• (disco) in reply to RaceProUK

    Any number with infinitely many digits and no algorithm with which to generate them.

    Here's the method to find one:

    0­. start from zero

    1. generate a random decimal digit
    2. append it at the end of the decimal digits of your number
    3. does your number have infinitely many digits? if no, jump to 1
    4. does any finite algorithm exist which always generates that exact sequence? if yes, jump to 0.
    5. you now have a number that can't be represented

    Note that you can never get past step 3.

    Step 4 exists for the chance that the random algorithm happened to generate a number such as infinitely many 3s with no other digits. There is a 0 percent chance of this happening, but it's still important to be thorough.

  • (disco) in reply to anotherusername
    anotherusername:
    Any number with infinitely many digits and no algorithm with which to generate them.

    Filed under: I thought of this before you posted it, but forgot to actually write the post, so I can't prove I thought of it first. :(

    In step 1, it is necessary that the digits be truly random, not pseudorandom, or else you could represent the number by describing the PRNG that generates the digits.

  • (disco) in reply to HardwareGeek

    True, but a PRNG which always generates that exact set of numbers given the exact same seed would therefore always fail the condition in step 4, resulting in another infinite loop.

  • (disco) in reply to anotherusername

    Yes, exactly. Any number that can be generated by the PRNG can be represented by the PRNG algorithm and seed, and therefore is not a member of the set of unrepresentable numbers.

  • (disco) in reply to anotherusername
    anotherusername:
    There is a 0 percent chance of this happening

    In probability theory, this is non-equivalent to "impossible", btw..

  • (disco) in reply to anotherusername
    anotherusername:
    does any finite algorithm exist which always generates that exact sequence?

    You might as well go straight to that and leave out the stuff with asking whether it has infinitely many digits: the finite numbers embed trivially within the infinite numbers.

  • (disco) in reply to PWolff
    PWolff:
    In probability theory, this is non-equivalent to "impossible", btw..

    How would you observe the difference?

  • (disco) in reply to dkf
    dkf:
    PWolff:
    In probability theory, this is non-equivalent to "impossible", btw..

    How would you observe the difference?

    What is the use of talking about infinities in a finite universe anyway?

  • (disco) in reply to PWolff
    PWolff:
    What is the use of talking about infinities in a finite universe anyway?

    That depends on what you mean by finite.

  • (disco) in reply to dkf

    Take a random number from the unit interval ([0,1]). The chance of the number being exactly equal to 0.5 is zero. But it is not impossible.

  • (disco) in reply to PleegWat
    PleegWat:
    The chance of the number being exactly equal to 0.5 is zero. But it is not impossible.

    Not if the random numbers are taken out of a finite set of representable numbers.

  • (disco) in reply to PleegWat
    PleegWat:
    The chance of the number being exactly equal to 0.5 is zero.

    The term you are looking for is “infinitesimal” and you need to ask what is the probability that the chance that something infinitesimally small is exactly equal to zero? ;)

  • (disco) in reply to dkf

    I learned zero.

  • (disco) in reply to PleegWat
    PleegWat:
    I learned zero.

    You learned wrong. :stuck_out_tongue:

  • (disco) in reply to dkf

    No. Zero is right. Infinitesimals don't exist in the field of real numbers. You have to extend that field, but I don't know if that is still measurable.

  • (disco) in reply to PWolff
    PWolff:
    No. Zero is right. Infinitesimals don't exist in the field of real numbers. You have to extend that field, but I don't know if that is still measurable.

    It's not like the original question makes much sense anyway. Exact equality isn't actually well-defined on the reals in the first place.

  • (disco) in reply to dkf

    (Do you mean "on the reals" or "on the surreal"?) On both sets equality is exactly defined, for all that I know. A real number can be defined as a set of rational numbers (or a pair of sets of rational numbers, which is equivalent). Surreal numbers similar.

  • (disco) in reply to PWolff
    PWolff:
    On both sets equality is exactly defined, for all that I know.

    OK, then is 0.5 equal to 0.4999… (with an infinite number of nines)? If not, how can you tell the difference?

    I assume that a number must have identity independent of its representation.

  • (disco) in reply to dkf
    dkf:
    OK, then is 0.5 equal to 0.4999… (with an infinite number of nines)?

    Yes. Yes, it is.

    There are more pages on the Internet discussing the question whether or not 1 equals 0.9999... (with an infinite number of nines) than you can read within a human lifetime.

