




    $begingroup$
    Actually, not even all native English speakers are "familiar with" the naming scheme of your first example: in British English, 15,249 is read out as "fifteen thousand two hundred and forty nine". Aside from that the form of your question amounts to: "please invent something nice and then tell me what its properties are". This is not a good MSE question.
    $endgroup$
    – Rob Arthan
    Dec 20 '18 at 22:54














12












12








12


4



$begingroup$


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 $Ltimes 26$
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.










share|cite|improve this question











$endgroup$




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 $Ltimes 26$
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.







elementary-number-theory elementary-set-theory logic order-theory puzzle






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 21 '18 at 15:45









Aloizio Macedo

23.6k23887




23.6k23887










asked Dec 20 '18 at 13:24









JDHJDH

32.6k680145




32.6k680145








  • 2




    $begingroup$
    I have a method for writing the names of numbers in alphabet: For $n$ just write a string of $n+1$ times the letter "a". So $0$ is "a", and "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa" is $345$. Now the order type is again $omega$!
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 13:35






  • 1




    $begingroup$
    This exact question was asked two weeks ago on r/math subreddit.
    $endgroup$
    – Wojowu
    Dec 20 '18 at 13:48






  • 1




    $begingroup$
    @Wojowu Great! I guess the puzzle is in the air.
    $endgroup$
    – JDH
    Dec 20 '18 at 13:53






  • 1




    $begingroup$
    @Holo: It will be, the string will just be indexed with a non-standard integer.
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 20:48






  • 3




    $begingroup$
    Actually, not even all native English speakers are "familiar with" the naming scheme of your first example: in British English, 15,249 is read out as "fifteen thousand two hundred and forty nine". Aside from that the form of your question amounts to: "please invent something nice and then tell me what its properties are". This is not a good MSE question.
    $endgroup$
    – Rob Arthan
    Dec 20 '18 at 22:54














  • 2




    $begingroup$
    I have a method for writing the names of numbers in alphabet: For $n$ just write a string of $n+1$ times the letter "a". So $0$ is "a", and "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa" is $345$. Now the order type is again $omega$!
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 13:35






  • 1




    $begingroup$
    This exact question was asked two weeks ago on r/math subreddit.
    $endgroup$
    – Wojowu
    Dec 20 '18 at 13:48






  • 1




    $begingroup$
    @Wojowu Great! I guess the puzzle is in the air.
    $endgroup$
    – JDH
    Dec 20 '18 at 13:53






  • 1




    $begingroup$
    @Holo: It will be, the string will just be indexed with a non-standard integer.
    $endgroup$
    – Asaf Karagila
    Dec 20 '18 at 20:48






  • 3




    $begingroup$
    Actually, not even all native English speakers are "familiar with" the naming scheme of your first example: in British English, 15,249 is read out as "fifteen thousand two hundred and forty nine". Aside from that the form of your question amounts to: "please invent something nice and then tell me what its properties are". This is not a good MSE question.
    $endgroup$
    – Rob Arthan
    Dec 20 '18 at 22:54








2




2




$begingroup$
I have a method for writing the names of numbers in alphabet: For $n$ just write a string of $n+1$ times the letter "a". So $0$ is "a", and "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa" is $345$. Now the order type is again $omega$!
$endgroup$
– Asaf Karagila
Dec 20 '18 at 13:35




$begingroup$
I have a method for writing the names of numbers in alphabet: For $n$ just write a string of $n+1$ times the letter "a". So $0$ is "a", and "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa" is $345$. Now the order type is again $omega$!
$endgroup$
– Asaf Karagila
Dec 20 '18 at 13:35




1




1




$begingroup$
This exact question was asked two weeks ago on r/math subreddit.
$endgroup$
– Wojowu
Dec 20 '18 at 13:48




$begingroup$
This exact question was asked two weeks ago on r/math subreddit.
$endgroup$
– Wojowu
Dec 20 '18 at 13:48




