This thread makes my head hurt ;_;
This thread makes my head hurt ;_;
The messenger is standing at the gate
Ready to let go
Ready for the crush
Too late for whispers
Too late for the blush
The past is mercy
When the future is aglow
"Peegee" is a firetruck. Weeooweeoo etc.
I smurfing knew I should have paid attention in pre-calc. I hated permutations and sequences.Originally Posted by o_O
Whether or not you run out of words depends upon whether or not you apply a set constraints to the selection of a word, such that a situation arises whereby there cannot be a valid name derived from existing ones. For example, if you said that a name is only valid provided that it doesn't contain a repeated string of two or more characters, you can easily see that there is a limited number of names you may choose. You can visualise it like this:
Take some alphabet of arbitrary size (we'll use 2, for argument's sake) = { A, B }.
Any number name can be created from this alphabet, but you can't have a two-character string repeated.
Now you take the set of all possible names - to speed things up I'll give examples by word length:
1: A
2: AA
3: AAA
4: AAAB
5: AAABB
6: AAABBA
7: AAABBBA
8: I challenge you to come up with an 8-letter one.
Mathematically, if you visualise each character of the alphabet as a state, as you sequentially read each letter of some string, there can be at most n state changes for an alphabet of size n before a repeat of a two digit string is inevitable. Similarly, if you restrict yourself to say actual English words, you only have 30,000 or so different numbers you can name.
If there are no such restrictions then there is an unlimited number of names. The proof is trivial: For word length n append character at position n-1 to string. In other words, you simply append the same character over and over. For an alphabet of size m and some arbitrary word length n, the total number of unique names is m<sup>n</sup>; since there is no limit on the number of letters in a word, every word length is valid, and the number of words rapidly increases in magnitude with word length. Therefore, for our English alphabet there is 26<sup>n</sup> for all possible word lengths n > 0.
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.Whether or not you run out of words depends upon whether or not you apply a set constraints to the selection of a word, such that a situation arises whereby there cannot be a valid name derived from existing ones. For example, if you said that a name is only valid provided that it doesn't contain a repeated string of two or more characters, you can easily see that there is a limited number of names you may choose. You can visualise it like this:
Take some alphabet of arbitrary size (we'll use 2, for argument's sake) = { A, B }.
Any number name can be created from this alphabet, but you can't have a two-character string repeated.
Now you take the set of all possible names - to speed things up I'll give examples by word length:
1: A
2: AA
3: AAA
4: AAAB
5: AAABB
6: AAABBA
7: AAABBBA
8: I challenge you to come up with an 8-letter one. :p
Mathematically, if you visualise each character of the alphabet as a state, as you sequentially read each letter of some string, there can be at most <i>n</i> state changes for an alphabet of size <i>n</i> before a repeat of a two digit string is inevitable. Similarly, if you restrict yourself to say <i>actual</i> English words, you only have 30,000 or so different numbers you can name.
If there are no such restrictions then there is an unlimited number of names. The proof is trivial: For word length <i>n</i> append character at posi<b></b>tion <i>n-1</i> to string. In other words, you simply append the same character over and over. For an alphabet of size <i>m</i> and some arbitrary word length <i>n</i>, the total number of unique names is <i>m<sup>n</sup></i>; since there is no limit on the number of letters in a word, every word length is valid, and the number of words rapidly increases in magnitude with word length. Therefore, for our English alphabet there is <i>26<sup>n</sup></i> for all possible word lengths <i>n > 0</i>. :p
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.
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.
Last edited by Peegee; 06-07-2008 at 04:23 PM.
I'm going to call the number 9674658498683497438964328697564243297067 theundeadhero.
...
But that would just be simple substitution. That's not a new idea! Bad PG!
True. I've segwayed this conversation towards boring territory. I win yet again.But that would just be simple substitution. That's not a new idea! Bad PG!
I award myself 7 epeen points and 4 internets.
