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Title: 111,777 Post by Johno-G on Jan 12th, 2003, 2:27pm "The least integer not nameable in fewer than nineteen syllables is 111,777" Given that 111,777 (one-hundred and eleven thousand, seven-hundred and seventy seven) contains 19 syllables, and no integer less than 111,777 can be expressed in nineteen syllables, what is wrong with the above statement? (assume we are using standard english, and that we are not counting negative numbers) |
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Title: Re: 111,777 Post by towr on Jan 12th, 2003, 2:34pm if the statement were true , you could use "The least integer not nameable in fewer than nineteen syllables" to name the number, which is 18 syllables.. I don't think this riddle should be in hard if can get it that fast.. It's a nice riddle though.. It remind me of "Why is every (natural) number special?" |
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Title: Re: 111,777 Post by Johno-G on Jan 13th, 2003, 12:38am I don't think I've heard the 'why is every natural number special' riddle... perhaps you could enlighten me? |
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Title: Re: 111,777 Post by towr on Jan 13th, 2003, 3:13am Well, suppose not every natural number is special.. One of those not-special numbers has to be the lowest one, which makes it special.. You can easily do the somethign similar for any whole number, and rational numbers (since there are 1-to-1 mappings from one to the other). |
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Title: Re: 111,777 Post by Icarus on Jan 13th, 2003, 4:37pm The english language has a finite (though large) number of syllables. Let N be this number. There are fewer than N19 sentences with less than 19 syllables in the English language (the vast majority of these syllable sequences are nonsense). Hence only finitely many natural numbers can be named in fewer than 19 syllables, and so there are infinitely many natural numbers that cannot be named in 19 syllables. The natural numbers are a well-ordered set. This means that any subset of the natural numbers (including the set of all natural numbers not nameable in fewer than 19 syllables) must have a least element. So, what is "the least integer not nameable in fewer than nineteen syllables"? |
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Title: Re: 111,777 Post by Johno-G on Jan 14th, 2003, 12:39am This is actually a paradox - there is no least integer not nameable in fewer than nineteen syllables. Think about it - let A represent the set of all integers that are not nameable in fewer than nineteen syllables. Take it as a given that 111,777 is the least integer with no fewer than nineteen syllables in it's name. (eg. it has nineteen syllables if you just say "one-hundred and eleven thousand, seven-hundred and seventy-seven") So, A would be {111777,b,c,...}, where b and c are integers greater than 111,777 that are not nameable in less than nineteen syllables, right? But, by saying "The least integer not nameable in fewer than nineteen syllables" you are naming the least integer in A, which means that 111,777 can be named in less than nineteen syllables, so it can't be in A. Now, b would be the least integer not nameable in fewer than nineteen syllables, but having this property makes it nameable in eighteen syllables, and so on... (also, since 111,777 is no longer in the set of integers not nameable in less than nineteen syllables, it does not adequateley fit the description, "the least integer not nameable in fewer than nineteen syllables", and so is not nameable in eighteen syllables, and hence returns to set A as the least member. Then, of course, it is, once again, not nameable in fewer than nineteen syllables...) |
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Title: Re: 111,777 Post by Icarus on Jan 14th, 2003, 6:11pm Ah - but then what is wrong with my proof that there is a least integer not nameable in 19 characters? Or are you claiming that this is a true, unsolvable paradox? |
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Title: Re: 111,777 Post by towr on Jan 14th, 2003, 11:52pm Actually, there is nothing wrong with naming for instance the number one with the sentence "The least integer not nameable in fewer than nineteen syllables".. It doesn't destroy the universe or anything.. The sentence needn't be true on a meta-level to be a name.. Since the naming is rather arbitrary any number could be the "The least integer not named in fewer than nineteen syllables", but none is "The least integer not nameable in fewer than nineteen syllables". You could not name 0, and only start naming everything after. You could skip the first gazillion numbers, and only start naming after that. You can name _any_ number in less than nineteen syllables, just not all at the same time/namespace.. Actually, by shifting the quantifier scope, 0 would be the right answer for natural numbers (since you could not-name it in 19 syllables).. |