1




1




$begingroup$
@Wojowu Great! I guess the puzzle is in the air.
$endgroup$
– JDH
Dec 20 '18 at 13:53




$begingroup$
@Wojowu Great! I guess the puzzle is in the air.
$endgroup$
– JDH
Dec 20 '18 at 13:53




1




1




$begingroup$
@Holo: It will be, the string will just be indexed with a non-standard integer.
$endgroup$
– Asaf Karagila
Dec 20 '18 at 20:48




$begingroup$
@Holo: It will be, the string will just be indexed with a non-standard integer.
$endgroup$
– Asaf Karagila
Dec 20 '18 at 20:48




3




3




$begingroup$
Actually, not even all native English speakers are "familiar with" the naming scheme of your first example: in British English, 15,249 is read out as "fifteen thousand two hundred and forty nine". Aside from that the form of your question amounts to: "please invent something nice and then tell me what its properties are". This is not a good MSE question.
$endgroup$
– Rob Arthan
Dec 20 '18 at 22:54




$begingroup$
Actually, not even all native English speakers are "familiar with" the naming scheme of your first example: in British English, 15,249 is read out as "fifteen thousand two hundred and forty nine". Aside from that the form of your question amounts to: "please invent something nice and then tell me what its properties are". This is not a good MSE question.
$endgroup$
– Rob Arthan
Dec 20 '18 at 22:54










1 Answer
1






active

oldest

votes


















9












$begingroup$

Let us consider the digit-pronunciation naming system, by which
one simply pronounces the digits of a number in order, so that
$7216$ is pronounced "seven two one six" and so on for any number.
Thus, we obtain a naming system of the numbers, and while it does
not extend the standard nomenclature, nevertheless I find it to be
perfectly sensible, providing a definite unique name for every
natural number. This naming system is sometimes actually used for
very large numbers, such as reading off the number on a credit
card, and it is also commonly used to help disambiguate small
numbers, such as $50$ and $15$. So I find it to be a reasonable
naming system.



Let us place the natural numbers in alphabetical order with respect
to this naming system. Thus, $882746$ appears alphabetically before
$87$, which appears before $8734$. Note that any prefix of a word
appears earlier in the alphabetical order.



Theorem. The order type of the natural numbers, in alphabetical
order with respect to the digit-pronunciation naming system, is
exactly $$omegacdot(1+mathbb{Q})+1.$$



Proof. That is, we have $1+mathbb{Q}$ many copies of $omega$, with a
final point on top.



I will analyze the naming system with respect to base ten, but a
similar analysis works regardless of the base.



Consider first the alphabetical order of the ten digits themselves:




eight, five, four, nine, one, seven, six, three, two, zero




Notice that these digit names are prefix-free — none of them
is an initial segment of another. Thus, when comparing the names of
two numbers, we will never be in a situation where part of one
digit is combined with part of another in order to make the
alphabetical comparison. Rather, the alphabetical order is the same
as the lexical order on the strings of digits themselves,
considered in the alphabetical digit order above.



The largest number of all, in the alphabetical order, is zero,
since no other number starts with the letter "z", and so this
number will appear as the very last entry alphabetically. This
explains the final $+1$ in the theorem claim.



The smallest number in alphabetical order, in contrast, is $8$,
since it begins with "e", and the only other numbers beginning with
"e" also begin with $8$, followed possibly by additional digits,
and thus will appear after the single-digit $8$.



The next number after $8$, alphabetically, is $88$ and then $888$
and $8888$ and so on. I claim that every number (except $0$) has an
alphabetical successor, which is simply to add a digit $8$ at the
end of the decimal representation of the number. For example, the
next number after $532876$ is $5328768$, because any other digit
sequence above the first number must either extend it or deviate
from one of those digits. But $5328768$ will be below any other
higher deviation or extension, and so it is a successor. Similarly,
$53287688$ and $532876888$ are the next few numbers, simply adding
more $8$'s at the end.



