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Peegee
06-06-2008, 10:05 PM
well he didn't say it verbatim but he'll concede that he invented the notion and agrees with my claim.

Here goes.

A number is loosely defined as a permutation of any amount of digits from the set [0-9]. It can also be associated with an english word, so that when you see the english word 'Three' you know it means '3'.

An english word for any number is defined as a permutation of any amount of characters from the set [a-z]. We could specify something like how a vowel must exist in the word, and since english words technically don't have a maximum length, it would just be another infinite set.

Now, we have two infinite sets. Like two lines with infinite dots, one can draw a 1:1 association between each number and each word. You'd run out of 'numbery' sounding words eventually, and will have to start using other permutations of the english alphabet. Eventually you'll run out of unique words, and have to 're-use' words, like the word 'word', or the word 'peegee', or 'seazy', etc.

Ergo, Rye, ryechu, pikachu, peegee, microphone, and camera are all numbers.

Thanks to Cz for hurting my head. >: (

rubah
06-06-2008, 10:08 PM
are you drunk peegies

or high?

Maybe in France they allow that sissy double posting stuff, but not 'round these parts! :p

Peegee
06-06-2008, 10:12 PM
No double poasting.

I'm neither.

Lawr
06-06-2008, 10:14 PM
I like cake.

Peegee
06-06-2008, 10:16 PM
I like cake.

cake and sagensyg would be numbers in my argument.

Denmark
06-06-2008, 10:34 PM
Why have I always perfectly understood you, PG?

Bonus: TRANSLATE THAT INTO NUMBERS!

Lawr
06-06-2008, 10:47 PM
I think I get it, like the number '17039' has no repeated numbers. And 'Person' has no repeated letters. Is that right? Or are you on some mushrooms or something?

rubah
06-06-2008, 10:48 PM
mullet, if you can't recognize an excellent example of double posting, i don't know if I want to be your friend anymore.

denffe is the number 491508 or 049158 or something.

holy crap I can't believe I just did that.

Cz
06-06-2008, 10:52 PM
See, I told you this was too deep for GC.

Peegee
06-06-2008, 11:05 PM
No I'm just poor at expressing myself properly.

Every number as an associated english word . one, two, three, four, seventy, sixty seven million, etc.

I'm arguing that since numbers are infinite, there must be an infinite amount of words to describe them. Since an infinite amount of things, where the thing is just any permutation of a finite set will just exhaust new/made up items in that set (like made up words for numbers) we'll have to use items in that set that we understand as other words, such as peegee and apple.

Lawr
06-06-2008, 11:09 PM
Ok I get you. So you're taking the stand point from the term "Googol" meaning 100 zeros. Sagensyg will be a number and 450 zeros after it.

Peegee
06-06-2008, 11:11 PM
Ok I get you. So you're taking the stand point from the term "Googol" meaning 100 zeros. Sagensyg will be a number and 450 zeros after it.

Right, except you aren't defined yet, but you and every other possible english word theoretically has a number attached to it.

rubah
06-06-2008, 11:25 PM
just say 'every number has a name'

sag just defined himself, i think.

Peegee
06-07-2008, 06:53 AM
just say 'every number has a name'

sag just defined himself, i think.

It's not official yet.

o_O
06-07-2008, 07:38 AM
well he didn't say it verbatim but he'll concede that he invented the notion and agrees with my claim.

Here goes.

A number is loosely defined as a permutation of any amount of digits from the set [0-9]. It can also be associated with an english word, so that when you see the english word 'Three' you know it means '3'.

An english word for any number is defined as a permutation of any amount of characters from the set [a-z]. We could specify something like how a vowel must exist in the word, and since english words technically don't have a maximum length, it would just be another infinite set.

Now, we have two infinite sets. Like two lines with infinite dots, one can draw a 1:1 association between each number and each word. You'd run out of 'numbery' sounding words eventually, and will have to start using other permutations of the english alphabet. Eventually you'll run out of unique words, and have to 're-use' words, like the word 'word', or the word 'peegee', or 'seazy', etc.

Ergo, Rye, ryechu, pikachu, peegee, microphone, and camera are all numbers.

Thanks to Cz for hurting my head. >: (

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

Blue Harvest
06-07-2008, 10:32 AM
This thread makes my head hurt ;_;

Rantz
06-07-2008, 03:00 PM
"Peegee" is a firetruck. Weeooweeoo etc.

Caraliz
06-07-2008, 04:11 PM
well he didn't say it verbatim but he'll concede that he invented the notion and agrees with my claim.

Here goes.

A number is loosely defined as a permutation of any amount of digits from the set [0-9]. It can also be associated with an english word, so that when you see the english word 'Three' you know it means '3'.