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Title: Re: 111,777 Post by Johno-G on Jan 15th, 2003, 2:36am Hmmm... I think I understand what you're saying, Towr, but tell me if I've obviously got it all wrong; The sentence asks for the least integer not nameable in fewer than nineteen syllables. Zero, however, is nameable in fewer than nineteen syllables. The sentence requires that you are not able to use less than nineteen syllables to name a number. Zero, however, can be named in less than nineteen. Have I understood what you were saying? |
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Title: Re: 111,777 Post by towr on Jan 15th, 2003, 8:51am That is one way to read the sentence.. And clairly you could give any number the name 'butterscotch-pudding', it may not make sense, but it's less than nineteen syllables.. You could however read the sentence differently. In which case it doesn't require that it is not possible to name the number in fewer than 19 syllables, but requiring it is possible not to name it in fewer than 19 syllables. And clairly, 0 being the least of the natural numbers could be named by something more than 19 syllables. Its like 'not(doing(something))', and '(not(doing))(something)', in the latter case you 'actively' refrain from doing something.. It's a rather contrived interpretation, but works linguisticly.. |
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Title: Re: 111,777 Post by Icarus on Jan 15th, 2003, 7:42pm Okay, but that is bypassing the central question here rather than answering it. Consider fixing a definite meaning on some subset of the English language (or whatever language you choose). Say that a number is "named" by a sentence, if the sentence uniquely specifies that number according to the meaning you have fixed. By my argument, the sentence "the least integer not nameable in fewer than nineteen syllables" must uniquely specify some number. And of course, the number that it specifies is both nameable and not nameable in fewer than 19 syllables. Failure to resolve this paradox will not make the universe explode. However it will make mathematics explode! And this one along with some other related paradoxes, including the far more famous Russell's Paradox, shook the mathematical world to the core when they were discovered at the end of the 19th century. You can prove anything from a paradox, so if even one exists in a theory, every possible statement in that theory is both true and false. So, the resolution of paradoxes is of prime importance. How is this one resolved? |
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Title: Re: 111,777 Post by SWF on Jan 15th, 2003, 7:53pm If the intent of this riddle is notice the paradox, sort of like "this sentence is false", then I agree with towr, that this belongs in the Easy section. Although I can see the intent, this problem is not worded such that there is a paradox. The way it is phrased asks for "naming" the number not describing it. Towr describes how that interpretation does not give a paradox. If describing it qualifies as "naming", then 111,777 can be described as "The product of 3, 19, 37 and 53". That is less than 19 syllables which would also make the original claim a false statement rather than a paradox. |
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Title: Re: 111,777 Post by Icarus on Jan 15th, 2003, 8:12pm It's not hard to spot the paradox. It is very hard to resolve the paradox. |
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Title: Re: 111,777 Post by towr on Jan 15th, 2003, 11:55pm The paradox comes from the use of metalanguage and normal language at the same time (or rather just not distinguishing).. I could name a dog 'cat', but it's still a dog. What it is doesn't change, but how it is named does. Likewise I can name some number 'the least integer that cannot be named in fewer than nineteen syllables' without it being the meta-level 'least integer that cannot be named in fewer than nineteen syllables'. It's not a paradox in mathematics, since mathematics seperatates langauge and meta-language.. In a way it _is_ Russel's paradox (if I remember correctly what that was).. And in the modern set-theory that isn't a paradox anymore, since it was made impossible.. Likewise this problem can't be defined in modern set-theory, because you're not allowed to mix up different levels anymore.. (as far as I know) |
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Title: Re: 111,777 Post by Icarus on Jan 16th, 2003, 7:22pm Quote:
You can name a cat "dog" only if you are allowed to ignore the meanings of the words when naming things. My point is that you still have the paradox without allowing this. To make this clear, rephrase it as "the least integer not describable in less than 19 syllables", with the word "describable" understood to mean that one provides a statement which is true only for that single integer. Russell's paradox concerns the set A = {B | B is a set and B is not an element of itself}. Either A is an element of itself or not. But if it is an element of itself, then by its definition, it is not an element of itself. Whereas if A is not an element of itself, by its definition, it must be an element of itself. There are two solutions to this predicament. the more commonly used Zermelo-Frankel Set Theory simply says that some relations cannot be used to define a set. That is, there is no such set as A. The other solution, Russell-Whitehead Set Theory, has complex rules for forming relations, which make it impossible to even say "B is not an element of itself". I would not say that the 19-syllable paradox is the same as Russell's paradox, even if they are related. Every actual paradox is just as closely related (at least all that I have ever heard of). ZF set theory avoids Russell's paradox by not allowing the existance of the necessary set, not by restricting what can be said. The only set involved in the 19-syllable paradox is one that must be allowed to exist if the set theory is to have any power. The only recourse is to define the grammar in such a way that the statement "the least integer not describable in less than 19 syllables" cannot be formed. |
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Title: Re: 111,777 Post by James Fingas on Jan 17th, 2003, 2:15pm I would say that these paradoxes are all exactly the same. All you're saying is, effectively: "This statement is false". If we assume the statement as an axiom, then we are able to deduce its negation. That applies to all of these paradoxes. The only logical thing to do is to say that such a statement is meaningless. To be more precise: 1) "This statement is false" -- the statement can be neither true nor false. 2) "The least integer not namable in fewer than nineteen syllables" -- this clause cannot refer to any specific integer. It neither does not does not identify the number 111,777. 3) "A = {B | B is a set, and B is not a member of itself}" -- A cannot refer to any specific set. That is to say, the contents of A are never determined exactly. A neither includes nor excludes itself. In all these cases, the nature of the problem prohibits a solution. Here's another one: "Why is there no correct answer to this question?" |
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Title: Re: 111,777 Post by Icarus on Jan 17th, 2003, 3:31pm But all actual paradoxes say effectively "This statement is false". That is the definition of "paradox". To say that all paradoxes are the same because they reduce to this, is like saying all dogs are the same because they are all canis domesticus. Actual paradoxes are not problems to be solved. They are pitfalls to be avoided. To resolve a paradox is to set things up so that it cannot occur. |
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Title: Re: 111,777 Post by Chronos on Jan 17th, 2003, 3:48pm The key here, as I see it, is that not every string of English words necessarily represents a number. The string "The color of grass", for instance, does not represent a number (unless we arbitrarily define it to mean some number, but let's not do that). Similarly, the string "The least integer not nameable in fewer than nineteen syllables" does not represent a number. Similarly, "the smallest non-special number" does not describe a number, either. One other minor nitpick (which is not really relevant), by the way: In standard English nomenclature for numbers, you leave out the "and". So 111,777 is "One hundred eleven thousand, seven hundred seventy-seven", which is only 17 syllables. Of course, one could re-word the riddle slightly to fix this problem. |
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Title: Re: 111,777 Post by Pietro K.C. on Jan 17th, 2003, 11:31pm "What is the least integer not nameable in fewer than nineteen syllables?" Maybe I'm missing something here, but I did not find it at all hard to resolve this "paradox". Upon reading it, the "solution" that came to mind was almost exactly the one described by towr. The main points, as I see them, are: * It is somewhat meaningless to say "not nameable", as towr pointed out; for I can call anything just "A". * Hence the puzzle, to be intelligible, should read "not named" instead of "not nameable"; for all names are conventions, implicitly assumed to have been set up prior to asking the "paradoxical" question. * Suppose we have carried out the naming of all naturals, i.e. defined an injective mapping from the positive integers to the set of strings of syllables (we need injectivity or some similar condition to insure that infinitely many numbers have names of nineteen syllables or more). This mapping could be suitably algorithmic, so as not to annoy the constructivists. Then we could ask two questions: 1. What number maps to the string "the least number not nameable in fewer than nineteen syllables"? 