Thus, every number except $0$ in the alphabetical order is followed
by a sequence of order type $omega$, which is obtained by simply
tacking on additional $8$s. And so the order will be a number of
copies of $omega$, plus one more point $0$ at the top.



Let me argue that those copies of $omega$ are themselves densely
ordered. If one number $m$ precedes another $n$ alphabetically, but
$n$ is not just adding $8$'s to the end of the decimal
representation of $m$, then either there is some alphabetically
upward deviation in the digits of $m$ to form $n$, or else $n$
extends the digits of $m$, but eventually using some digits other
than $8$. It is easy to see that we can find another number in
between, which also won't be just adding $8$s.



Perhaps it is easiest to see this by example. The number $7536$ is
alphabetically prior to $752$, since "three" is alphabetically
earlier than "two". In between these numbers, we can find $75366$,
which has it own copy of $omega$ arising from $753668$, $7536688$,
$75366888$ and so on.



Thus, the blocks of $omega$ obtained by appending $8$'s are
themselves densely ordered: between any two of them we can find
another.



Notice that there is a very first such block of $omega$ in the
alphabetical order the numbers, namely, the block consisting of
$8$, $88$, $888$ and so on, which appears at the very beginning of
the numbers in alphabetical order.



There is in contrast no largest block, before the final $0$,
because if we are given any number $n$, we can append some other
digits other than $8$ to the end of the decimal representation, and
thereby find another copy of $omega$ above $n$ in the alphabetical
order.



Thus, the $omega$ blocks arising from appending $8$'s are
themselves densely ordered, with a first such block and no last
such block. Since there are only countably many numbers, we must
have exactly $1+mathbb{Q}$ many such blocks of size $omega$. And
with the final point $0$ at the very top, it follows that the order
type of the natural numbers in the digit-pronunciation naming
system is precisely $$omegacdot(1+mathbb{Q})+1,$$ as claimed.
$Box$.



Several of us had discussed this problem over beers last night in
Münster, including Stefan Hoffelner and Stefan Mesken, following my talk at the Münster Logic
Oberseminar. Stefan Hoffelner had suggested that we consider the digit-pronunciation naming system.



Let me say finally that it seems to me that the features of the
digit-pronunciation naming system will appear essentially in all
the naming systems, and so I expect this kind of analysis to be
able to extend to the other nomenclatures, with perhaps slightly
different endpoint effects.






share|cite|improve this answer











$endgroup$













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    $begingroup$

    Let us consider the digit-pronunciation naming system, by which
    one simply pronounces the digits of a number in order, so that
    $7216$ is pronounced "seven two one six" and so on for any number.
    Thus, we obtain a naming system of the numbers, and while it does
    not extend the standard nomenclature, nevertheless I find it to be
    perfectly sensible, providing a definite unique name for every
    natural number. This naming system is sometimes actually used for
    very large numbers, such as reading off the number on a credit
    card, and it is also commonly used to help disambiguate small
    numbers, such as $50$ and $15$. So I find it to be a reasonable
    naming system.



    Let us place the natural numbers in alphabetical order with respect
    to this naming system. Thus, $882746$ appears alphabetically before
    $87$, which appears before $8734$. Note that any prefix of a word
    appears earlier in the alphabetical order.



    Theorem. The order type of the natural numbers, in alphabetical
    order with respect to the digit-pronunciation naming system, is
    exactly $$omegacdot(1+mathbb{Q})+1.$$



    Proof. That is, we have $1+mathbb{Q}$ many copies of $omega$, with a
    final point on top.



    I will analyze the naming system with respect to base ten, but a
    similar analysis works regardless of the base.