    Edit (added): The problem is that the concept of real numbers as such can't be grasped by the human mind any more than a fourth spatial dimension; we have to rely entirely on definitions, theorems, and other abstractions.

  • (disco) in reply to dkf

    Those are different representations for the same number.

  • (disco) in reply to PleegWat

    Actually, no they aren't. As I understand it, there does exist a consistent framework for working with infinitesimals as distinct from the value whose limit they approach, in something called 'hyperreal' numbers, though I could not even begin to understand much less explain them. As a practical matter, infinitesimals are the basis of differential calculus (at least in Newton's original formulation - Leibnitz had a different form, but it was basically the same concept, and formulations of the calculus that avoid limits didn't appear until the late 20th century), which has proven to be pragmatically useful if nothing else.

  • (disco) in reply to ScholRLEA

    You can use extensions of the field of the real numbers to do calculus in the "intuitive" way Leibniz did (I'm not so sure about Newton's "fluents"), and it can be easier to do calculus rigorously with them instead of the usual limit approach.

    Nevertheless, the hyperreals are still an extension, that is, not the same as the real numbers. Just like integer numbers are an extension of the natural numbers, rational numbers of the integers, complex of the real, and so on. (Every extension adds a layer of abstraction that makes it a bit more difficult for the human mind to wrap around that contraption.)

    There is no way to distinguish between 0.4999... (infinite number of 9s) and 0.5 within the real numbers, therefore we call them equal.

    In the same way, we call 2 * 1/2 the same as 1, for in the theory of rational numbers, there can't be a way to distinguish them, although it is clear that, in nature, there is no way to put two half eggs together to one complete egg. (Or, even if we could, we can't pull three apples out of a bag containing exactly two apples, then put one apple back, and have an empty bag.)

    In hyperreal numbers, surreal numbers or other similar extensions of the real numbers, there is a defined difference between a number that one would naïvely write as 0.49999... (infinite number of 9s) on the one hand and 0.5 on the other hand. But that number is not represented by the sequence 0.49999... but rather by (the equivalence class of) the sequence {0, 0.4, 0.49, 0.499, ...}.

  • (disco)

    OK, you are correct, it is outside of the set of reals. I defer to you're greater knowledge regarding the representations, as I am not familiar with that aspect of hyperreals.

  • (disco) in reply to PWolff
    PWolff:
    Not if the random numbers are taken out of a finite set of representable numbers.

    There are infinitely many representable numbers in any given interval. (And infinitely more non-representable numbers; in fact the majority of them are non-representable, but the number of representable numbers is nonetheless infinite.)

    dkf:
    Exact equality isn't actually well-defined on the reals in the first place.

    In computing, maybe, but why wouldn't exact equality be defined on reals?

    dkf:
    OK, then is 0.5 equal to 0.4999… (with an infinite number of nines)? If not, how can you tell the difference?

    0.5 and 0.4999... are two different ways of representing the same exact number in base-10. So yes, 0.5 is exactly equal to 0.4999...

  • (disco) in reply to ScholRLEA
    ScholRLEA:
    hyperreals

    Why is a set of numbers that are less real than ordinary real numbers described as hyperreal? :trolleybus:

  • (disco) in reply to HardwareGeek
    HardwareGeek:
    ScholRLEA:
    hyperreals

    Why is a set of numbers that are less real than ordinary real numbers described as hyperreal?

    "hyperreal" is a contraction of "hyperactive" and "for real".

  • (disco) in reply to Dreikin
    Dreikin:
    urkerab:
    Has anyone actually discovered one yet? If so, which was the first to be discovered?
    Such a number would be describable, wouldn't it?
    That's the joke...
    Dreikin:
    (The reasoning is basically:
    • The cardinality of the set of all possible representations of numbers is ℵ₀.
    • The cardinality of the set of real numbers is ℵ₁.
    • ℵ₀ < ℵ₁.
    • Therefore, some real numbers can not be represented (i.e. described).
    • There are more real numbers that can not be represented than can be represented.)
    So what you're saying is that although we know that most numbers are undescribable, we'll never have an example of one?
  • (disco) in reply to urkerab
    urkerab:
    So what you're saying is that although we know that most numbers are undescribable, we'll never have an example of one?

    what? like π?

    we can describe it, and we know how to calculate it, and we've calcualted it far enough that the error for a circle the size of the observable universe would be IIRC about 5 angstroms (a very small error indeed) but we'll never be able to completely and accurately represent the actual number

  • (disco) in reply to urkerab
    urkerab:
    So what you're saying is that although we know that most numbers are undescribable, we'll never have an example of one?

