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Nov-04-06
 | | Sneaky: How many of these following statements are true?
1. At least one of these statements is false.
2. At least two of these statements are false.
3. At least three of these statements are false.
4. At least four of these statements are false.
5. At least five of these statements are false.
6. At least six of these statements are false.
7. At least seven of these statements are false.
8. At least eight of these statements are false.
9. At least nine of these statements are false.
10. All ten of these statements are false.
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| Nov-04-06 | | positionalbrilliancy: None of them. |
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Nov-04-06
| | Sneaky: Sorry, doesn't work. Consider, you claim that none of them are true. So your saying that all 10 of those statements are false, right? So you're agreeing with statement #10 then, right? Doesn't that make it true? This one is so frustrating. The recursive nature of it all makes me go bonkers and yet once you get your head around it, it's all very clear. |
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| Nov-04-06 | | positionalbrilliancy: One of them is true, number 10. |
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Nov-04-06
| | Sneaky: But if #10 is true, then it must be true that "All ten of these statements are false." which means that #10 is false. But we just said it's true! Oy!! http://www.geocities.com/transactoi...
<Does not compute! Does not compute!> |
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| Nov-04-06 | | azaris: None of the statements are valid propositions in propositional calculus, so they have no truth value. |
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| Nov-04-06 | | technical draw: I have a list of one hundred digits taken from a true random source. (list A). I have another list of one hundred digits taken from another true random source. (list B). I decide to merge both lists alternating from one list to another and I produce a list of two hundered digits. (list C). Question: Is list C a true random list? |
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| Nov-04-06 | | themadhair: <Sneaky> Logically 10 → 9 → 8 → 7 → 6 → 5 → 4 → 3 → 2 → 1 (note that → means implies). So 10 must be false otherwise all statements would be necessarily true and a paradox would result. Now if 9 were true than so also would statements 8 through 1 be true - again resulting in a paradox. Working backwards from 10 in this way we see that statements 10 to 6 must be false in order to avoid a paradox. This allows statements 1 to 5 to be true by inspection. <None of the statements are valid propositions in propositional calculus, so they have no truth value.> As a mathematician even I must shun this assertion that mathematics contains any truth. Mathematics is an idealised system built using the tools of logic and rigour. It is without error and imperfection. As a consequence it does not have a basis in the real world - and as such cannot have any inherent truth. To me, suggesting that mathematics contains truth is tantamount to numerology. |
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| Nov-04-06 | | themadhair: <technical draw> Yes. |
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| Nov-04-06 | | azaris: <technical draw> Define "true random list". |
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| Nov-04-06 | | azaris: <As a mathematician even I must shun this assertion that mathematics contains any truth. Mathematics is an idealised system built using the tools of logic and rigour. It is without error and imperfection. As a consequence it does not have a basis in the real world - and as such cannot have any inherent truth. To me, suggesting that mathematics contains truth is tantamount to numerology.> I don't think I claimed anywhere that mathematics contains inherent truth. But for logical deduction like yours to work, we must work inside a system like propositional calculus where statements have truth values and valid logical deductions ascertaining the truth values of other statements can be made. The problem with statements like "this statement if false" or "the next statement is true" is that they work on a meta level that exists on a higher level from the usual logical propositions that we work with. Since this can easily cause paradoxes, we limit ourselves to propositions that only work on the object level. So no self- or outer-referencing propositions, please. |
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| Nov-04-06 | | technical draw: <azaris> A true random list is a list of numbers (digits) where each digit has equal chance of coming out, then replacement and another event until one hundred events. (as opposed to a pseudo random list that is derived from a formula) <themadhair> Care to elaborate, or do you have a source?. Actually, I'm studying this problem and would like others input. |
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| Nov-04-06 | | azaris: <technical draw> So it's a realization of n independent uniform distributions on the set 0,1,...,9? I suppose you could interpret the shuffled union of two shorter "random lists" as one long "random list". If the individual digits are independent from each other it should not matter in which order you draw them. |
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| Nov-04-06 | | themadhair: <I don't think I claimed anywhere that mathematics contains inherent truth.> I admit to misreading and misinterpreting your post and retract my comment. <The problem with statements like "this statement if false" or "the next statement is true" is that they work on a meta level that exists on a higher level from the usual logical propositions that we work with.> Point taken, but wasn't <sneaky>'s puzzle well-defined and self-contained with the question <How many of these following statements are true?> attached to his header? The puzzle boils down to finding which statements need to be true/false in order to avoid a paradox - surely that is a task within the means of logic? |
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| Nov-04-06 | | themadhair: <technical draw> What <azaris> said. When you are generating your 100 number lists you have already made the assumption that combining 100 seperate single number lists preserves randomness. So why should combining two lists be any different? Only if the method used was somehow biased - which in this case is not. |