    Consider first the alphabetical order of the ten digits themselves:




    eight, five, four, nine, one, seven, six, three, two, zero




    Notice that these digit names are prefix-free — none of them
    is an initial segment of another. Thus, when comparing the names of
    two numbers, we will never be in a situation where part of one
    digit is combined with part of another in order to make the
    alphabetical comparison. Rather, the alphabetical order is the same
    as the lexical order on the strings of digits themselves,
    considered in the alphabetical digit order above.



    The largest number of all, in the alphabetical order, is zero,
    since no other number starts with the letter "z", and so this
    number will appear as the very last entry alphabetically. This
    explains the final $+1$ in the theorem claim.



    The smallest number in alphabetical order, in contrast, is $8$,
    since it begins with "e", and the only other numbers beginning with
    "e" also begin with $8$, followed possibly by additional digits,
    and thus will appear after the single-digit $8$.



    The next number after $8$, alphabetically, is $88$ and then $888$
    and $8888$ and so on. I claim that every number (except $0$) has an
    alphabetical successor, which is simply to add a digit $8$ at the
    end of the decimal representation of the number. For example, the
    next number after $532876$ is $5328768$, because any other digit
    sequence above the first number must either extend it or deviate
    from one of those digits. But $5328768$ will be below any other
    higher deviation or extension, and so it is a successor. Similarly,
    $53287688$ and $532876888$ are the next few numbers, simply adding
    more $8$'s at the end.



    Thus, every number except $0$ in the alphabetical order is followed
    by a sequence of order type $omega$, which is obtained by simply
    tacking on additional $8$s. And so the order will be a number of
    copies of $omega$, plus one more point $0$ at the top.



    Let me argue that those copies of $omega$ are themselves densely
    ordered. If one number $m$ precedes another $n$ alphabetically, but
    $n$ is not just adding $8$'s to the end of the decimal
    representation of $m$, then either there is some alphabetically
    upward deviation in the digits of $m$ to form $n$, or else $n$
    extends the digits of $m$, but eventually using some digits other
    than $8$. It is easy to see that we can find another number in
    between, which also won't be just adding $8$s.



    Perhaps it is easiest to see this by example. The number $7536$ is
    alphabetically prior to $752$, since "three" is alphabetically
    earlier than "two". In between these numbers, we can find $75366$,
    which has it own copy of $omega$ arising from $753668$, $7536688$,
    $75366888$ and so on.



    Thus, the blocks of $omega$ obtained by appending $8$'s are
    themselves densely ordered: between any two of them we can find
    another.



    Notice that there is a very first such block of $omega$ in the
    alphabetical order the numbers, namely, the block consisting of
    $8$, $88$, $888$ and so on, which appears at the very beginning of
    the numbers in alphabetical order.



    There is in contrast no largest block, before the final $0$,
    because if we are given any number $n$, we can append some other
    digits other than $8$ to the end of the decimal representation, and
    thereby find another copy of $omega$ above $n$ in the alphabetical
    order.



    Thus, the $omega$ blocks arising from appending $8$'s are
    themselves densely ordered, with a first such block and no last
    such block. Since there are only countably many numbers, we must
    have exactly $1+mathbb{Q}$ many such blocks of size $omega$. And
    with the final point $0$ at the very top, it follows that the order
    type of the natural numbers in the digit-pronunciation naming
    system is precisely $$omegacdot(1+mathbb{Q})+1,$$ as claimed.
    $Box$.



    Several of us had discussed this problem over beers last night in
    Münster, including Stefan Hoffelner and Stefan Mesken, following my talk at the Münster Logic
    Oberseminar. Stefan Hoffelner had suggested that we consider the digit-pronunciation naming system.