    Correct. If a real number can be described by, say, a decimal number with infinitely many digits, there must be a sequence that describes those digits, an algorithm for producing those digits. If those algorithms are themselves of finite length, they must be enumerable and so countable (i.e., mappable to the integers; see the works of Church, Turing, Gödel and Kolmogorov for details).

    Hence we know there must be real numbers with infinitely many digits whose digit production algorithm is itself infinite, i.e., that their complexity and their representation can be considered to be unified. This might be achieved by making the sequence truly random I guess. Moreover, there must be a lot of such numbers…

  • (disco) in reply to accalia
    accalia:
    urkerab:
    So what you're saying is that although we know that most numbers are undescribable, we'll never have an example of one?
    what? like π?

    we can describe it

    Doesn't that defeat the purpose of an undescribable number?
  • (disco) in reply to accalia
    accalia:
    like π?

    No, because for all that that is a transcendental number, there's an algorithm which can calculate the digits of it; it's algorithmic complexity is much lower than that of a truly random number.

  • (disco) in reply to accalia
    accalia:
    urkerab:
    So what you're saying is that although we know that most numbers are **undescribable**, we'll never have an example of one?
    what? **like π**?

    we can describe it

    You seem to know the answer to your second question.

  • (disco) in reply to urkerab
    urkerab:
    Doesn't that defeat the purpose of an undescribable number?

    ah. you mean that sort of number.

    yeah.... good luck with that. inorder to identify the number we must be able to describe it. So in order to create an indescribeable number you would need to create a number such that the algorithm describing it is on the order of $n^{10^{100^{g_{64}}}}$ complexity

    (and because mathjax was disabled...... here's a png render of that formula) [image]

  • (disco) in reply to accalia
    accalia:
    So in order to create an indescribeable number you would need to create a number such that the algorithm describing it is on the order of $n^{10^{100^{g_{64}}}}$ complexity

    That sounds like something describable.

    http://scienceblogs.com/goodmath/2009/05/15/you-cant-write-that-number-in/

  • (disco) in reply to boomzilla
    boomzilla:
    That sounds like something describable.

    true, but good luck writing the algorithm for more than a couple of digits.... that's one hell of an exponent on the digit count.

    it would literally be more space efficient to just write the number out and never stop writing.

  • (disco) in reply to accalia
    accalia:
    we'll never be able to completely and accurately represent the actual number

    In base 10, that may be true. In base π, on the other hand... :stuck_out_tongue:

  • (disco) in reply to boomzilla

    [quote="boomzilla, post:187, topic:50418] http://scienceblogs.com/goodmath/2009/05/15/you-cant-write-that-number-in/ [/quote] One of the comments just blew my mind by throwing Cantor's diagonal argument into the list of describable numbers. Is the resulting number on the list? Surely I just described it...

  • (disco) in reply to accalia
    accalia:
    true, but good luck writing the algorithm for more than a couple of digits.... that's one hell of an exponent on the digit count.

    Yes, but how difficult it is to do isn't important to whether it's describable or not.

    urkerab:
    One of the comments just blew my mind by throwing Cantor's diagonal argument into the list of describable numbers. Is the resulting number on the list? Surely I just described it...

    Did you really write outdescribe all those numbers when you made your list?

  • (disco) in reply to boomzilla
    boomzilla:
    Did you *really* write outdescribe all those numbers when you made your list?
    Oh no! Not only can't I describe any indescribable numbers, but now I find I can't describe my list of describable numbers, which surely means that some of those numbers aren't actually describable after all?
  • (disco) in reply to urkerab
    urkerab:
    Not only can't I describe any indescribable numbers, but now I find I can't describe my list of describable numbers, which surely means that some of those numbers aren't actually describable after all?

    Write an algorithm to create the list of describable numbers. Job done!

  • (disco)

    Surely the very definition of 'indescribable numbers' is, in fact, a description? So indescribable numbers can in fact be described, and as such are not actually indescribable numbers, given that you can describe them as indescribable.

