Standard numbering

The standard numbering of the countable-infinite set of character strings is that, under the assumption of any alphabet with finite power and unambiguous character numbering (where the numbers produce the total stock of all characters ), the way of enumeration (where each number produces exactly one word ), which is exactly the bijective enumerability (where every possible string exactly one number produced) reverses that for all the words of any length obeyed the optimum convention that ${\displaystyle \Sigma ^{*}}$ ${\displaystyle \Sigma :=\{\sigma _{1}\ldots \sigma _{m}\}}$ ${\displaystyle m:=|\Sigma |\in \mathbb {N} }$ ${\displaystyle \sigma \in \Sigma ^{\{1\ldots m\}}}$ ${\displaystyle 1\leq i\leq m}$ ${\displaystyle \sigma _{i}\in \Sigma }$ ${\displaystyle \nu _{\sigma }\in (\Sigma ^{*})^{\mathbb {N} }}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \nu _{\sigma }^{n}\in \Sigma ^{*}}$ ${\displaystyle \alpha _{\sigma }\in \mathbb {N} ^{\Sigma ^{*}}}$ ${\displaystyle w\in \Sigma ^{*}}$ ${\displaystyle \alpha _{\sigma }^{w}\in \mathbb {N} }$ ${\displaystyle \sigma _{i_{1}}\ldots \sigma _{i_{L}}\in \Sigma ^{2+}}$ ${\displaystyle L\in \mathbb {N} }$

${\displaystyle \alpha _{\sigma }^{\sigma _{i_{1}}\ldots \sigma _{i_{L}}}=\sum _{p=1}^{L}i_{p}m^{L-p}}$

example

Be with . ${\displaystyle \Sigma =\{1,2\}}$ ${\displaystyle (1,2)=(\sigma _{1},\sigma _{2})}$

The elements of the set can be listed systematically: ${\displaystyle \Sigma ^{*}}$

${\displaystyle \Sigma ^{*}=\{\epsilon ,1,2,11,12,21,22,111,112,121,122,211,212,221,222,\ldots \}}$

The i-th word in the list always appears . ${\displaystyle \nu _{\sigma }^{i}}$

${\displaystyle \nu _{\Sigma }^{8}=112}$corresponds . ${\displaystyle \alpha _{\sigma }^{112}=\alpha _{\sigma }^{\sigma _{1}\sigma _{1}\sigma _{2}}=1\cdot 2^{2}+1\cdot 2^{1}+2\cdot 2^{0}=8}$

The latter can be checked quickly with the help of a Haskell line interpreter:

strings chars = [] : [ string ++ [char] | string <- strings chars, char <- chars ]

zip [0..16] (strings "12")

[(0, ""), (1, "1"), (2, "2"), (3, "11"), (4, "12"), (5, "21"), (6 , "22"), (7, "111"), (8, "112"), (9, "121"), (10, "122"), (11, "211"), (12, " 212 "), (13," 221 "), (14," 222 "), (15," 1111 "), (16," 1112 ")]

It becomes clear that our prevailing place value system does not result in a standard numbering in the sense of the above definition in view of the leading zeros to be skipped :

zip [0..12] (strings "0123456789")

[(0, ""), (1, "0"), (2, "1"), (3, "2"), (4, "3"), (5, "4"), (6 , "5"), (7, "6"), (8, "7"), (9, "8"), (10, "9"), (11, "00"), (12, " 01 ")]

zip [0..12] (strings "1234567890")

[(0, ""), (1, "1"), (2, "2"), (3, "3"), (4, "4"), (5, "5"), (6 , "6"), (7, "7"), (8, "8"), (9, "9"), (10, "0"), (11, "11"), (12, " 12 ")]

drop 99 $ zip [0..121] (strings "123456789X")

[(99, "99"), (100, "9X"), (101, "X1"), (102, "X2"), (103, "X3"), (104, "X4"), ( 105, "X5"), (106, "X6"), (107, "X7"), (108, "X8"), (109, "X9"), (110, "XX"), (111, "111"), (112, "112"), (113, "113"), (114, "114"), (115, "115"), (116, "116"), (117, "117 "), (118," 118 "), (119," 119 "), (120," 11X "), (121," 121 ")]

drop 90 $ zip [0..100] (strings "123456789")

[(90, "99"), (91, "111"), (92, "112"), (93, "113"), (94, "114"), (95, "115"), ( 96, "116"), (97, "117"), (98, "118"), (99, "119"), (100, "121")]