    Let me say finally that it seems to me that the features of the
    digit-pronunciation naming system will appear essentially in all
    the naming systems, and so I expect this kind of analysis to be
    able to extend to the other nomenclatures, with perhaps slightly
    different endpoint effects.






    share|cite|improve this answer











    $endgroup$


















      9












      $begingroup$

      Let us consider the digit-pronunciation naming system, by which
      one simply pronounces the digits of a number in order, so that
      $7216$ is pronounced "seven two one six" and so on for any number.
      Thus, we obtain a naming system of the numbers, and while it does
      not extend the standard nomenclature, nevertheless I find it to be
      perfectly sensible, providing a definite unique name for every
      natural number. This naming system is sometimes actually used for
      very large numbers, such as reading off the number on a credit
      card, and it is also commonly used to help disambiguate small
      numbers, such as $50$ and $15$. So I find it to be a reasonable
      naming system.



      Let us place the natural numbers in alphabetical order with respect
      to this naming system. Thus, $882746$ appears alphabetically before
      $87$, which appears before $8734$. Note that any prefix of a word
      appears earlier in the alphabetical order.



      Theorem. The order type of the natural numbers, in alphabetical
      order with respect to the digit-pronunciation naming system, is
      exactly $$omegacdot(1+mathbb{Q})+1.$$



      Proof. That is, we have $1+mathbb{Q}$ many copies of $omega$, with a
      final point on top.



      I will analyze the naming system with respect to base ten, but a
      similar analysis works regardless of the base.



      Consider first the alphabetical order of the ten digits themselves:




      eight, five, four, nine, one, seven, six, three, two, zero




      Notice that these digit names are prefix-free — none of them
      is an initial segment of another. Thus, when comparing the names of
      two numbers, we will never be in a situation where part of one
      digit is combined with part of another in order to make the
      alphabetical comparison. Rather, the alphabetical order is the same
      as the lexical order on the strings of digits themselves,
      considered in the alphabetical digit order above.



      The largest number of all, in the alphabetical order, is zero,
      since no other number starts with the letter "z", and so this
      number will appear as the very last entry alphabetically. This
      explains the final $+1$ in the theorem claim.



      The smallest number in alphabetical order, in contrast, is $8$,
      since it begins with "e", and the only other numbers beginning with
      "e" also begin with $8$, followed possibly by additional digits,
      and thus will appear after the single-digit $8$.



      The next number after $8$, alphabetically, is $88$ and then $888$
      and $8888$ and so on. I claim that every number (except $0$) has an
      alphabetical successor, which is simply to add a digit $8$ at the
      end of the decimal representation of the number. For example, the
      next number after $532876$ is $5328768$, because any other digit
      sequence above the first number must either extend it or deviate
      from one of those digits. But $5328768$ will be below any other
      higher deviation or extension, and so it is a successor. Similarly,
      $53287688$ and $532876888$ are the next few numbers, simply adding
      more $8$'s at the end.



      Thus, every number except $0$ in the alphabetical order is followed
      by a sequence of order type $omega$, which is obtained by simply
      tacking on additional $8$s. And so the order will be a number of
      copies of $omega$, plus one more point $0$ at the top.



      Let me argue that those copies of $omega$ are themselves densely
      ordered. If one number $m$ precedes another $n$ alphabetically, but
      $n$ is not just adding $8$'s to the end of the decimal
      representation of $m$, then either there is some alphabetically
      upward deviation in the digits of $m$ to form $n$, or else $n$
      extends the digits of $m$, but eventually using some digits other
      than $8$. It is easy to see that we can find another number in
      between, which also won't be just adding $8$s.



      Perhaps it is easiest to see this by example. The number $7536$ is
      alphabetically prior to $752$, since "three" is alphabetically
      earlier than "two". In between these numbers, we can find $75366$,
      which has it own copy of $omega$ arising from $753668$, $7536688$,
      $75366888$ and so on.



      Thus, the blocks of $omega$ obtained by appending $8$'s are
      themselves densely ordered: between any two of them we can find
      another.



      Notice that there is a very first such block of $omega$ in the
      alphabetical order the numbers, namely, the block consisting of
      $8$, $88$, $888$ and so on, which appears at the very beginning of
      the numbers in alphabetical order.