I still like cake.Whether or not you run out of words depends upon whether or not you apply a set constraints to the selection of a word, such that a situation arises whereby there cannot be a valid name derived from existing ones. For example, if you said that a name is only valid provided that it doesn't contain a repeated string of two or more characters, you can easily see that there is a limited number of names you may choose. You can visualise it like this:
Take some alphabet of arbitrary size (we'll use 2, for argument's sake) = { A, B }.
Any number name can be created from this alphabet, but you can't have a two-character string repeated.
Now you take the set of all possible names - to speed things up I'll give examples by word length:
1: A
2: AA
3: AAA
4: AAAB
5: AAABB
6: AAABBA
7: AAABBBA
8: I challenge you to come up with an 8-letter one.
Mathematically, if you visualise each character of the alphabet as a state, as you sequentially read each letter of some string, there can be at most <i>n</i> state changes for an alphabet of size <i>n</i> before a repeat of a two digit string is inevitable. Similarly, if you restrict yourself to say <i>actual</i> English words, you only have 30,000 or so different numbers you can name.
If there are no such restrictions then there is an unlimited number of names. The proof is trivial: For word length <i>n</i> append character at posi<b></b>tion <i>n-1</i> to string. In other words, you simply append the same character over and over. For an alphabet of size <i>m</i> and some arbitrary word length <i>n</i>, the total number of unique names is <i>m<sup>n</sup></i>; since there is no limit on the number of letters in a word, every word length is valid, and the number of words rapidly increases in magnitude with word length. Therefore, for our English alphabet there is <i>26<sup>n</sup></i> for all possible word lengths <i>n > 0</i>.
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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| = 10<sup>1</sup> = 10</i>, but <i>|W| = 26<sup>1</sup> = 26</i>.
Similarly, <i>|N| = 10<sup>2</sup> = 100</i>, but <i>|W| = 26<sup>2</sup> = 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| = 10<sup>n</sup></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| = 26<sup>n</sup></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>∑(10<sup>i</sup>)</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>∑(26<sup>i</sup>)</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.
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,...}).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.
I have made this argument many times to many people.
Oh, did I mention I'm awesome?
3628800 (in 26-ecimal notation) is not every conceivable number, peegies! stop putting artificial limits on things.
I think what peg was saying was that if each element in a size 10 subset of {a-z} is assigned a value corresponding to an element in {0-9}, then we can form a string representing any conceivable number, but still have a word, since they're letters. They wouldn't be pronounceable, obviously.
Really, it's simply regular base-10 with a different set of characters substituted in, just like Rob said.
Dear PG,
95% of the time, I literally have no idea what you're talking about.
Sincerely,
Miriel
lol at your naming convention of just appending x letters of n to a string of x value.
We should totally get on each other's msn lists. You shall never be bored again.Dear PG,
95% of the time, I literally have no idea what you're talking about.
Sincerely,
Miriel
The set {a..z} is bigger than the set {0..9}. But the set of all possible strings using arbitrary amounts of elements of {a..z} and the set of all possible strings using arbitrary amounts of elements of {0..9} are equally infinite. They have the same cardinality i.e. same size. Both are countable sets.
Consider this: which set is bigger, the set of all integers when you write them in binary, or the set of all integers when you write them in decimal? It's clear that they are the same set, written down using different representations.
The alphabet, a-z, could be considered a way of representing numbers in base 26. There's a 1:1 mapping from the set of all integers written in binary to the set of all integers written in decimal, base 26, or any other base. They're all equally infinite.
Both are infinite, but the set of real numbers is bigger than the set of integers, in the sense that there is no 1:1 mapping from real numbers to integers. In any mapping of integers to real numbers, there will always be at least one real number that has no corresponding integer. Per Cantor's diagonal argument.
The fact that integers are a subset of real numbers doesn't matter. Cardinality of infinite sets isn't a straightforward topic. Which set has more elements, the set of odd integers, or the set of all integers? Turns out they have the same cardinality, even though the set of odd numbers is a subset of the set of integers.
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