2. What is the least number such that its assigned name is nineteen syllables long, or more? Given the naming scheme, one would be hard pressed to see paradox in either question. * The original "paradox" rests upon confusion between language and metalanguage (as towr said) - particularly, in a confusion between name and property. On one hand, you can call 69 "the least number not named in fewer than nineteen syllables", and then ask what number "the least number not nameable in fewer than nineteen syllables" denotes (names). Or, you can carry out the naming and then search for the number with the property of being "the least number not named in fewer than nineteen syllables". These are two very different things: the first answer is most definitely "69", but the second is absolutely NOT. * What is incoherent and "paradoxical" is carrying out a naming, singling out "the least number not named in fewer than nineteen syllables", replacing its previous name with "the least number not named in fewer than nineteen syllables", starting over, and being amazed that this process does not end. It is not startling to discover that we cannot fix a number with the property of being "the least number not named in fewer than nineteen syllables", a property involving names, if the naming scheme itself is not fixed. * No doubt, given a naming scheme, one can refer to "the least number not named in fewer than nineteen syllables" as just that; but that takes place in the metalanguage, that is, externally to the naming scheme. The expression "not named" actually means "not named in the naming scheme". It all rests on the numbers having fixed names, really. * All the "paradox" argument says is that, in any naming scheme, the number with the property of being "the least number not named in fewer than nineteen syllables" cannot have "the least number not named in fewer than nineteen syllables" as a name. Well, duh. * I am hard pressed to even acknowledge this as a paradox - I rank it with calculation mistakes. Would anyone call the inscription "7 X 8 = 54", to be found in many a first-grader's notebook, a paradox? My opinion, in this case at least, is that no modifications need be made in the grammar or the mathematics "to set things up so that it cannot occur" (in Icarus's words), any more than we need to fix arithmetic to stop first-graders from making mistakes. * In a different view, keeping the original wording of the puzzle, it is clear that "not nameable in fewer than X syllables" actually means "the standard English number-naming algorithm yields a result of X or more syllables", because, as remarked before, one could name any number "Bond, James Bond". Therefore, one could argue that part of the "paradoxical" nature of this puzzle is the ambiguous use of "not nameable" - it's like saying, "all names must consist solely of strings of various lengths of the letter 'A' ", and then magically calling something "B". Unless, of course, we are willing from the start to consider any string a possible name, in which case the preceding *'s may be applied. Am I missing something very big? Because I don't see the mistake or the difficulty in my (and towr's) explanation. |
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Title: Re: 111,777 Post by BNC on Jan 18th, 2003, 3:42pm The way I read the riddle, “not nameable” meant “not addressable”. I mean, you may “address” or identify numbers in different ways. For example, I can tell you “the least prime number larger than 10” or “eleven” – as addressing goes, they are synonyms. But if 111,777 is indeed “The least integer not nameable in fewer than nineteen syllables”, the way I se, we do have a problem. Because then you may address it as “The least integer not nameable in fewer than nineteen syllables”, thus yielding the name=address incorrect. Just my 2c. |
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Title: Re: 111,777 Post by Icarus on Jan 18th, 2003, 8:06pm Yes, Pietro, you are missing something big. The paradox is not directly in the puzzle as it is stated, or particularly, as you are interpreting it. You and towr are interpreting "name" in a way that avoids the actual paradox. Since you find no paradox in what this says by your meaning, you say there is no paradox there. This is like saying that since the self-referential sentence "This sentence is true", is not paradoxical, self-reference must not lead to paradox, without giving thought to that other sentence which is clearly paradoxical. This is why I rephrased it to use the word "describable" instead of "nameable". The gist of this paradox is to uniquely describe an object in such a way that it cannot have that description. In this it is indeed very