      There is in contrast no largest block, before the final $0$,
      because if we are given any number $n$, we can append some other
      digits other than $8$ to the end of the decimal representation, and
      thereby find another copy of $omega$ above $n$ in the alphabetical
      order.



      Thus, the $omega$ blocks arising from appending $8$'s are
      themselves densely ordered, with a first such block and no last
      such block. Since there are only countably many numbers, we must
      have exactly $1+mathbb{Q}$ many such blocks of size $omega$. And
      with the final point $0$ at the very top, it follows that the order
      type of the natural numbers in the digit-pronunciation naming
      system is precisely $$omegacdot(1+mathbb{Q})+1,$$ as claimed.
      $Box$.



      Several of us had discussed this problem over beers last night in
      Münster, including Stefan Hoffelner and Stefan Mesken, following my talk at the Münster Logic
      Oberseminar. Stefan Hoffelner had suggested that we consider the digit-pronunciation naming system.



      Let me say finally that it seems to me that the features of the
      digit-pronunciation naming system will appear essentially in all
      the naming systems, and so I expect this kind of analysis to be
      able to extend to the other nomenclatures, with perhaps slightly
      different endpoint effects.






      share|cite|improve this answer











      $endgroup$
















        9












        9








        9





        $begingroup$

        Let us consider the digit-pronunciation naming system, by which
        one simply pronounces the digits of a number in order, so that
        $7216$ is pronounced "seven two one six" and so on for any number.
        Thus, we obtain a naming system of the numbers, and while it does
        not extend the standard nomenclature, nevertheless I find it to be
        perfectly sensible, providing a definite unique name for every
        natural number. This naming system is sometimes actually used for
        very large numbers, such as reading off the number on a credit
        card, and it is also commonly used to help disambiguate small
        numbers, such as $50$ and $15$. So I find it to be a reasonable
        naming system.



        Let us place the natural numbers in alphabetical order with respect
        to this naming system. Thus, $882746$ appears alphabetically before
        $87$, which appears before $8734$. Note that any prefix of a word
        appears earlier in the alphabetical order.



        Theorem. The order type of the natural numbers, in alphabetical
        order with respect to the digit-pronunciation naming system, is
        exactly $$omegacdot(1+mathbb{Q})+1.$$



        Proof. That is, we have $1+mathbb{Q}$ many copies of $omega$, with a
        final point on top.



        I will analyze the naming system with respect to base ten, but a
        similar analysis works regardless of the base.



        Consider first the alphabetical order of the ten digits themselves:




        eight, five, four, nine, one, seven, six, three, two, zero




        Notice that these digit names are prefix-free — none of them
        is an initial segment of another. Thus, when comparing the names of
        two numbers, we will never be in a situation where part of one
        digit is combined with part of another in order to make the
        alphabetical comparison. Rather, the alphabetical order is the same
        as the lexical order on the strings of digits themselves,
        considered in the alphabetical digit order above.



        The largest number of all, in the alphabetical order, is zero,
        since no other number starts with the letter "z", and so this
        number will appear as the very last entry alphabetically. This
        explains the final $+1$ in the theorem claim.



        The smallest number in alphabetical order, in contrast, is $8$,
        since it begins with "e", and the only other numbers beginning with
        "e" also begin with $8$, followed possibly by additional digits,
        and thus will appear after the single-digit $8$.



        The next number after $8$, alphabetically, is $88$ and then $888$
        and $8888$ and so on. I claim that every number (except $0$) has an
        alphabetical successor, which is simply to add a digit $8$ at the
        end of the decimal representation of the number. For example, the
        next number after $532876$ is $5328768$, because any other digit
        sequence above the first number must either extend it or deviate
        from one of those digits. But $5328768$ will be below any other
        higher deviation or extension, and so it is a successor. Similarly,
        $53287688$ and $532876888$ are the next few numbers, simply adding
        more $8$'s at the end.



