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Posers and Puzzles

Posers and Puzzles

  1. Donation Acolyte
    Now With Added BA
    20 Nov '04 12:01
    One day, a mischievous deity chooses a number uniformly at random from [0,1], and imparts to you knowledge of exactly what this number is, without telling you any of its special properties. In doing so, he says "This is the ineffable number. If you come to understand the number, I shall strike you dead instantly. However, unless and until that time comes, you will not die."

    We'll say that you 'understand' the number if you can provide a mathematical definition which uniquely determines the number. For example, people 'understand' numbers like 7, pi, the Riemann zeta function evaluated at e, and even weird numbers like omega, which is a constant derived from Turing's halting problem: http://mathworld.wolfram.com/ChaitinsConstant.html

    After a while you decide immortality isn't all it's cracked up to be, and so try to end it all by understanding the number. You don't care how long it takes - if you have to persuade generations of mathematicians to help you out until a definition is produced, then so be it.

    What is the probability that, despite your best efforts, there is no way you can avoid living forever?
  2. Standard member TheMaster37
    Kupikupopo!
    20 Nov '04 14:12
    This deity didn't speak to me about such a thing, so i am safe when i try to list some properties of when you CANNOT descibe the number by it's properties;

    1- the decimal expansion of the number cannot be finite, for then the number is a unique sum of several fractions.

    2- the expansion cannot contain any form of regularity, even when you take numbers as Pi into account.

    Given the fact that a human mind cannot know an infinite expansion like that i have my doubts to the solvability of this problem as stated in this thread.

    Another problem i come across is that in the normal [0, 1] numbers are unique. They all have their place in the ordered 'line' of [0, 1].

    Any real can be approximated as accurate as you want with fractions. Define a real as the unigue number closed in between two determined rows of fractions and you have your property.
  3. 20 Nov '04 18:51 / 1 edit
    Originally posted by Acolyte
    One day, a mischievous deity chooses a number uniformly at random from [0,1], and imparts to you knowledge of exactly what this number is, without telling you any of its special properties. In doing so, he says "This is the ineffab ...[text shortened]... e your best efforts, there is no way you can avoid living forever?
    There are an uncountable number of numbers between 0 and 1. Each human (except me perhaps) lives for a finite time, and there are and will ever be a finite number of humans - so they will be no help, because they can each only understand a countable set of numbers (their brains run at a finite rate). As for me, even though I will live for ever, at any finite point in time I can only have understood a countable set of numbers. Therefore there are an uncountable number of numbers that will never be thought of - and sadly the probability is 100% that my number is one of those.
  4. Standard member royalchicken
    CHAOS GHOST!!!
    20 Nov '04 21:13 / 1 edit
    Originally posted by Acolyte
    One day, a mischievous deity chooses a number uniformly at random from [0,1], and imparts to you knowledge of exactly what this number is, without telling you any of its special properties. In doing so, he says "This is the ineffab ...[text shortened]... e your best efforts, there is no way you can avoid living forever?
    First of all, he's given you a 'special property', that of 'ineffability', so you're dead to begin with.

    What constitutes exact knowledge of this number? Every real number has a decimal expansion, which can either be finite, periodic or non-periodic. In either of the first two cases, you are guaranteed to know a special property, and it is possible to know a special property in the third case (for example ''The number is log(2)''. However, special properties (those which you can understand, in any case), are expressible as countably infinite or finite strings of finite symbols. It's not too hard to prove that a countably infinite set of finite things is countably infinite, so unless my first paragraph is more than a joke (in which case the probability is zero), there are only countably many real numbers with special properties, so your probability of living forever is 'infinitesimally close' to 1 (heh; let's call it 0.9 rec, because, to borrow a sentiment ''FOR FVCK'S SAKE 2<3 SO 0.9 rec < 1!!!!!11''.

    EDIT My post is basically the same as IAAT's.
  5. Donation Acolyte
    Now With Added BA
    21 Nov '04 02:45
    *Instead of 'special properties' I should have said 'special mathematical properties', though in fact the answer would be the same if I had said 'given any subset X of [0,1] which you understand, you know (as a kind of 'sixth sense' which is unmathematical) whether the ineffable number is in X or not'.

    royalchicken and iamatiger are both right about the probability. However, I'll make the following comments:

    Mathematicians can understand uncountable sets of numbers, eg R. However the set of all numbers which are individually understandable is countable.

    In my book, a definition which requires an infinite string of symbols to write down doesn't make a lot of sense - to be understandable, it must be possible to specify something uniquely using a finite string of symbols. Without this condition all real numbers are understandable, for instance, because their decimal expansions amount to countably long definitions. TheMaster37's definition falls down for this reason.


    The point is was trying to convey with this puzzle is the concept of specialness in mathematics - that there are sets out there which are 'special' in themselves, even though 'most' of the elements in them are not special in that we cannot say 'this is the unique element which has property P'. The most worrying sets of all are the ones where NONE of the elements are understandable, even if the set itself is understandable, for example the set of subsets of R which are not Lebesgue measurable. Many problems of this kind are dealt with using the Axiom of Choice, giving rise to theorems of the form 'there exists an x such that...' where it is impossible to give a single example of such an x.
  6. Standard member TheMaster37
    Kupikupopo!
    22 Nov '04 15:46
    You CAN annotate a countable set of conditions in a finite string of symbols.