  • (disco) in reply to urkerab
    urkerab:
    throwing Cantor's diagonal argument into the list of describable numbers

    That might depend a bit how we look at it. (I haven't thoroughly thought through it yet)

    The only way Cantor's diagonal argument makes sense, is to have a countably infinite set of real numbers.

    urkerab:
    Is the resulting number on the list? Surely I just described it...

    Yes, you did.

    And thereby proved that

    • if the set of describable numbers is infinite, it must be uncountably infinite

    • if the set of describable numbers is finite, it must be infinite,

    Therefore the set of describable numbers is is actually uncountably infinite.


    We can't do more for we can't apply Cantor's diagonal argument to uncountably infinite sets - because this argument numbers an infinite set of lines by the natural numbers, so the set of lines is of course countably infinite.


    We could try the set of numbers that can be described with finite length descriptions. This is countably infinite, too (see the proof that the set of polynomials with rational coefficients is countably infinite). So this set won't do either (see above).

    So we could either take the set of numbers that is describable with countably infinitely long descriptions. Whether this is consistent or not, depends on whether we take the axiom of choice for granted or not (and thereby the possibility of well-ordering this set).

    If we assume that we can well-order that set, we can talk about the first number in the well-order that is outside that set, so this can't be done either. Implication: the class of describable numbers is not a set (see the proof that the class of ordinal numbers is not a set).


    What remains is, we talk about numbers that can be described with a description of a (previously) given maximal length. This is consistent, even with the possibiliby of well-ordering any set, for saying "The first number outside that set" implicitly contains the definition of "that set" and is therefore longer.

    I think this is very closely related to programming languages that allow loops only if the maximum number of loop passes is defined before the loop is entered. Those languages are not Turing complete, though.

    Well, I just wrote what came on my mind, I apologize in case it was too complicated.

  • (disco) in reply to RaceProUK
    RaceProUK:
    Surely the very definition of 'indescribable numbers' is, in fact, a description?

    No, that's more of a nickname. :smiley:

    A number (with an infinite number of digits) is probably best considered to be indescribable if it has no shorter description (i.e., algorithm for digit production) than the number itself. Thinking about what such a beast might look like is brain-bendingly hard, but “it looks really random” is a really good approximation: less random things probably have a finite algorithm for generating them.

  • (disco) in reply to RaceProUK
    RaceProUK:
    Surely the very definition of 'indescribable numbers' is, in fact, a description?

    Nope, it's a definition of the set, not a definition of the elements within.

  • (disco) in reply to RaceProUK
    RaceProUK:
    Surely the very definition of 'indescribable numbers' is, in fact, a description?

    I can't decide if you're serious or not. Given past math discussions, I'll assume you are.

    Yes, that is a description. Just not the type of description that would put the number into the set of describable numbers.

  • (disco)

    Pedants! [image]

  • (disco) in reply to RaceProUK
    RaceProUK:
    Pedants!

    Look...it's math, for crying out loud. It's the only proper way to talk about it. Even if you insist on calling it maths.

  • (disco) in reply to boomzilla
    [image] Seems my trolling was a little *too* subtle...

    Edit: And even when I post I was taking the piss, people are still biting...

  • (disco) in reply to RaceProUK
    RaceProUK:
    indescribable numbers can in fact be described

    The set of indescribable numbers can. Not the individual numbers that are elements of that set.

    But otoh, in mathematics, you can never be sure there isn't a contradiction somewhere, so it might be that we actually were able to prove that at least one number of the set of indescribable numbers can be described if only we knew how. Wouldn't be the first time we had to rethink an entire branch of mathematics.


    Maybe if we take/assume the axiom of choice as granted (and therefore that the set of indescribable numbers can be well-ordered), we can run into a contradiction:

    We take any well-order of that set, and then we talk about the first number of that well-order (which exists by the definition of a well-order), which is a description of a number that is element of that set. This is a contradiction indeed.

    Implication: Either there are no indescribable numbers (which can't be the case, as we have strong evidences if not a proof), or the class of indescribable numbers is not a set (which would imply that the class of numbers is not a set either, for the class of describable numbers is a set), or there exists a well-order of that set but we can't describe it (which implies there exists a well-order of real numbers, but there is no way to describe one).

  • (disco) in reply to PWolff

    The best part? IIUC, there is a good chance that the question of whether indescribable numbers exist as a sub-set of numbers is undecidable. What fun!

Leave a comment on “Listicle”

Log In or post as a guest

Replying to comment #:

« Return to Article