        Thus, every number except $0$ in the alphabetical order is followed
        by a sequence of order type $omega$, which is obtained by simply
        tacking on additional $8$s. And so the order will be a number of
        copies of $omega$, plus one more point $0$ at the top.



        Let me argue that those copies of $omega$ are themselves densely
        ordered. If one number $m$ precedes another $n$ alphabetically, but
        $n$ is not just adding $8$'s to the end of the decimal
        representation of $m$, then either there is some alphabetically
        upward deviation in the digits of $m$ to form $n$, or else $n$
        extends the digits of $m$, but eventually using some digits other
        than $8$. It is easy to see that we can find another number in
        between, which also won't be just adding $8$s.



        Perhaps it is easiest to see this by example. The number $7536$ is
        alphabetically prior to $752$, since "three" is alphabetically
        earlier than "two". In between these numbers, we can find $75366$,
        which has it own copy of $omega$ arising from $753668$, $7536688$,
        $75366888$ and so on.



        Thus, the blocks of $omega$ obtained by appending $8$'s are
        themselves densely ordered: between any two of them we can find
        another.



        Notice that there is a very first such block of $omega$ in the
        alphabetical order the numbers, namely, the block consisting of
        $8$, $88$, $888$ and so on, which appears at the very beginning of
        the numbers in alphabetical order.



        There is in contrast no largest block, before the final $0$,
        because if we are given any number $n$, we can append some other
        digits other than $8$ to the end of the decimal representation, and
        thereby find another copy of $omega$ above $n$ in the alphabetical
        order.



        Thus, the $omega$ blocks arising from appending $8$'s are
        themselves densely ordered, with a first such block and no last
        such block. Since there are only countably many numbers, we must
        have exactly $1+mathbb{Q}$ many such blocks of size $omega$. And
        with the final point $0$ at the very top, it follows that the order
        type of the natural numbers in the digit-pronunciation naming
        system is precisely $$omegacdot(1+mathbb{Q})+1,$$ as claimed.
        $Box$.



        Several of us had discussed this problem over beers last night in
        Münster, including Stefan Hoffelner and Stefan Mesken, following my talk at the Münster Logic
        Oberseminar. Stefan Hoffelner had suggested that we consider the digit-pronunciation naming system.



        Let me say finally that it seems to me that the features of the
        digit-pronunciation naming system will appear essentially in all
        the naming systems, and so I expect this kind of analysis to be
        able to extend to the other nomenclatures, with perhaps slightly
        different endpoint effects.






        share|cite|improve this answer











        $endgroup$



        Let us consider the digit-pronunciation naming system, by which
        one simply pronounces the digits of a number in order, so that
        $7216$ is pronounced "seven two one six" and so on for any number.
        Thus, we obtain a naming system of the numbers, and while it does
        not extend the standard nomenclature, nevertheless I find it to be
        perfectly sensible, providing a definite unique name for every
        natural number. This naming system is sometimes actually used for
        very large numbers, such as reading off the number on a credit
        card, and it is also commonly used to help disambiguate small
        numbers, such as $50$ and $15$. So I find it to be a reasonable
        naming system.



        Let us place the natural numbers in alphabetical order with respect
        to this naming system. Thus, $882746$ appears alphabetically before
        $87$, which appears before $8734$. Note that any prefix of a word
        appears earlier in the alphabetical order.



        Theorem. The order type of the natural numbers, in alphabetical
        order with respect to the digit-pronunciation naming system, is
        exactly $$omegacdot(1+mathbb{Q})+1.$$



        Proof. That is, we have $1+mathbb{Q}$ many copies of $omega$, with a
        final point on top.



        I will analyze the naming system with respect to base ten, but a
        similar analysis works regardless of the base.