An english word for any number is defined as a permutation of any amount of characters from the set [a-z]. We could specify something like how a vowel must exist in the word, and since english words technically don't have a maximum length, it would just be another infinite set.

Now, we have two infinite sets. Like two lines with infinite dots, one can draw a 1:1 association between each number and each word. You'd run out of 'numbery' sounding words eventually, and will have to start using other permutations of the english alphabet. Eventually you'll run out of unique words, and have to 're-use' words, like the word 'word', or the word 'peegee', or 'seazy', etc.

Ergo, Rye, ryechu, pikachu, peegee, microphone, and camera are all numbers.

Thanks to Cz for hurting my head. >: (

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 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. :p
I fucking knew I should have paid attention in pre-calc. I hated permutations and sequences.

Peegee
06-07-2008, 04:14 PM
well he didn't say it verbatim but he'll concede that he invented the notion and agrees with my claim.

Here goes.

A number is loosely defined as a permutation of any amount of digits from the set [0-9]. It can also be associated with an english word, so that when you see the english word 'Three' you know it means '3'.

An english word for any number is defined as a permutation of any amount of characters from the set [a-z]. We could specify something like how a vowel must exist in the word, and since english words technically don't have a maximum length, it would just be another infinite set.

Now, we have two infinite sets. Like two lines with infinite dots, one can draw a 1:1 association between each number and each word. You'd run out of 'numbery' sounding words eventually, and will have to start using other permutations of the english alphabet. Eventually you'll run out of unique words, and have to 're-use' words, like the word 'word', or the word 'peegee', or 'seazy', etc.

Ergo, Rye, ryechu, pikachu, peegee, microphone, and camera are all numbers.

Thanks to Cz for hurting my head. >: (

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

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.

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.

theundeadhero
06-07-2008, 05:44 PM
I'm going to call the number 9674658498683497438964328697564243297067 theundeadhero.

Denmark
06-07-2008, 06:31 PM
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.

But that would just be simple substitution. That's not a new idea! Bad PG!

Peegee
06-07-2008, 07:45 PM
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.

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.

I award myself 7 epeen points and 4 internets.

Lawr
06-07-2008, 10:05 PM
well he didn't say it verbatim but he'll concede that he invented the notion and agrees with my claim.

Here goes.

A number is loosely defined as a permutation of any amount of digits from the set [0-9]. It can also be associated with an english word, so that when you see the english word 'Three' you know it means '3'.

An english word for any number is defined as a permutation of any amount of characters from the set [a-z]. We could specify something like how a vowel must exist in the word, and since english words technically don't have a maximum length, it would just be another infinite set.

Now, we have two infinite sets. Like two lines with infinite dots, one can draw a 1:1 association between each number and each word. You'd run out of 'numbery' sounding words eventually, and will have to start using other permutations of the english alphabet. Eventually you'll run out of unique words, and have to 're-use' words, like the word 'word', or the word 'peegee', or 'seazy', etc.

Ergo, Rye, ryechu, pikachu, peegee, microphone, and camera are all numbers.

Thanks to Cz for hurting my head. >: (

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

I still like cake.

o_O
06-08-2008, 04:45 AM
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.

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.

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 :p), 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.


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.

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,...}). :p

Dolentrean
06-08-2008, 05:50 AM
I have made this argument many times to many people.

rubah
06-08-2008, 07:18 PM
3628800 (in 26-ecimal notation) is not every conceivable number, peegies! stop putting artificial limits on things.

o_O
06-09-2008, 01:05 AM
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. :p

Really, it's simply regular base-10 with a different set of characters substituted in, just like Rob said. :p

Miriel
06-09-2008, 02:08 AM
Dear PG,

95% of the time, I literally have no idea what you're talking about.

Sincerely,

Miriel

Peegee
06-09-2008, 02:10 AM
lol at your naming convention of just appending x letters of n to a string of x value.


Dear PG,

95% of the time, I literally have no idea what you're talking about.

Sincerely,

Miriel

We should totally get on each other's msn lists. You shall never be bored again.

Dr Unne
06-09-2008, 02:36 AM
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).

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.


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.

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.

Relapse
06-09-2008, 04:28 AM
?

i'll take the number 777544121123587.

The Unknown Guru
06-09-2008, 05:20 AM
In hexadecimal, beef is a valid number.

o_O
06-09-2008, 05:30 AM
So is feedfacebeef. :p

Blue Harvest
06-09-2008, 11:03 AM
We should totally get on each other's msn lists. You shall never be bored again.

I want Peegee's msn :(

Huckleberry Quin
06-09-2008, 12:54 PM
......................

Breine
06-09-2008, 05:19 PM
I think I understand this..?

The Unknown Guru
06-10-2008, 01:10 AM
So is feedfacebeef. :p

Epic win.