close to Russell's paradox, but this paradox accomplishes it in a different fashion. The paradox has nothing to do with names as such. To set up this paradox, you need the following: 1) a condition C on relations R, 2) a relation M(x) which implies the relation "C(R) => NOT R(x)" and which satisfies condition C. 3) a proof that there exists x such that M(x). In this case C is the condition that the relation have fewer than 19 syllables in English (similar variants in other languages). This paradox was one of several discovered in the investigation of paradoxes instigated by Russell's discovery of his. This was an extremely hard time for mathematicians who were looking at the foundations of Mathematics, because as I have already said, one paradox in mathematics would bring the whole of it down, and they had found real paradoxes in the most fundamental parts of mathematics. So, yes, this paradox is a very big thing. But you are right about this: The grammar of mathematics does not need changed to solve this paradox. The reason, though, is that the grammar of mathematics ALREADY HAS BEEN CHANGED. This is not a new paradox. It was resolved at the turn of the last century, and you and towr have already refered to the vital concept behind its resolution. A concept that was not truly developed until this and similar paradoxes were discovered. So, I ask you this: What is the difference between "mathematics" and "meta-mathematics"? |
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Title: Re: 111,777 Post by Pietro K.C. on Jan 19th, 2003, 3:45pm Hmm... I'm not sure I entirely follow the formal argument, with the relations and conditions. Could you perhaps clarify it a bit? When you say "relation R(x)", you mean a unary relation? In that case, the argument would be as follows: * M(x) => ( C(M) => NOT M(x) ) (by #2 in Icarus's post) * M(x) (because of the proof of existence, #3 in Icarus's post) -------- * C(M) => NOT M(x) * C(M) (#2 in Icarus's post) -------- * NOT M(x) - conclusion and contradiction! Do I grasp it correctly? Hmm, I think I do, because now I see very clearly the relation to Russell's paradox. Ah! How bright doth the light presently shine in my mind! :) I see now what you mean by the self-reference example. You are entirely right, of course. Whoa. I'll give this some more thought before I spout further nonsense. :) |
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Title: Re: 111,777 Post by towr on Jan 20th, 2003, 12:01am M(x) => ( C(M) => NOT M(x) ) (M(x) AND C(M)) => NOT M(x) NOT (M(x) AND C(M)) OR M(x) NOT M(x) OR NOT C(M) |
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Title: Re: 111,777 Post by Icarus on Jan 20th, 2003, 5:23pm on 01/19/03 at 15:45:37, Pietro K.C. wrote:
Now that's an idea! Maybe I should try it sometime! ;) It amazes me over and over again that posts that I sweated over and rewrote several times, until I was sure that they were paragons of clarity, should prove so obtuse when I reread them the next day! I'll not try to clarify the paradox again. You have the gist of it. (towr, note by the last part of #2, we have C(M), and by #3, we have M(x). Or, if I have misunderstood why you stopped your post where you did, I apologize.) This paradox was resolved by the introduction of the concept of meta-mathematics. Eliminate relational variables (variables whose values are relations, rather than objects), and paradoxes of this type cannot be stated, because condition #2 requires C be a statement within the theory, but whose arguments are relations. But we still need to be able to talk about types of relations, and give general rules for handling relations. So we push conditions on relations a level up, and create a mathematical theory about the mathematical theory - meta-mathematics! This still allows us the freedom to discuss things like "relations with less than 19 syllables" (at least if the concept were rigorously defined), but not the freedom to talk about "the least integer describable in less than 19 syllables). As soon as we have that much freedom, we are doomed. This does not solve Russell's paradox however, which is part of why I insist that the two paradoxes are not the same. Russell's paradox uses the equivalence between unary relations and sets in "naive" set theory (every unary relation corresponds to the set of all objects that satisfy it) to create a 19-syllable type paradox within the confines of the theory itself, even after the introduction of meta-mathematics. The resolution of Russell's paradox comes only by breaking the equivalence. |
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Title: Re: 111,777 Post by towr on Jan 21st, 2003, 12:04am on 01/20/03 at 17:23:54, Icarus wrote:
but if you want, NOT (M(x)) OR NOT (C(M)) = NOT (M(x) AND C(M)) given M(x) AND C(M) so (NOT (M(x) AND C(M))) AND (M(x) AND C(M)) == FALSE |
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