        Consider first the alphabetical order of the ten digits themselves:




        eight, five, four, nine, one, seven, six, three, two, zero




        Notice that these digit names are prefix-free — none of them
        is an initial segment of another. Thus, when comparing the names of
        two numbers, we will never be in a situation where part of one
        digit is combined with part of another in order to make the
        alphabetical comparison. Rather, the alphabetical order is the same
        as the lexical order on the strings of digits themselves,
        considered in the alphabetical digit order above.



        The largest number of all, in the alphabetical order, is zero,
        since no other number starts with the letter "z", and so this
        number will appear as the very last entry alphabetically. This
        explains the final $+1$ in the theorem claim.



        The smallest number in alphabetical order, in contrast, is $8$,
        since it begins with "e", and the only other numbers beginning with
        "e" also begin with $8$, followed possibly by additional digits,
        and thus will appear after the single-digit $8$.



        The next number after $8$, alphabetically, is $88$ and then $888$
        and $8888$ and so on. I claim that every number (except $0$) has an
        alphabetical successor, which is simply to add a digit $8$ at the
        end of the decimal representation of the number. For example, the
        next number after $532876$ is $5328768$, because any other digit
        sequence above the first number must either extend it or deviate
        from one of those digits. But $5328768$ will be below any other
        higher deviation or extension, and so it is a successor. Similarly,
        $53287688$ and $532876888$ are the next few numbers, simply adding
        more $8$'s at the end.



        Thus, every number except $0$ in the alphabetical order is followed
        by a sequence of order type $omega$, which is obtained by simply
        tacking on additional $8$s. And so the order will be a number of
        copies of $omega$, plus one more point $0$ at the top.



        Let me argue that those copies of $omega$ are themselves densely
        ordered. If one number $m$ precedes another $n$ alphabetically, but
        $n$ is not just adding $8$'s to the end of the decimal
        representation of $m$, then either there is some alphabetically
        upward deviation in the digits of $m$ to form $n$, or else $n$
        extends the digits of $m$, but eventually using some digits other
        than $8$. It is easy to see that we can find another number in
        between, which also won't be just adding $8$s.



        Perhaps it is easiest to see this by example. The number $7536$ is
        alphabetically prior to $752$, since "three" is alphabetically
        earlier than "two". In between these numbers, we can find $75366$,
        which has it own copy of $omega$ arising from $753668$, $7536688$,
        $75366888$ and so on.



        Thus, the blocks of $omega$ obtained by appending $8$'s are
        themselves densely ordered: between any two of them we can find
        another.



        Notice that there is a very first such block of $omega$ in the
        alphabetical order the numbers, namely, the block consisting of
        $8$, $88$, $888$ and so on, which appears at the very beginning of
        the numbers in alphabetical order.



        There is in contrast no largest block, before the final $0$,
        because if we are given any number $n$, we can append some other
        digits other than $8$ to the end of the decimal representation, and
        thereby find another copy of $omega$ above $n$ in the alphabetical
        order.



        Thus, the $omega$ blocks arising from appending $8$'s are
        themselves densely ordered, with a first such block and no last
        such block. Since there are only countably many numbers, we must
        have exactly $1+mathbb{Q}$ many such blocks of size $omega$. And
        with the final point $0$ at the very top, it follows that the order
        type of the natural numbers in the digit-pronunciation naming
        system is precisely $$omegacdot(1+mathbb{Q})+1,$$ as claimed.
        $Box$.



        Several of us had discussed this problem over beers last night in
        Münster, including Stefan Hoffelner and Stefan Mesken, following my talk at the Münster Logic
        Oberseminar. Stefan Hoffelner had suggested that we consider the digit-pronunciation naming system.



        Let me say finally that it seems to me that the features of the
        digit-pronunciation naming system will appear essentially in all
        the naming systems, and so I expect this kind of analysis to be
        able to extend to the other nomenclatures, with perhaps slightly
        different endpoint effects.







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        edited Jan 12 at 16:15

























        answered Jan 12 at 8:49









        JDHJDH

        32.6k680145




        32.6k680145






























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