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| Nov-04-06 | | azaris: <Point taken, but wasn't <sneaky>'s puzzle well-defined and self-contained with the question <How many of these following statements are true?> attached to his header? The puzzle boils down to finding which statements need to be true/false in order to avoid a paradox - surely that is a task within the means of logic?> Ah, but the logician wishes to drain the fun out of everything and ensure that the logical system we use to make deductions is consistent. While things might be made to work in this example, we usually forbid such propositions. And if our system of logic does not allow for such propositions, how can be use logic to deduce the correct answer? |
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| Nov-04-06 | | technical draw: <azaris> Tks for your input. <it should not matter in which order you draw them> is precisely the question. Once drawn do they lose their random characteristics? |
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| Nov-04-06 | | azaris: <technical draw> The realization of a random variable (like your "random digit") is not in any way random, it's just a number that stays the way it is once drawn. Consider this experiment. You want to produce a number 0-99 by drawing both digits from a uniform distribution. But before the draw, you flip a coin to see which digit you draw first. It is easy to see that the result of the coin flip in no way alters the distribution of the resulting number. |
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| Nov-04-06 | | technical draw: OK. Thanks all for considering my problem. I will use your input for futher study. |
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| Nov-04-06 | | Ilgiz Tashkhodzhaev: <Ah, but the logician wishes to drain the fun out of everything and ensure that the logical system we use to make deductions is consistent.> Ah, but this not inductively within self-containing set, so formal logic refutes such conclusion! I ask how to true logician can finds deductions implicit in puzzle, truth value is higher than 0 but less than 1. This is problem with propositional analysis in modal logic. Thanks, I.T. |
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Nov-04-06
| | Sneaky: <themadhair> <Sneaky> <Working backwards from 10 in this way we see that statements 10 to 6 must be false in order to avoid a paradox. This allows statements 1 to 5 to be true by inspection.> That's right, or I should say, that was my conclusion as well. The first half are true, the second half are false. Any other configuration leads to a paradox. |
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Nov-04-06
| | Sneaky: <I have a list of one hundred digits taken from a true random source. (list A). I have another list of one hundred digits taken from another true random source. (list B). I decide to merge both lists alternating from one list to another and I produce a list of two hundered digits. (list C).> Like everybody here I think that list C is truly random, but wait, let's try to muddy the waters by stretching what technical draw stated. Suppose I sent two employees into the field and tell them "bring me back lists of random numbers." One guy decides to go into the Casino and watch a certain die. He writes down the numbers that it shows: 1, 5, 6, 3, 5, 4, 2, 6, etc. The other empoyee, by sheer coincidence, decides to watch the exact same die, and starts at the exact same toss. The only difference is that he records the BOTTOM, i.e. the side that touches the felt, instead of the side that shows face-up. Now when you interweave the two lists, they form couplets that always add up to 7. If my list starts "1, 5, 6, 3..." then his starts "6, 2, 1, 4..." When we shuffle them together the result is far from random, and yet you can argue that both lists, in themselves, are purely random. |
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| Nov-05-06 | | azaris: <Now when you interweave the two lists, they form couplets that always add up to 7. If my list starts "1, 5, 6, 3..." then his starts "6, 2, 1, 4..." When we shuffle them together the result is far from random> There is no such concept of randomness or non-randomness in probability theory. Every single list is equally likely to appear as an outcome of this random event, even "1, 1, 1,...". You won't get anywhere with probability if you invent your own intuitive concepts and try to make heads or tails (pun intended) out of them. Many have tried and failed. |
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| Nov-05-06 | | themadhair: <Sneaky> Your 'counter-example' is slightly flawed. <technical draw>'s random list required that <A true random list is a list of numbers (digits) where each digit has equal chance of coming out>. This breaks down with your dice roll list for the subtle reason that if you specify list one as being random (and it is) than list two ISN'T random since it is derived from list one. In other words, given list one I could say with 100% certainty the contents of list two - ergo it isn't random if list one is random. In terms of probability theory two events A and B are independant if P(A∩B) = P(A)P(B). Clearly in this case the two events are not independant (in fact P(A∩B) = P(A)=P(B)) so composing them in any fixed way cannot be a true random list. <There is no such concept of randomness or non-randomness in probability theory.> That may be so, but in <technical draw>'s question the concept of 'a true random list' he (she?) introduced was well-defined within the context of his (her?) question. While we may use ideas from probability theory to better understand the question, we in no suppose that we assume there exists a concept of randomness within probability theory. Although I agree that <technical draw>'s attempts to produce such a theory have a very low probability of success - but I can't say such a scenario is impossible. <While things might be made to work in this example, we usually forbid such propositions.> Has the logician admitted with this comment that <sneaky>'s puzzle is *gasp* logically sound? |
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| Nov-07-06 | | technical draw: Here's an easy one for y'all:
You flip a coin 5 times. What has a greater chance of occuring: 1. First 3 are heads
2. First 4 are heads
Easy, no? |
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