4 ms·

Can somebody explain to me how is this useful? Is it used for anything? Or is it pure theoretical concept with interesting emerging properties.

tialaramex 7y ago
The constants themselves would perhaps be useful but they're non-computable so we can't find out what they are anyway. The idea is useful yes.
- me_me_me 7y ago
  > The idea is useful yes. Well that's the crux of my question, how can it be useful wen its completely undefined.
  - jerf 7y ago
    The idea is mathematically useful. If you're asking about it's practical use, it has none.
  - deleted 7y ago
    [deleted]
  - balfirevic 7y ago
    It is defined, very rigorously actually. But it is uncomputable.
- fxj 7y ago
  The value of a Chaitin constant is highly machine-dependent. In some cases, it can even be proved that not a single bit can be computed (Solovay 2000). Chaitin constants Omega_U are perhaps the most obvious specific example of uncomputable numbers. They are also known to be transcendental. Calude et al. (2002) computed the first 64 bits of Chaitin's constant Omega_U for a certain universal Turing machine as Omega_U = 0.0000001000000100000110..._2 (2) = 0.00787499699... http://mathworld.wolfram.com/ChaitinsConstant.html http://mathworld.wolfram.com/ChaitinsConstant.html
  - tromp 7y ago
    That machine is NOT universal in the required sense, since each data bit is given in ASCII and contributes 7 bits to the program length.
mFixman 7y ago
I find this constant to be the most intuitive way to define uncomputable numbers, which itself is the most intuitive way to define uncountable sets.
- frutiger 7y ago
  > which itself [uncomputable numbers] is the most intuitive way to define uncountable sets Can you explain your intuition for this step?
  - mFixman 7y ago
    One of the ways to define the set of computable numbers is as the set of numbers for which there exists a program that, given an integer, gives you the n'th digit of this number. This set is countable: each one of these programs represents a single number, and you can use Gauss to transform any program into a integer and vice-versa. What's amazing is that almost every real number you heard about is part of this set: pi, e, and √2 can all be nicely put into a well-defined countable set. The set of the reals is uncountable, which means that almost every number is uncomputable like Chaitin's constant.
    - frutiger 7y ago
      You have described how the set of computable numbers are countable. But you have then posited "the set of the reals is uncountable" without any further explanation as to how that is the case. Your explanation helps one understand why computable numbers are countable (and that is for sure a useful insight), but says nothing about uncountable sets. I still feel the diagonal argument is the clearest intuition for defining an uncountable set.
      - mFixman 7y ago
        I was taught that real numbers were countable before learning what computables numbers where. "There's a set of countable numbers that includes every real number you know and doesn't include handwavy concepts" was news to me. Personally, I never found the diagonal argument intuitive. The proof with the nested open intervals in a sequence was easier for me to understand, but that's probably because I have more of a background in CS.
fxj 7y ago
It might have some relevance in information compression. He was also working on the halting problem (see lisp code below) and writing programs in LISP that were proving his theorems. http://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/turing.l http://jillian.rootaction.net/~jillian/science/chaitin/www.c... He also writes in his book: http://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch7.html http://jillian.rootaction.net/~jillian/science/chaitin/www.c... So in a way, in all three cases, Gödel, Turing, and I, we already have a new ``biological'' complicated mathematics, the mathematics of the third millennium, or at least of the 21st century. [As a child I used to dream that I was in the far future, in a library, desperate to see how it had all turned out, desperate to see what science had achieved. And I would take a volume off the shelf and open it, and all I could see were words, words, words, words that made no sense at all... Writing this book brings back long-forgotten thoughts and the unusual lucidity I experience when my research is going well and everything seems inevitable.]

tialaramex 7y ago

The constants themselves would perhaps be useful but they're non-computable so we can't find out what they are anyway. The idea is useful yes.

me_me_me 7y ago
> The idea is useful yes. Well that's the crux of my question, how can it be useful wen its completely undefined.
- jerf 7y ago
  The idea is mathematically useful. If you're asking about it's practical use, it has none.
- deleted 7y ago
  [deleted]
- balfirevic 7y ago
  It is defined, very rigorously actually. But it is uncomputable.
fxj 7y ago
The value of a Chaitin constant is highly machine-dependent. In some cases, it can even be proved that not a single bit can be computed (Solovay 2000). Chaitin constants Omega_U are perhaps the most obvious specific example of uncomputable numbers. They are also known to be transcendental. Calude et al. (2002) computed the first 64 bits of Chaitin's constant Omega_U for a certain universal Turing machine as Omega_U = 0.0000001000000100000110..._2 (2) = 0.00787499699... http://mathworld.wolfram.com/ChaitinsConstant.html http://mathworld.wolfram.com/ChaitinsConstant.html
- tromp 7y ago
  That machine is NOT universal in the required sense, since each data bit is given in ASCII and contributes 7 bits to the program length.

mFixman 7y ago

I find this constant to be the most intuitive way to define uncomputable numbers, which itself is the most intuitive way to define uncountable sets.

frutiger 7y ago
> which itself [uncomputable numbers] is the most intuitive way to define uncountable sets Can you explain your intuition for this step?
- mFixman 7y ago
  One of the ways to define the set of computable numbers is as the set of numbers for which there exists a program that, given an integer, gives you the n'th digit of this number. This set is countable: each one of these programs represents a single number, and you can use Gauss to transform any program into a integer and vice-versa. What's amazing is that almost every real number you heard about is part of this set: pi, e, and √2 can all be nicely put into a well-defined countable set. The set of the reals is uncountable, which means that almost every number is uncomputable like Chaitin's constant.
  - frutiger 7y ago
    You have described how the set of computable numbers are countable. But you have then posited "the set of the reals is uncountable" without any further explanation as to how that is the case. Your explanation helps one understand why computable numbers are countable (and that is for sure a useful insight), but says nothing about uncountable sets. I still feel the diagonal argument is the clearest intuition for defining an uncountable set.
    - mFixman 7y ago
      I was taught that real numbers were countable before learning what computables numbers where. "There's a set of countable numbers that includes every real number you know and doesn't include handwavy concepts" was news to me. Personally, I never found the diagonal argument intuitive. The proof with the nested open intervals in a sequence was easier for me to understand, but that's probably because I have more of a background in CS.

fxj 7y ago

It might have some relevance in information compression. He was also working on the halting problem (see lisp code below) and writing programs in LISP that were proving his theorems. http://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/turing.l http://jillian.rootaction.net/~jillian/science/chaitin/www.c... He also writes in his book: http://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch7.html http://jillian.rootaction.net/~jillian/science/chaitin/www.c... So in a way, in all three cases, Gödel, Turing, and I, we already have a new ``biological'' complicated mathematics, the mathematics of the third millennium, or at least of the 21st century. [As a child I used to dream that I was in the far future, in a library, desperate to see how it had all turned out, desperate to see what science had achieved. And I would take a volume off the shelf and open it, and all I could see were words, words, words, words that made no sense at all... Writing this book brings back long-forgotten thoughts and the unusual lucidity I experience when my research is going well and everything seems inevitable.]