    An easy example of this is 0;

    We call 0 the one and only real number R wich satisfies the following condition; -|1/n| < R < |1/n| for all numbers in N*

    Above i gave a countable set of requirements to be met for this 0. As you can see, the list of symbols i used i quite finite.
  7. Standard member Nemesio
    Ursulakantor
    24 Nov '04 06:25
    Originally posted by Acolyte
    One day, a mischievous deity chooses a number uniformly at random from [0,1], and imparts to you knowledge of exactly what this number is, without telling you any of its special properties. In doing so, he says "This is the ineffable number. If you come to understand the number, I shall strike you dead instantly. However, unless and until that tim ...[text shortened]... s the probability that, despite your best efforts, there is no way you can avoid living forever?
    .204857408754?



    Nemesio
  8. Donation Acolyte
    Now With Added BA
    25 Nov '04 12:07
    Originally posted by TheMaster37
    You CAN annotate a countable set of conditions in a finite string of symbols.

    An easy example of this is 0;

    We call 0 the one and only real number R wich satisfies the following condition; -|1/n| < R < |1/n| for all numbers in N*

    Above i gave a countable set of requirements to be met for this 0. As you can see, the list of symbols i used i quite finite.
    Of course it's possible in special cases to express a countable or even uncountable number of conditions, eg 'A is a subset of B' is equivalent to saying, for every element x of A, 'x is in B'. But your 'determined rows of fractions' are not, in general, finitely expressible.
  9. Standard member Fat mans revenge
    Pennywise Says Hi
    25 Nov '04 18:57
    This may seem kind of simple, but wouldn't a mathematical property of a number that you already know simply be 1-X=your number? Or
    Your number + X =1?

    Forgive my ignorance if I misuderstood something, but this is what I thought of when I read the question.
  10. Donation Acolyte
    Now With Added BA
    25 Nov '04 23:27 / 1 edit
    Originally posted by Fat mans revenge
    This may seem kind of simple, but wouldn't a mathematical property of a number that you already know simply be 1-X=your number? Or
    Your number + X =1?

    Forgive my ignorance if I misuderstood something, but this is what I thought of when I read the question.
    Ah, but what is X? You can either use x,y,z etc as 'dummy' variables (as below) or they each have to be defined in turn. You have to give some definition that specifies the ineffable number uniquely, ie you describe it, in terms of numbers you've already defined, as having some property that no other number has. For example, 1 is the one and only number x for which x*y = y for all choices of y, so this amounts to a definition of the number 1.

    That might sound a bit stringent, but bear in mind that we can do this for all rationals and even some irrationals: for example, now that we've defined 1, we can say:

    3/4 is the product of the number (1+1+1) with the multiplicative inverse of (1+1+1+1) (ie the unique x such that (1+1+1+1)*x = 1).

    and

    e is the limit of the series 1 + 1/1 + 1/(1+1)! + 1/(1+1+1)! + ...
    (... is allowed as long as the continuation of the series is obvious - I could have written this very precisely, but that requires more notation)

    and it's fairly easy to show that all the 'the's I use in these definitions refer to uniqueness (the hardest one is 'e is the limit of...', which takes a little bit of analysis). If you can do this kind of thing, there can be no doubt whatsoever about what the number is, so in a sense you have 'classified' it.
  11. Standard member Fat mans revenge
    Pennywise Says Hi
    26 Nov '04 00:19
    Well, in this situation X is the difference between your number and 1.

    But as you put it, I can select any number between 0 and 1. An easy example is the number 0.5. This is the only number that can be multiplied by 2 to become one. That simple.

    Again, I feel like I have missed something.
  12. Standard member TheMaster37
    Kupikupopo!
    26 Nov '04 13:00
    Originally posted by Fat mans revenge
    Well, in this situation X is the difference between your number and 1.

    But as you put it, I can select any number between 0 and 1. An easy example is the number 0.5. This is the only number that can be multiplied by 2 to become one. That simple.

    Again, I feel like I have missed something.
    Let's give your number a name, say #

    What you just said is to define # by a number X.

    Then you were asked what X was, you answered X is the difference between # and 1. You're going in cicles.

    What does an apple look like? It looks like a peach. Ok, but what does a peach look like? A peach looks like an apple.

    An alien following the discussion still does not know what an apple looks like...
  13. 26 Nov '04 23:51 / 2 edits
    Even More Mischievous Deity: The ineffable number is the smallest/largest number in [0,1] that cannot be described in less than twenty words.
  14. Donation Acolyte
    Now With Added BA
    27 Nov '04 00:14
    Originally posted by THUDandBLUNDER
    Even More Mischievous Deity: The ineffable number is the smallest/largest number in [0,1] that cannot be described in less than twenty words.
    That's like saying 'the number in [0,1] that squares to 10'. It's a bad definition because such a number doesn't exist; in other words, the set of numbers in [0,1] which cannot be described in less than twenty words has no least or greatest element. There's no shortage of sets like that, though: take the open interval (0,1), for example.
  15. Standard member Fat mans revenge
    Pennywise Says Hi
    27 Nov '04 02:31
    Originally posted by TheMaster37
    Let's give your number a name, say #

    What you just said is to define # by a number X.

    Then you were asked what X was, you answered X is the difference between # and 1. You're going in cicles.

    The reason why I come to this conclusion is that I already know what the number is. As said in the original problem. If I already know what the number is, then I can substitute that in for #. And the only variable left is X.

    Perhaps my misperception on this problem is that there is a specific number that the original poster is trying to get us to come up with. In that case, I can see no way of finding that number(I'm not a mathematician or anything)